Discussion on ladder numbers and the special zero algebra
by Jim Adams, 14th November 2013
 2013
Abstract. This work sketches for the general reader and undergraduate mathematician
aspects of ladder numbers and the special zero algebra, by introducing an algebra for
infinitesimals compatible with non-standard analysis, and for infinities.
Implicitly, the inconsistency of the real number system has been thought a possibility
since the work of Gödel [Gö31]. Using a definition of relative countability to challenge
Cantor's diagonal argument, we show this example is not generic, so that methods of
transfinite induction acting on the density of ℚ in ℝ demonstrate the inconsistency of
analysis, as currently axiomatised. We provide an alternative.
The theory developed is applied to limits and convergence, transcendence, including
results on antitone Galois connections, and to differential calculus and K theory.
0
The special zero algebra equates 0 to a set, the set of special real numbers.
Table of contents
The questions and answers are divided into fourteen sessions.
Session one – prologue, is an introduction, mainly philosophical in outlook.
Foundations
Session two – ladder algebras and nonstandard analysis, provides a technical
introduction to ladder numbers, firstly relating it to nonstandard analysis, then deriving
the algebra, including with superexponential extensions. The tetration problem is solved
for exponential algebra D1 in the circumstance of ladder numbers, and we discuss
representability and constructability.
Session three – the transfer principle, applies model theory to ladder numbers.
Session four – ladder number counting and constructability, proves that all sets of
numbers obtained iteratively from ℕ are countable.
Session five – ordering and the axiom of choice, defines various types of order,
showing that ladder numbers are not well-ordered.
Session six – Cantor diagonalisation, deconstructs the Cantor diagonal argument.
Session seven – transfinite induction and analysis, demonstrates that the real number
system is inconsistent, as currently axiomatised.
Session eight – ordinals and cardinals, redefines the idea of cardinals in the context of
ladder numbers.
Applications
Session nine – limits and convergence, converts the standard proofs of the BolzanoWeierstrass and Heine-Borel theorems to ladder number theorems.
Session ten – ladder number transcendence, accepts transcendental numbers in 𝕌.
Session eleven – Galois theory and infinite ladder automorphisms, shows there exist,
for example, transcendental solutions to polynomial equations.
Session twelve – ladder calculus, converts standard calculus to ladder calculus.
Session thirteen – further mathematical applications, mentions automated theorem
proving, matrix infinitesimals and K theory.
Session fourteen – the special zero algebra (SZA), is mainly independent of the rest
0
1
and introduces an algebra for 0 and 0 as types of sets.
Dependency diagram for the sessions.
one
Foundations
two
three
four
Applications
nine
five
six
ten
seven
eleven
fourteen
twelve
thirteen
eight
Meaning of symbols.
𝔸, the set of algebraic numbers in roots of positive whole numbers,
3
for example [7√+2 – √+5]/2.
ℂ, the set of complex numbers, a + bi, i = √−1.
ℕ, the set of natural numbers, whole numbers > 0 or > 0.
𝕌, the set of Eudoxus numbers – the replacement for the reals. As we define them, they are
countable. They do not contain infinitesimals or ordinal infinities.
ℤ, the set of positive, zero or negative integers.
𝛆, an infinitesimal, which may be multiplied by Eudoxus coefficients.
𝛆n, n = 2, 3, ... a hyperinfinitesimal.
𝛀, an ordinal infinity, which may be multiplied by Eudoxus coefficients.
𝛀n, n = 2, 3, ... a hyperinfinity.
𝛆. 𝛀 = 1
LU, the set of Eudoxus ladder numbers, for example
... + a𝛆2 + b𝛆 + c + d𝛀 + e𝛀2 + ...,
where the coefficients a, b, c, d, e  𝕌.
LR, the set of real ladder numbers, where the coefficients are uncountable in number.
Set S is countable with respect to set T if there exists at least one bijective mapping between
S and T, but there may exist other mappings which are not bijective.
Cardinal MS  MT if there exists no bijective map between elements of S and T.
0
Ж (zheh), the set 0 where memory of 0 is retained.
1
Ч (cheh), the infinity 0 where memory of 0 is retained.
Context
I have put together this work as a dialogue, partly as a technique with illustrious antecedents
which I am not trying to emulate, but also as a way of bringing out concerns so that the
programme on ladder numbers is brought into focus. The response to questions raised is
intended to clarify this programme both in the mind of the reader and even in my own mind.
Introductory material is written in a style open to the general reader.
First session – prologue
We discuss the nature of the mathematical enterprise, and how it relates to innovations
introduced in this paper. Some philosophical stances taken in mathematics and
metamathematics are introduced, including my own non-constructivist point of view.
Q: Why did you choose the words ladder number?
A: Having invited criticism of work on this idea, I received comments, so that this work has
gone through a number of major iterations. It was clear to me that, arising from this critique,
the adoption of the words real number for these ideas was unacceptable, and a new term had
to be found.
Q: So what are the new characterisations of number theory that you are adopting?
A: This is not entirely new; as a programme it was mapped out by Leibniz. There are a
number of interests for me here. This includes questions on limits and the approach via
sequences and ultrafilters in non-standard analysis, and its reformulation as an algebra of
infinitesimals and infinities. Well, that is one explanation of what I am talking about, but
there are other aspects too. I think there are three other areas – the axiom of choice and wellordering, the axioms of countability, and lastly the question of what you can talk about, so
this includes questions on constructability.
Q: So you differ in all these areas from other mathematicians?
A: I must be a mathematical nonconformist, in thinking the last two of these areas need
challenging, but I am a reluctant revolutionary, because my attitude has been when
challenging the status quo in mathematics, that this is there for a reason, and there is a
substantial amount of work that has been peer reviewed and has been accepted and mulled
over for a very long time, so you have your work cut out if you are trying to do that.
Perhaps I need to make a philosophical point here about what mathematics is. I think it is
about sure reasoning with respect to an axiom system. But we have found in mathematics, for
instance with the parallel postulate in geometry, that the axiom system is a bit arbitrary.
There is often long discussion about what are the appropriate concepts and definitions, but
once you have that, the mathematics is certain. So I am saying that all mathematics is about
relative truth – a relativity with respect to the axiom system which may be varied. It is then
the problem of physics to establish which axiom systems operate in practice.
It seems rather evident, that if you are free to adopt different assumptions, then you can reach
different conclusions.
I also want to state that for a long time I have been perturbed by the basis, and consistency of,
the real number system. This is in an absolute state – consistent or inconsistent.
In its domain of discourse, first-order logic quantifies over elements. The work of Gödel
demonstrates there is no countable proof that will show the consistency of first-order
arithmetic (but consistency or inconsistency is still there), and the work of Gentzen, using
combinatorial methods, showed that there exist proofs with an uncountable number of stages
(in the sense of Cantor, which I do not accept) which demonstrate this consistency [Ad15a].
He showed consistency is provable over the base theory of primitive recursive arithmetic
with the additional principle of quantifier-free Cantorian transfinite induction up to the
ordinal ε0. We might ask whether there is a similar proof of the consistency of analysis, a
hope expressed by Gentzen himself.
To analyse the consistency of analysis further, we will extend the ordinal infinity  to
encompass an algebra acting on a new ordinal infinity 𝝮, and deconstruct some Cantorian set
theoretic ideas based on enumerability.
Q: Before we go on to further descriptions of nonstandard analysis, infinity and infinitesimals
in the next session, you mentioned constructability as an issue. What is contentious here?
A: This is an important question, because it goes into philosophical issues of how we
interpret mathematics, what it is, and what is its relation to the world or the universe.
It is possible to assert, but given the finite nature of mankind difficult to prove, that the
universe is finite. There is an ‘ultrafinitist’ school of mathematics, to which I used to belong
but no longer do so, which posits that since the universe is finite all mathematics must be
embedded in a framework that is countable and finite. Thus, we abandon the axiom of
induction, because all proofs must stop at a large number less than the number of states in the
universe.
So I would describe this as a ‘normative’ statement of what mathematics is about, and what it
should confine itself to – in other words ‘what is there’.
It may be that the universe is actually a lot larger than we think. It could even satisfy the
property of being infinite. Even if that were not the case, conceptually it might be possible to
embed the correct description of the universe in a model which is infinite. So the statement
would be that we are not allowed to think in that way.
Connected with the situation of mathematical reasoning existing in the human brain are two
aspects – one is that mathematics is a set of rules for symbolic manipulation – a syntax, and
the other is that it is a set of rules for mapping the syntax onto the physical world as examples
– a meaning of the mathematics. My point of view is that even if you specify normative rules
for syntax manipulation and restrict discussion of what it is proper to discuss mathematically
of the world, at least culturally you are going, eventually, to get rebellion – this is after all
what some artistic movements are about. So, taking sides, I think it is unnecessary to be
unduly prescriptive.
A second point here is that the world exists outside the human, and although, so far as is
currently known, mathematics can only be appreciated by our species, questions of existence
are, in the world, largely independent of human cognition and behaviour. The current defect
of physics, perhaps derived from the human condition – that processes of measurement define
the science, so that what is not measured is claimed not to exist – with consequential
interpretation of the world and the promotion of ‘observables’ as the basis of physics rather
than, in the phrase of John Bell [Be87] ‘be-ables’, is reflected in mathematics by a conflict
between statements of existence and processes of proof.
Let me clearly put forward the point of view, that relative to the axiom system mathematical
statements can be true or false (actually, independence of the postulates needs to be
demonstrated), whereas it may not be known whether or not there is a proof. The reverse
implication needs care; in Galois theory, proofs of unprovability of the quintic by finite
radical formulas for commutative polynomial rings are known, yet for a polynomial of degree
n any selected set of n solutions containing radicals can be chosen.
The situation is mirrored in mathematical intuitionalism, which has implementation in logics
more general than the Boolean, but to which is appended often ideas of constructability. The
process seems to be, if it cannot be constructed, it is not there and it cannot be discussed. My
position is that existence is independent of constructability. Further, that induction can be
extended beyond the natural numbers.
I did not think this was contentious, since mathematical ideas abound in the extension of
number systems beyond the positive whole numbers, but when this process is applied to the
existence of extended methods of proof, such as transfinite induction, I have found it has met
with resistance, as impermissible mathematics, especially if the process demonstrates for
uncountable ℝ, that ℚ is dense in ℝ is a contradiction.
Q: What is the special zero algebra?
A: The SZA predates work on ladder numbers, was to begin with independent of it, and
involves a special algebra for zero, including
1
0
and
0
0
as objects similar to sets rather than
numbers. For me work on this idea loosened up the Cantor axioms for infinities, and provided
the precursor of the ladder algebra.
Foundations
Second session – ladder algebra and nonstandard analysis
The usual approach to nonstandard analysis is delineated. Infinitesimals are defined, and
whether or not they are present in decimal expansions of the Eudoxus number replacement,
𝕌, of the reals and in the algebraic numbers, 𝔸, is discussed. Hyperinfinitesimals are
introduced, and then in a similar way ordinal infinities and hyperinfinities. The axioms for
these are extended to exponentiation under the special and distinct algebra D1, and likewise
under superexponential extensions. Finally, representable infinitesimals are discussed.
Q: Do you accept the idea of ultrafilters?
A: For the reader, we are talking about nonstandard analysis here, which defines hyperreal
numbers *ℝ from a well-behaved algebra of sequence equivalence classes, the members of
which can be identified by a non-finite index set. We run into trouble if
(a0, a1, a2, ...) < (b0, b1, b2, ...)  (a0 < b0) & (a1 < b1) & (a2 < b2) ...,
since some entries of the first sequence may be bigger than the corresponding entries of the
second sequence, and some others may be smaller, so we do not distinguish between two
sequences a and b under this equivalence relation if a ≤ b and b ≤ a. In particular, we define a
zero sequence. As a result, the equivalence classes of sequences that differ by some sequence
declared zero will form a field which is called a hyperreal field. A consistent choice of index
sets is given by any free ultrafilter F on the natural numbers ℕ.
The idea is to single out a family F of subsets of ℕ and to define that a  0 if and only if the
sequence belongs to F. Any family of sets that satisfies (1) – (3) below is called a filter. If (4)
also holds, F is called an ultrafilter (because it cannot be enlarged as a filter). A free ultrafilter
does not contain any finite sets.
1)
Any set containing a set that  F also  F.
2)
The intersection of any two sets that  F also  F.
3)
The empty set   F, because then everything becomes zero; every set  .
4)
From two complementary sets, one  F.
Ultrafilters are fine, provided one accepts other ideas on real numbers, in particular that the
axiom system adopted for these – with or without the axiom of choice – is the only one
possible. My approach uses infinitesimals, and ultrafilters use sequences, but in the nature of
things the infinitesimal 𝝴  an ultrafilter F. The difference for ladder numbers is in the
designation of other characteristics – including that 𝝴 has an algebra.
As traditionally taught, real numbers (we will later adopt the term Eudoxus number for a
modification of these) are defined by Cauchy sequences
x1, x2, x3, ...,
in which for every positive Eudoxus number u there is a positive integer k, such that for all
natural numbers m, n > k
|xm – xn| < u,
where the vertical bars denote the absolute value. In a similar way we can define Cauchy
sequences of rational or complex numbers.
Q: What are the notions to be introduced by the algebra of ladder numbers?
A: There are two related ideas, one connected with infinities, and the other to infinitesimals.
These ideas have to be algebraically linked to be coherent.
When infinitesimals are absent, the new Eudoxus designation of the reals which we will
subsequently describe and which diverges from the standard, will be denoted by 𝕌.
We will denote ℝ as the set of real numbers, LU as the corresponding set of countable ladder
numbers and LR as uncountable ladder numbers. As well as ℕ, there are other countable sets:
ℤ the set of positive, zero or negative integers, ℚ the rational numbers, and 𝔸 the real
algebraic numbers in radicals derived from ℚ. In general, if a set X contains 0 it may be
denoted by X0, if it does not include 0 we may designate it as X0, and if it is positive as X+.
To preview what follows, we will distinguish between the ordinal infinity, 𝝮N, as selected
from a condition, which complies with an algebra in which 𝝮N  𝕌, similarly for the
infinitesimal 𝝴N, and the cardinal infinity of ℕ, denoted M0, which refers to a bijective
mapping between sets, in which there is also no allocation describing directly and bijectively
the number of elements in ℕ by M0  ℕ, because then M0 + 1 would also describe the
cardinality of ℕ.
An objective of ladder arithmetic is to define the ordered r-tuple (a, b𝝮N, ... x𝝮Nr,) for which
composition of r-tuples extends the features of a vector space.
Q: What is the relation of these sets and this algebra to infinitesimals?
A: We first need a definition of an infinitesimal. In modern notation Euclid states, as was
known to Archimedes
m  ℕ and   ℝ+, n  ℕ0: n > m,
and since this was stressed by the ancients, we infer they had considered the contrary.
The Archimedean property appears in Book V of Euclid’s elements as Definition 4:
“Magnitudes are said to have a ratio to one another which can, when multiplied, exceed one
another”. Because Archimedes’ Axiom V of ‘On the sphere and cylinder’ credited it to
Eudoxus of Cnidus it is also known as the Eudoxus axiom.
We have called any positive 𝝴e, for which we select a representative 𝝴, such that
m  ℕ0, NOTn  ℕ: 𝝴en > m,
an infinitesimal, excluding {𝝴e} from 𝕌. Archimedes used infinitesimals in heuristic
arguments, although he denied that these were finished mathematical proofs.
When infinitesimals are present, so they are not Eudoxus numbers, we can adopt the idea that
a ‘number’  = 3.14159 ... cannot be algorithmically distinguished over ℕ from certain
members of a set of ladder numbers, and on taking the limit, this set – which becomes
separated in its elements by infinitesimals – maintains the fact that it is an equivalence class
of numbers, and not a single entity. Algorithmically it is then possible to take a
‘representative’ for  – call it * – like it is possible to fix the representative for the base
point of a vector space, but it is always possible to find, say, two values 1* and 2*, which
are not equal and are separated by an infinitesimal.
However, we reason with respect to not necessarily constructible states, specifying only one
  𝕌, where all infinitesimals are zeroised, noting from the subsequent discussion on
representable infinitesimals, that countable formulas for  may contain infinitesimal terms.
The situation is different for algebraic numbers. For  we have no terminating formulas
which determine  – they are all convergent algorithms over ℕ. But for, say, √2, there is
quite a simple formula, (√2)2 = 2. The consequence of this is that √2 is indeed a number and
not algorithmically an equivalence class of numbers, because to use the argument of
reasoning via contradiction, if you say there are two values of √2, say √2 and √2 + 𝝴, where
𝝴 is an infinitesimal, then on squaring the second, 2 + 2√2𝝴 + 𝝴2  2.
The natural numbers do not possess infinitesimals. We define the positive rationals ℚ+ to
consist for natural numbers m, n  ℕ of numbers in the form m/n, so that ℚ does not possess
infinitesimals. Since 𝔸 is obtained from ℚ as finite linear combinations of elements of ℚ
multiplied by finite positive Eudoxus radicals, it follows that 𝔸 does not possess
infinitesimals either.
If we consider an infinitesimal, 𝝴, we may generate from each 𝝴 further infinitesimals, say via
the natural numbers with n  ℕ, by forming n𝝴. We denote this set by 𝝴N.
We can define hyperinfinitesimal numbers, in the following example as a positive number 
 𝝴N, such that for every  that belongs to the natural infinitesimals 𝝴N, there does not exist a
ρ  𝝴N, with ρ > . So for example (𝝴N)2 is a hyperinfinitesimal.
Since LU contains infinitesimals, it follows that 𝝴L contains the hyperinfinitesimals specified
above, if 𝝴L exists, and so on for further hyperinfinitesimal levels, inductively.
The ladder arithmetic now defines the ordered infinitesimal r-tuple (a, b𝝴N, ... x𝝴Nr,) for which
composition of r-tuples extends the features of a vector space.
We link the idea of infinite 𝝮 numbers with infinitesimals 𝝴, by defining as a structural
relation 1/𝝮Nj = 𝝴Nj. Using the vector space analogy we now define a scalar product given by
(a, b𝝴N, ... x𝝴Nr,).(c, d𝝮N, ... y𝝮Nr,) = (ac + bd + ... + xy).
However, we do not need to be limited in this way. We can define the natural product by
ordinary multiplication under the relation, for j and k, 𝝮Nj𝝮Nk = 𝝮N(j+k). This algebra is seen
to contain the scalar product algebra.
Q: What are the axiomatics of the 𝝮 algebra?
A: The infinity 𝝮 can be represented in a similar manner to the infinitesimal 𝝴:
m  ℕ and 𝝮e, NOTn  ℕ: 𝝮e < mn,
where 𝝮e is an equivalence class of infinities, in which we can choose a standard instance,
denoted by 𝝮. Here 𝝮 is derived as an instance taken from maximal elements outside the set
ℕ.
For integers ℤ and numbers p, q, where [p]ℤ = pℤ, we define the operation + by
[p]ℤ + [q]ℤ = [p + q]ℤ,
and multiplication . by
[p]ℤ.[q]ℤ = [p.q]ℤ.
We can define equations, for example
[p + q𝝮 + 𝝮]ℤ – [q𝝮 – 𝝮r]ℤ + [𝝮]ℤ.[𝝮]ℤ = [p + 𝝮r + 𝝮 + 𝝮2]ℤ
in the 𝝮 algebra.
Q: What is the algebra of exponentials here?
A: This is a non-standard one if we adopt the exponential algebra D1 given in [Ad14a] and
referred to in [Ad14b] and [Ad14c] – note the fourth axiom
(a) = a()
(ai) = a(i)
(a)i = a(i)
(ai)i = a(i).
We have no qualms in defining
ei = cos 𝝮 + i sin 𝝮.
The specification of the algebra of some complex superexponential operators is currently
considered an unsolved problem. We investigate this now.
In order to develop notions (semantics) utilising superexponentiation, it is necessary to
develop a suitable notation (syntax), so that meanings can be expressed by symbolic
manipulations – ‘meaning’ being specific instances of mappings from symbolic or other
representational manifestations to the objective world.
To this end we introduce the superexponentiation  symbol. We adjoin a number, say n, to
this operation, so that when n = 1 we are dealing with addition, when n = 2 with
multiplication, and n = 3 with exponentiation. The higher-order superexponentiation
operations, for n > 3, will be described inductively by stepping down to the (n – 1) case, and
so on.
Since exponentiation for ladder numbers is not in general associative, that is, very often
(a(bc))  ((ab)c), we have decided on a representation that describes a regular nesting
of brackets, so that when this regular nesting occurs, we may dispense with brackets.
In particular, we introduce n to indicate nesting on the left, for example
(((a n b) n c) … n d)  a n b n c … n d.
We introduce an alternative notation for the above, which we will use sparingly, for example
when n is a complicated expression, for emphasis, removing ambiguity, or calculation rather
than display. This is
<n for n.
For nesting on the right, we introduce a completely analogous notation, namely
(a n … (b n (c n d)))  a n ... b n c n d,
and the equivalent non-superscript notation containing
ah n> bh+1.
The n = 4 operation is known as tetration. We desire to develop a superexponentiation
algebra which like exponential algebra D1 does not branch as real values derived from
imaginary ones, except obtained from ein.
For n > 2 the exponential operations for this superexponentiation 𝝮 algebra we specify as
satisfying for left nesting
(a) n𝝮 = a(<n-1)
(ai) n𝝮 = ai(<n-1)
(a) ni𝝮 = ai(<n-1)
(ai) ni𝝮 = ai(<n-1).
Where we are not dealing with 𝝮 but an m  𝕌, substitute m for it.
The complex D1 binomial theorem has been defined in [Ad14a], in which
(a + bi)(c + di) = (a + bi)c(a + bi)di,
without abnormal complications which occur in the standard approach, and compatible with
the Euler relation
ei = cos  + i sin .
For n = 3, that is, exponentiation, we can evaluate
(a)n 𝝮, (ai)n 𝝮, (a)n i𝝮 and (ai)n i𝝮
by Euler relations or extended binomial techniques, where in D1
ii = ei/2 = i.
In this manner, we claim generalised to 𝝮 operations, all complex intermediate relations
between 1 and those for the operations n and n can be specified inductively for n  ℕ.
Q: Are infinitesimals and infinities directly representable?
A: Note that we can represent 𝝴 as a difference in LU. In binary, if we choose
𝝴 = 1.000 ... – 0.111 ...,
where the ... is interpreted, say for the first term, for every n  ℕ, the nth place after the
binary point is 0, then taking 1.000 ... as a natural number, for example
𝝴2 = (1.000 ...)2 – 10(0.111 ...) + [0.111 ...]2 = -0.111 ... + [-1 + 1.111 ... + 𝝴2],
and this 𝝴 is less than any positive rational number.
There is the additional point here that two expansions of the same number, in the traditional
approach without infinitesimals, may have different representations. Thus, assuming that
infinitesimals are not present
1.0000 ... = 0.11111 ... = 0.1010 ... + 0.0101 ...
in binary, provided we admit all representable numbers, say derived within a ring.
If to the contrary we are dealing with ladder numbers with infinitesimals in LU or LR, then
suppose we were to assume in decimal notation that
1.0000 ... – 0.9999 ... = 𝝴
for some infinitesimal 𝝴, then
1
m(9 – 0.1111 ...) =
m𝛆
9
,
so that the right hand side is again an infinitesimal. Indeed, we are then led to the conclusion
that
1
3
 0.3333 ...
and say if at position n  ℕ in a decimal expansion we have an entry rn, then
rn
r
– n+1 + 𝝴
10n 10n+1
and
r
r
- 10nn + 10n+1
n+1 + 𝝴
contain representable infinitesimals, the sum of which contains a representable infinitesimal.
Q: If infinitesimals are representable, is it consistent that infinitesimals can be derived from a
higher rung of the ladder number?
A: This situation is inherent in standard analysis, except that different representations are put
in the same equivalence class. If we adopt as an axiom that these representations map
bijectively to ladder numbers, then for some non-finite representations we are forced to
descend a rung to infinitesimals and can represent one rung from another.
If we assume that the rung of numbers is countable, then the rung of infinitesimals derived
from numbers is countable too. The pair of rungs, numbers and infinitesimals, is countable in
conjunction, and so does not violate the theorem we will prove that only countable sets can
be derived from ℕ.
Further, this idea also applies to infinities. For the 𝝮 rung, 𝝮  ℕ, and a divergent series
summed over all ℕ in the number rung possesses a type of 𝝮 property, that is it does not
belong to an element of ℕ and is strictly greater than such an element.
Third session – the transfer principle
The implementation of meanings in a language and a brief sketch with references to the
transfer principle is given.
Q: As a general principle, how are meanings implemented in the systems we will meet?
A: We are able to assert that there exists at least one consistent system, and that is the world
we live in. The proof of the consistency of the world is not derived by abstract symbolic
reasoning, but by virtue of being in the world, whilst we are alive. This consistency is
permanent, irrespective of the existence or absence of observers who are able to assert it. The
problem is then to create at least one consistent symbolic representation so that the world
maps onto it. This includes the programme of physics. At the same time, we may be able to
create other axiom schemas which are internally consistent, but which have no ascertained
meaning as maps to the world. These are allowable mathematical or metamathematical
constructs.
The question arises, given a local description of an axiom schema which is valid, whether
there exists only one globalisation, that is, only one extension of the states and rules of
inference of the schema which is consistent. If that were not the case, there would exist at
least two globalisations for which in conjunction the inconsistency “A and NOTA” holds. Our
process will be to select one globalisation which is valid.
Q: What is the meaning of this 𝝮 algebra?
A: Thus far we have introduced an allocation without semantics to
𝝮, n𝝮, n, 𝝮n, 𝝮,
etc., and the algebra works irrespective of its interpretation. We have for the following
distinct sets under the partial order given by 
2ℤ  ℤ  ℚ  𝕌,
and the question arises, is there a consistent semantic allocation between the distinct sets we
have introduced and these 𝝮 values? We should have
Set
ℕ, ℤ
[p]ℤ
ℚ
𝕌, base n
Allocation
𝝮
p𝝮
𝝮2
n
.
The 𝝮 designation for ℕ and ℤ is 𝝮N and 𝝮Z respectively, for ℚ it is 𝝮Q = 𝝮N  𝝮N0 or 𝝮Z 
𝝮Z0. We can evaluate n by the binomial theorem in n = (n – 1) + 1 and 𝝮.
Q: So for instance √𝛀 is allowable?
A: This would be logical, and 𝝮1/n, 𝝴1/𝝴, 𝝮 𝝮𝝮, etc., with order interrelationships.
For allocations we implement the law of continuity introduced by Leibniz. It is the principle
that "whatever succeeds for the finite, also succeeds for the infinite”, restated in model theory
as the transfer principle, which states that all statements of some language that are true for
some structure are true for another structure.
In specifying a language L we have a mapping between its names and the objects to which
the names refer. The names belong to the language but in general the objects do not. Many
objects can have one name, so that for a name n and an object f(n), we specify that f -1(n)  n
is surjective.
From names we may form other syntactical structures, , of which formulas are one example.
The simplest formulas are called atomic formulas, and well-formed formulas are built up
from these. For names ni and index sets i, j and k, denote formulas by j(ni).
The meanings expressible in the language have domain the j(ni) and their transformations
Tk. We form the commutative diagram
j(ni)

f(j(ni))


Tk(j(ni))  Tk(f(j(ni))) = f(Tk(j(ni))).
The globalisation we will select will obey in a specific way the transfer principle, by forming
a model M of the set of sentences designating the superexponential algebra in the language
L, and forming an extension (or restriction) of the model M, as we have already done for the
infinities derived from 𝝮 or alternatively its concomitant infinitesimal 𝝴.
The transfer principle is discussed further in [Rob63] and [HL85].
Fourth session – ladder number counting and constructability
The results of this session will be used in sessions six and seven, and mark the first stage at
which the results differ from standard conceptions. We define countability using the phrase
‘at least one’ bijection – that is, there can be circumstances where there is no bijection. We
show, generalised to superexponential extensions, that all iterative operations on ℕ give
countable results.
Q: How can ladder numbers be counted and constructed?
A: S.C. Kleene defines that a set is enumerable if there is a 1-1 correspondence, that is, an
injective map which is onto, from the natural numbers to the elements of the set. We prefer to
define and emphasise that a set is countable if there exists at least one bijection, in other
words a map which is both injective and surjective, between the natural numbers and the set.
We may form the set ℕ2 = ℕ  ℕ𝝮N, and even though 𝝮N  ℕ, this set is countable, as we
will see later, in a similar sort of way that the rationals ℚ are deemed countable. Then ℕ2 has
a number of elements 𝝮N2, and we may form ℕ3 = ℕ2  ℕ2𝝮N2.
Generally, we may split ℕ into nℕ, nℕ + 1, ..., nℕ + (n – 1) and match each of these sets
countably nℕ  ℕ1, nℕ + 1  ℕ2𝝮1, ..., nℕ + (n – 1)  ℕn𝝮n-1. Thus ℕn is countable.
The set ℕN = {ℕ1, ℕ2, ...} has a number of elements 𝝮 which does not belong to ℕ1, ℕ2, etc.
but we may form ℕN  ℕN𝝮 and this is also countable. Denote ℕN by ℕℕ = ℕ2. Then we
may form countable ℕn, countable ℕN, and so on iteratively for superexponentiation
operations.
In this way, we see all constructable sets derived from ℕ are bijectively countable.
Q: Well, I think the previous comment was lacking in detail; the proof that all constructible
entities derived from the natural numbers are countable needs further justification. Can you
provide it? This is in violation of current mathematical understanding, that 2‫א‬0 is uncountable.
A: Firstly, I think we need to go into a proof given by Cantor, which is correct, that the
rationals are countable. We set up the array which is traversed countably by the arrows shown
in the following diagram. The number of elements in the array is the number of elements in
ℕ2 = ℕ  ℕ.
This is mapped onto the rationals by taking pairs (m, n) obtained from rows and columns in
the array and mapping
(m, n)  m/n  ℚ0.
1
2
3
4
5
...
1
2
3
4
5
...
1
2
3
4
5
...
1
2
3
4
5
...
1
2
3
4
5
...
If it is claimed that 0  ℚ, we can obtain ℚ by appending it to the beginning of the list, a
method we will use for Cantor’s diagonal argument.
Then we say ℚ is countable, because we have demonstrated a bijection
ℕ  ℕ2  ℚ,
so that on substituting ℕ2 for ℕ we obtain by induction, which is a proof system defined by
ℕ, a bijection
ℕ  ℕp ,
where p  ℕ.
We wish to establish a bijection
ℕ  ℕN.
What is meant by the expression on the right hand side? Suppose we have a set of sets
{ℕ1, ℕ2, ... ℕp, ...}.
Is this countable? We can for instance demonstrate the bijections
ℕ  p + ℕ  p + 𝝮ℕ  ℕp.
As before this set, ℕN, has a number of elements, 𝝮 writing now exponentially for the
superscript, which does not belong to ℕN, but appending 𝝮 to ℕp results in a set which does
not belong to ℕN but is bijectively countable to ℕN, for example by appending 𝝮 to the
beginning. Then the transfer to ℕN is countable, because we have the bijections
ℕ  ℕ  𝝮p  ℕp  ℕp  𝝮  ℕN.
Q: I don’t think this proves ℕN is countable. It is an assertion.
A: To try an alternative method, we can map bijectively
{ℕ1, ℕ2, ... ℕp, ...}  {ℕ, 2ℕ , ... pℕ, ...},
then on the right hand side, this set is at most as big as ℕ  ℕ = ℕ2, which we have proved is
countable.
We now claim, if we may be permitted to introduce an order relation into
1 < 𝝮 < 𝝮, etc.,
that
2 < 𝝮,
and that since all sets constructible from ℕ, 𝝮 and 𝝮 are countable, so are sets constructed
from ℕ and 2. In particular, the set of all permutations of ℕ, with infinity
𝝮! = (𝝮 – 1)(𝝮 – 2) ... < 𝝮
is countable.
It is clear that as all sets obtainable as exponential operations are countable, and since
superexponentiation is obtained inductively from exponentiation, that all sets obtained from
the aforementioned method are countable under superexponential extensions.
Fifth session – ordering and the axiom of choice
This session defines various types of order, and shows that ladder numbers are not wellordered. The failure of the standard proof ‘ℚ is dense in ℝ’ is then demonstrated for ladder
numbers.
Q: What are the definitions involving order?
A: A preorder for elements p and p has at most one binary operation p < p, which is
reflexive, p < p, and transitive, p < p and p < p implies p < p. A partial order is a preorder
with the axiom that p < p and p < p implies p = p. A linear order is a partial order in which
either p < p or p < p.
Q: What is the relationship of this idea to well-ordering?
A: A linear ordering < of a set P is a well-ordering if every nonempty subset of P has a least
element. Thus ℕ and closed sets in ℚ+ are well-ordered, but ℤ and ℚ are not. Our sets obey
second order logic where the axiom of choice holds but well-ordering does not, developing a
result of Paul Cohen that there is a model of ZF in which bounded subsets of the real numbers
cannot be well-ordered [Co63], [Co64], [Co66].
Q: You said there are three areas you are investigating. The axiom of choice is one of them.
What have you to say about it?
A: A choice function is a function f, defined on a collection X of nonempty sets, such that for
every set s in X, f(s) is an element of s. With this concept, the axiom of choice (AC) can be
stated: For any set X of nonempty sets, there exists a choice function f defined on X. That
you can dispense occasionally with this axiom is well known, which is independent of the
Zermelo-Fraenkel (ZF) axioms of set theory. ZFC is ZF with AC. Sets can obey second order
logic, which also quantifies over relations and functions, where AC holds but the wellordering principle does not and this leads to different real number systems, as was
demonstrated by Paul Cohen. I don’t think I am demonstrating any deviancy here – but some
early teaching systems support the equivalence of well-ordering with the axiom of choice, as
the only possibility. Ladder numbers abandon well-ordering, so this is a very related idea.
A linear ordering (P, <) is complete if every nonempty bounded subset has a least upper
bound (l.u.b.). If we restrict ourselves to a set less than r, where 𝝴 is an infinitesimal and r is
not, the set of all infinitesimals that are not hyperinfinitesimals when subtracted from r does
not have the l.u.b. of r or any other number.
Thus some bounded subsets of the ladder numbers we are considering are not well-ordered.
This designation is in accordance with the result of Cohen.
If the infinitesimal 𝝴 has a lowest value, then 1/𝝴 has a highest value. We can construct a set
L]Z] derived from ℤ which is bounded as ℤ  (-ℕ + 𝝮N), internally wholly separated by
ladder integers and countably infinite. The absence of this set in standard theory is an
essential component in the proof that ℚ is dense in ℝ.
The standard proof that ℚ is dense in ℝ proceeds by proving that
(i) If x  ℚ and y  ℚ, then x + y  ℚ, and xy  ℚ.
(ii) If x  ℚ and y  ℝ \ ℚ, then x + y  ℝ \ ℚ, and xy  ℝ \ ℚ provided x  0.
(iii) If x  ℚ and y  ℚ with x < y, then there exists a z  ℚ such that x < z < y.
(iv) If x  ℚ and y  ℚ with x < y, then there exists a z  ℝ \ ℚ such that x < z < y.
(v) If x  ℝ \ ℚ with x > 0, then there exists a z  ℚ such that 0 < z < x.
(vi) For every r  ℝ there exists an n  ℤ such that n > r.
This is dependent on the completeness axiom already given: every nonempty subset S of ℝ
that is bounded above has a l.u.b. belonging to ℝ. Using the completeness axiom, we can
prove that ℤ is actually unbounded above.
(vii) (The Eudoxus property of ℝ). The set ℤ of integers is unbounded above in ℝ. There then
follows
(viii) (Density of ℚ in ℝ). If x  ℝ and y  ℝ with x < y, then there exists a z  ℚ such that
x < z < y.
Sixth session – Cantor diagonalisation
The definition of relative countability is used to demonstrate that the Cantor diagonal
argument is ineffective. Three scenarios are now available; the replacement of the reals is
the countable Eudoxus numbers, 𝕌, the countable ladder numbers LU, or the uncountable
ladder numbers LR.
Q: You wish to modify the theory for infinities – presumably the continuum hypothesis.
What have you to say here?
A: No, my ideas are completely different, and need enumerating. The criticism of Cantorian
set theory goes back to Kronecker, and later, intuitionalism, but these are also not my points
of view. Firstly, I want to discuss countability. I want to mention that the argument which I
am going to give cannot at this juncture be generally accepted by mathematicians, because it
leads to conclusions which are not generally accepted. I will start with an example. Consider
the mapping
2ℤ  ℤ
in which the integer 2z in 2ℤ is mapped to 2z in ℤ. This is an injective mapping, because
there are gaps in ℤ, namely the odd numbers, which are not covered in the mapping. Now
consider the mapping
2ℤ  ℤ
in which the integer 2z in 2ℤ is mapped to z in ℤ. This is a bijective mapping, both surjective
and injective, and there are no gaps as a consequence in ℤ.
My definition of countability with respect to 2ℤ of ℤ is that there exists at least one bijective
mapping from 2ℤ to ℤ.
Note, in this definition, that any number of mappings 2ℤ  ℤ which are not bijective do not
demonstrate the uncountability of ℤ with respect to 2ℤ. What we have to demonstrate is the
existence of at least one mapping with no gaps to show countability, or the absence of any
such example to demonstrate uncountability.
My notation is that ℕ0 is the natural numbers with 0, and ℕ0 is the natural numbers starting
from 1, and ℕ can be either or both of these. I also describe countability with respect to ℕ by
the contraction ‘countable’. Then ℕ is countable, the mapping being ℕ to ℕ such that
numbers n map bijectively to the same n.
The reader may now see what I am driving at, or driving to, in the following argument due to
Cantor. We wish to demonstrate the uncountability of the real numbers, ℝ. Assume we can
arrange the real numbers in a list that is countable, determined by ℕ0. We will ignore the
integer parts of these real numbers, and just discuss their expansions in base two arithmetic
after the binary point.
We will assume for the purposes of argument that we can suppress numbers containing an
irreducible component with infinitesimal representation from the list. Cantor’s claim,
accepted by most mathematicians, is that any such list of real numbers indexed by ℕ cannot
be set up, and that therefore ℝ is uncountable. Here are the two lists
ℕ0
ℝ
1
r1
2
r2
3
r3
...
...
Generate a real number, r0, which is not in the ℝ list and is not equal to r1 in the first position
after the binary point, not equal to r2 in the second position after the binary point, and so on.
Then r0 does not belong to the list.
According to the definition we have already given, this does not demonstrate the
uncountability of ℝ. It demonstrates one example of ℝ which is not in the ℕ list. But for
uncountability of ℝ we have stated that we must prove the absence of any matching lists, and
we have not done so.
I state in parenthesis, we may add r0 to position 0 in ℕ0, and the generated example then
becomes, by the initial assumption, countable.
Though we have not shown by our procedures that the set of real numbers is countable, using
the proof theory of [Ad15a], the assumption of countability or uncountability is an
independent axiom. We will classify these scenarios in the seventh session.
I presume most mathematicians would feel uncomfortable with the postulate that there is only
one type of infinity. This is what I am claiming for the situation where the set of uncountable
infinitesimals is empty.
This postulate now sheds new light on the proof, commonly given, that ℚ is dense in ℝ. Both
ℚ and 𝕌 are countable. We have a further assertion – that when infinitesimals are added to ℝ,
then ℚ is not dense in ℝ. Clearly we are not adding infinitesimals to ℚ.
Q: A lot of people have difficulty with the Cantor argument, but the diagonal argument is
generic, so all diagonalisable enumerations are of this type. Further, the set of all these
enumerations is uncountable. For instance, we can add the counterexample to the list, and
then perform the diagonal argument again, generating new counterexamples.
A: You have not defined generic, so we need to do so. I assume by this you mean all real
number listings indexed by ℕ result in real numbers extraneous to the list, by a diagonal
argument. This is not so.
There are two lines of reasoning, one without a diagonal argument, and one with it.
Without a diagonal argument, the first item in the list is countable, so this is not extraneous to
the list, or to use the terminology, generic.
With a diagonal argument, since by hypothesis ℕ and ℝ are countable, and the set ℝ contains
all real numbers, it is possible at diagonal position n to choose an r  ℝ so that this
corresponds with the first element in the list. This could be, for example, a real number which
corresponds at all positions up to and including n with the first item in the list. Then the item
derived by this modified diagonal argument can correspond to the first item in the list, so
when the number of positions in r is mapped bijectively to ℕ, not all real numbers generated
by a diagonal argument are generic, but not if the number of positions in r cannot be mapped
bijectively to ℕ.
The generated set of counterexamples you gave is either countable or uncountable. List the
first counterexample generated as x1. From x1 in a new list generate a new counterexample
x2. We wish to show this set is uncountable. Then under the counter assumption that the set is
listable bijectively by ℕ, the set of counterexamples generated in this manner is ℕ, and there
does not seem to be a way of providing an alternative procedure that would contradict this.
I don’t know what this is meant to prove, because we only ever have one exception, and
when we add this to the list and apply the Cantor diagonal argument again, the first exception
now belongs to the list so that the number of exceptions is only ever one, and this is
countable.
To summarise this, the exception list we have generated is countable, where we have
generated one exception to a bijective map. Exceptions to bijective maps do not prove
uncountability with respect to ℕ, for instance the map
ℕ  ℕ: n  n, n < m
n  n + 1, n > m
does not demonstrate uncountability of ℕ because there is an exception at m in the codomain,
and the above rule is not the only one we can select. What we have to demonstrate is that no
bijective maps exist, and we have not done so.
There are several further issues raised from this reasoning. Firstly, if I have a list
ℕ0
ℝ
1
r1
2
r2
3
r3
...
...
and I permute the elements of ℝ, then on using Cantor’s diagonal argument, I generate new
counterexamples to the mapping ℕ  ℝ being bijective. If we claim the number of
counterexamples generated in this way is uncountable then the corresponding set of all
permutations of ℕ is uncountable, under the assumption that some bijection is possible. The
result of these permutations is still ℕ, which is countable.
So the question now is, can we derive an uncountable set from ℕ? I have demonstrated the
answer is no in the third session, and this issue is extraneous to the argument employed in the
Cantorian diagonal reasoning, which did not reference such issues.
Secondly, the representation of reals we have used is binary, so the diagonal argument
generates only one counterexample, on reversing the binary bits 0  1 and 1  0 at each
diagonal position. But we may wish to represent real numbers in a different base than 2, say
base k. Then we have k – 1 choices at each diagonal position different from the real number
we are inspecting. I think it is clear that if the mapping ℕ  ℝ is countable with ℝ in binary,
it is also countable with ℝ to base k. This is an argument made in the second session, by
splitting ℕ into kℕ + j, j < k, so that we map from k copies of ℕ to ℝ, and this is countable.
Application of the diagonal argument can be illustrated in the finite case. Consider listing all
values of the finite sequence:
(a, a)
(a, b)
(b, a)
(b, b)
By the diagonal argument choose from the first two items the item with the first element from
the first item the value b = NOTa, and the second element from the second item the value a =
NOTb. Then we cannot claim that the item (b, a) does not exist in the list, because a unique
diagonal does not exist for the list and (b, a) is not in the chosen diagonal. No correct
argument shows the list is not finite.
The ramifications of the deconstruction of the diagonal argument impinge on multiple areas
of mathematics, and it has not been our plan to address all of these aspects here.
Seventh session – transfinite induction and analysis
The inconsistency of standard and nonstandard analysis is demonstrated. Countability,
accessibility and questions of density are discussed.
Q: What is the corresponding situation for analysis?
A: Suppose that infinitesimals are absent from the rationals. We reason that if between every
pair of ladder numbers there exists a rational number  ℚ then infinitesimals are absent from
LU, so LU = 𝕌, and conversely. Indeed, if we choose two ladder numbers separated by an
infinitesimal, then there exists a ladder number half way between them. The two intervals we
have created with the interposed ladder number each contain by hypothesis a rational number,
and the difference between distinct rational numbers would then be an infinitesimal, a
contradiction.
Conversely, if infinitesimals are absent from LU = 𝕌, then consider ladder numbers r and s,
with s greater than r. A rational number q exists less than s – r. Consider a grid marked off
q
with intervals 2, with 0 one of the grid marks. Then r and s are separated by a rational
number.
So far we have not gone beyond countability with respect to ℕ. We now introduce the
assumption that there exists a set LR of ladder numbers which is not countable with respect to
ℕ. This is an extension of the algebra we had been previously considering.
We will prove that for LR  𝕌 in uncountable analysis, infinitesimals must be present.
We state that a true proposition holds irrespective of the existence or absence of a method of
proof.
We now define a set S is countable with respect to LR (traditionally called uncountable) if
and only if there exists a bijection from S to LR, LR is not finite and there is no bijection LR
 ℕ. Such mappings exist, for example LR  LR. Then there exists no algorithm using
bijections to prove there cannot be a rational number between every pair of LR ladder
numbers if we restrict ourselves to countability with respect to ℕ, but assuming by hypothesis
there exist at least m rationals interposed between any m + 1 distinct ladder numbers, there
does exist such an impossibility proof under countability with respect to LR. Since the
existence or absence of infinitesimals is not contingent on proof algorithms, if the ladder
numbers are selected, they must include ladder numbers LR separated by infinitesimals. This
can be rephrased as ℚ is not dense in LR.
We can go further in these definitions, by introducing countable or uncountable sets of
uncountable ladder numbers, in which a representative LR is uncountable and there is no
bijection LR  LR. Moreover, we might expect an axiom that the sets LR satisfy an order
relation between their 𝝮 designations and for instance that of ℕ and LR.
Zeroth shell of accessibility, finite proofs
First shell, countable infinite proofs
Second shell, uncountable infinite proofs
Depicted in the diagram above are three shells. In the zeroth shell there reside finite proofs.
Not all proofs in the first shell, for proofs over ℕ, that is, induction as commonly understood,
are available to proofs by finite methods. There is a hierarchy of shells of accessibility, the
inner for which do not contain some proofs which are available to shells which enclose them.
A model may be constructed where a proof which halts in an outer shell does not halt in an
inner shell, after a countability sequence in the outer shell not available to the inner one.
Q: Could you summarise how ladder algebra differs from the usual conceptions for the real
numbers, ℝ?
A: These theorems are contingent on the idea of density. ℝ is a topological space, provided
with a function assigning to each x in ℝ a non-empty collection n(x) of subsets of ℝ called
neighbourhoods of x. A subset S is called dense in ℝ if and only if the only closed subset of
ℝ containing S is ℝ itself. This can be expressed by saying the interior of the complement of
S is empty.
When ℚ and 𝕌 are countable, then ℚ is dense in 𝕌 and the irrational numbers in 𝔸 are
another dense subset, showing that 𝕌 may have several disjoint dense subsets.
Let LU be the set of ladder numbers where the algebra on its hyperinfinitesimals and
hyperinfinities has countable coefficients on these. So ℚ does not possess infinitesimals and
LU does. Then ℚ is not dense in LU, since otherwise the closed subset of LU containing ℚ
would be deficient in its hyperinfinitesimals.
When ℝ is uncountable, since this is not derivable from ℕ and thereby ℚ, the number of
places of ℝ is uncountable, and when a place in ℝ is at an uncountable distance from its
beginning, then two different values at this place cannot be subtracted and then multiplied by
a value in ℕ to give a value between elements in ℚ, since to backtrack from an uncountable
place in ℝ by a value in ℕ, this member in ℝ continues to exist at an uncountable place in ℝ,
otherwise a finite sum of elements in ℕ could reach an uncountable number. Thus, when ℝ is
uncountable, it contains infinitesimals, and ℚ is not dense in ℝ.
We have introduced the mutually exclusive possibilities that ℝ is uncountable and contains
infinitesimals, so that it is a ladder number LR with an algebra in uncountable coefficient
infinitesimals, otherwise ℝ = LU or ℝ = 𝕌. Since the uncountable version of ℝ may be
subdivided into an uncountable number of respectively finite or countably infinite subsets, its
LR can be inductively transcribed to an uncountable number of countable coefficient
hyperinfinitesimals. In no case does this correspond to the description of ℝ as traditionally
given, and thus the usual designation of ℝ is inconsistent.
Eighth session – ordinals and cardinals
We now introduce a necessarily redesigned nomenclature for cardinal infinities, MSj.
Q: Throughout our discussions concerning radical departures from the standard, we have not
yet mentioned the elephant in the living room. In the algebra you have presented, ordinal
infinity, , is conflated with the cardinal 0. Further, there has been no discussion of the
standard algebra in which 1 +    + 1. How do you justify this?
A: Yes, Einstein once stated that a theory should be as simple as possible but no simpler. We
should not conflate ordinals and cardinals, but the algebra demonstrated here operates on two
levels, one is as an 𝝮-algebra in which the features from which 𝝮 is derived are preserved,
and the coarser algebra based on equivalence classes which is essentially one which may be
thought of as operating on cardinals. So for cardinals we will now decide to adopt the symbol
Mn, based on the understanding that our allocation of how cardinals relate to some sets differs
from the standard. It is possible to collapse the 𝝮-algebra so that it refers to the equivalence
class based on the size of an M0 infinity. In standard systems 0 contains order information
not present in M0, but an algebra for 𝝮 satisfies, as a non-natural number, 𝝮 + 1 = 1 + 𝝮, and
this is a set of distinguished pairs, which has an order defined via its construction, 1  ℕ 
𝝮.
This is the structure of our axiomatics, which may indeed be altered, and we will show some
examples of how this novel axiom system may be modified in ways that are again nonstandard.
There are two conditions, the first for 𝝮 and the second under which Mn operates. The first is
as an irreducible element, which is not a number, where 𝝮 may be combined with scalars a
and b to form a + b𝝮. The algebra of b𝝮j is normally that of a module. The vector space
bj𝝮j can then be formed with basis 𝝮j, or its extended algebra.
The second, is as a measure, valuation, function or morphism, in the first instance from the
set ℕ  M0. The existence of ℕ is guaranteed in ZF and is known as the axiom of infinity.
An assumption it is possible to incorporate is that for sets S1, S2, ... Sn we can form their
respective unique maximal cardinal valuations MS1, MS2, ... MSn so that there is a partial order
relation amongst the MSj, say
S1  S2  ...  Sn



MS1 < MS2 < ... < MSn.
We know there exists a bijective map of elements to themselves for an arbitrary set S. Thus S
 S defines MS, which is then a maximal mapping. For a set S there may be no constructive
proof that such a map exists (or does not exist) for S  S; this is only available via the larger
cardinal. Nevertheless, we can state that such an MS exists or does not exist, mutually
exclusively, and can prove theorems based on this fact. Indeed, we have already done so.
It is not necessary to introduce an algebra in which both addition and multiplication are
noncommutative, which is done however in [Ad14b]. We would only wish to do so if there
were a model of infinities which impelled us to adopt this. The ordered pair (1, 𝝮) differs
from the ordered pair (𝝮, 1) and this is essentially the only type of difference that exists
between 1 +  and  + 1. We will therefore treat the  algebra as arising trivially from
reorderings of the 𝝮j basis in series such as bj𝝮j.
Q: The 𝝮 algebra you have given is non-standard and contradicts the algebra for ,
commonly given, for example  +  =  and 2 = . How do you explain this?
A: There is an answer that the Cantorian ordinal algebra here and the one we have been
considering can be subsumed under a common algebra, since the Cantorian algebra is
equivalent to a (mod 1) algebra for rings under 𝝮. We know the Cantorian algebra for
exponential  differs from such a (mod 1) algebra. I think this is a bit arbitrary, but it is
possible. We now see that the Cantorian algebra is a type of 𝝮 algebra under the constraint of
being (mod 1) under addition and multiplication for 𝝮, but not exponentiation. One might
expect infinitesimals to be treated in the same way. It would be consistent to have numbers
behaving in this way too, if we adopted it.
It is possible to introduce algebras for infinities and infinitesimals corresponding to
congruence arithmetic. Then for some p in this algebra
𝝮k+p = 𝝮k (mod p).
Q: What happens when p = 1?
A: This algebra is feasible, but note that its infinitesimals are the same as its infinities! In
such an algebra, I do not think we have a model of the standard reals.
The idea is more interesting when we consider ℤY rather than ℕ, defined when 𝝮 is now a
number, and 𝝮 = -𝝮 are identified  ℤY, so that 𝝮 + 1 = -𝝮 + 1 in this infinite cyclic algebra.
Applications
Ninth session – limits and convergence
Convergent and divergent infinite series and limits are discussed. The Bolzano-Weierstrass
theorem is then proved in 𝕌.
Q: You have only touched in passing on the characterisation of limits in this theory. What is
happening here, and are there any fundamental observations which need to be made about
infinite series?
A: Yes, there are fundamental observations. To begin with, we need to introduce in this
system a notation for closed sets, [ ], including their end points, and open sets, ] [, without
end points, which is often introduced axiomatically in standard theory.
The infinitesimal 𝝴, which we say is on the (-1)th rung, is of the same algebraic type as the
infinity 𝝮 on the first rung, and neither are numbers, which occupy the zeroth rung of the
ladder number.
For a series to be well defined, its sum within a ladder rung multiplied by its complex
conjugate must have an upper bound  ℕ. For a convergent infinite series we can make in
general a non computable choice by setting its infinitesimals to zero.
Different protocols of summation, say for positive and negative terms each of which is
separately divergent, may give rise to different evaluations of infinite series.
If the sum is divergent, when its sum is indexed over ℕ for the entire rung, the totality of its
solutions does not belong wholly to the rung. We then find that there are multiple ways in
which its value in the rung may be specified. This is because there is no unique sum for 𝝮.
Relative values of different infinite divergent sums may be compared, but this is contingent
on agreement of the order in which the comparison is made.
It is of little utility in this situation to mimic the convergent case, and set all values in the
rung to zero. Although we can formally manipulate an algebra involving numbers, 𝛆 and 𝝮,
there is an issue here on the precise evaluation, which does not exist within 𝕌, of both 𝛆 and
𝝮 by infinite series of numbers. Thus whereas we state that 𝛆 and 𝝮 exist, there is the
practical issue of determining coefficients of 𝛆 and 𝝮 under differing infinite sums.
Making a virtue of a necessity, we can use this situation to define the value of a divergent
sum in the ℕ part of the ladder as being a free variable, so that, for the sum of values in ℕ, its
possibilities range over all n  ℕ. To make a choice of this n, n then reverts to a bound
variable.
These considerations may be extended to series over other rungs.
Briefly, I will give two examples on limits.
Consider the series
1
Xm = ∑m
n=1( 2n ).
We know it is close to 1. Here is a diagrammatic indication of a proof.
1
¾
½
0
We say
lim Xm = 1.
m∞
If we were to specify that the limit exists on the zeroth rung, then the limit in this case is
outside the set of its members.
It is clear inductively that Xm is always < 1, so that for a half-closed, half-open interval Xm 
[0, 1[ and Xm  1. We might wish to define
lim Xm = [(1 – 𝝴), 1[,
m∞
where 𝝴 is a first-order infinitesimal. It is possible to expand or contract this set outside the
value 1. Intuitively, this sum is lower than 1 by 1/2n, and we wish to substitute 1/2 for 1/2n
in the limit, but it is not necessarily the case that this is 𝝴; we can have 2  𝝮.
If we take the divergent series on the zeroth rung
2 – 4 + 8 – 16 ...,
then we say the intermediate evaluation of the sum as (-2)m-1 is formally greater than -𝝮 and
less than +𝝮, so that we can write
n
m-1
-∑m
 [-𝝮, +𝝮],
n=1(-2) = (-2)
or even as belonging to the open set
(-2)m-1  ]-𝝮, +𝝮[.
The identical algebraic structure we have developed between 𝝮 and 𝝴 means this divergent
series on infinities is of a formally equivalent algebraic type with inclusions reversed and on
taking the multiplicative inverse of the series, to a convergent series on infinitesimals. For
example, with only inclusions reversed and 0 < 𝝴 < 2-n
-n
𝝴  ]0, 1 – ∑m
n=1 2 [.
Infinite series on setting infinitesimals to zero now possess the property for limits that, for
instance
1
3
= 0.3333 ... or 1.000 ... = 0.999 ...
whereas this was previously not the case, so that some unequal numbers in their infinite
decimal expansions now fit in the same equivalence class. Thus in the limit 1 – 𝛆𝛆 = 0.
Q: We have not yet discussed in sufficient detail the impact of these ideas on analysis. What
happens to convergence and compactness, in particular the Bolzano-Weierstrass and HeineBorel theorems?
A: A fundamental proposition in analysis states that an increasing sequence with a least upper
bound (l.u.b.) converges to a limit. For instance the sequence
1 3 7 15
0, 2, 4, 8, 16, ...
tends to 1.
The Bolzano-Weierstrass theorem states that a bounded sequence of real numbers has a
convergent subsequence. There are two such subsequences below.
1, -1, 1, -1, 1, -1, ...
The Bolzano-Weierstrass theorem applies to 𝕌n analysis in Euclidean space, En.
A compact space occurs when any infinite sequence of points sampled from the space must
eventually, infinitely often, get arbitrarily close to some point of the space.
The Heine-Borel theorem states that sets of real numbers are compact if and only if they are
closed and bounded. In essence it extends the Bolzano-Weierstrass theorem to Hausdorff
spaces, for instance 𝕌, where distinct points have disjoint neighbourhoods, including to
metric spaces where the distance between points may not be Pythagorean. The Heine-Borel
theorem does not apply in general to topological spaces, which contain Hausdorff spaces.
Subspaces and products of Hausdorff spaces are Hausdorff, but there are exceptions for
quotient spaces of Hausdorff spaces, like LU or LR. In fact, every topological space may be
realised as the quotient of some Hausdorff space. The existence of unique limits for filters
implies that a space is Hausdorff.
If an ordered set S has the property that every nonempty subset of S having an upper bound
also has a least upper bound, then S is said to have the l.u.b. property. The set 𝕌 of Eudoxus
numbers has the l.u.b. property. Similarly, the set ℤ of integers obeys it.
Any well-ordered set also has the l.u.b. property, and the empty subset also has a l.u.b., the
minimum of the whole set.
We have seen that infinitesimals are not well-ordered, so that infinitesimals and as a
consequence ladder numbers LU or LR do not possess the l.u.b. property.
A further example is given by the rationals, ℚ. Let S be the set of all rational numbers q such
that q2 < 2. Then S has an upper bound but no l.u.b. in ℚ. If we suppose p  ℚ is the l.u.b., a
contradiction is immediately deduced because between any two x and y  𝕌 (including √2
and p) there exists some rational p', which itself would have to be the least upper bound (if p
> √2) or a member of S greater than p (if p < √2).
First we prove the Bolzano-Weierstrass theorem when n = 1 for 𝕌n, in which case the
ordering on 𝕌 can be put to good use. Indeed we have the following result.
Lemma: Every sequence {xp} in 𝕌 has a monotone subsequence.
Proof: Let us call a positive integer p a peak of the sequence if m > p implies x p > x m that is,
if xp is greater than every subsequent term in the sequence. Suppose first that the sequence
has infinitely many peaks, p1 < p2 < p3 < … < pj < …. Then the subsequence {xpj}
corresponding to these peaks is monotonically decreasing, and we are done. So suppose now
that there are only finitely many peaks, let k be the last peak and p1 = k + 1. Then p1 is not a
peak, since p1 > k, which implies the existence of a p2 > p1 with xp2 > xp1. Again, p2 > k is not
a peak, hence there is p3 > p2 with xp3 > xp2. Repeating this process leads to an infinite nondecreasing subsequence xp1 < xp2 < xp3 ..., as desired. 
Now suppose we have a bounded sequence {xp} in 𝕌; by the Lemma there exists a monotone
subsequence, necessarily bounded. Our aim is to show (the monotone convergence theorem)
that this subsequence must converge.
The sequence {xp} must contain a monotone subsequence (an increasing subsequence or a
decreasing subsequence). Since we know that the whole sequence is bounded above and
below, the same certainly applies to any subsequence, and so if we have a monotone
subsequence then we must prove we have a convergent subsequence, using the l.u.b.
property.
If an increasing sequence {xp} is bounded above, then it is convergent and the limit is
l.u.b.{xp}. Since {xp} is non-empty and by assumption, it is bounded above, then, by the l.u.b
property of 𝕌, c = l.u.b.{xp} exists and is finite. Now for every e > 0, there exists an xk such
that xk > c – e, since otherwise c – e is an upper bound of {xp}, which contradicts c being
l.u.b.{xp}. Then since {xp} is increasing
p > k, |c – xp| = c – xp < c – xk < e,
hence by definition, the limit of {xp}is l.u.b.{xp}.
Finally, the general case can be easily reduced to the case of n = 1 as follows: given a
bounded sequence in 𝕌n, the sequence of first coordinates is a bounded real sequence, hence
has a convergent subsequence. We can then extract a second subsequence on which the
second coordinates converge, and so on, until in the end we have passed from the original
sequence to a subsequence n times, which is still a subsequence of the original sequence, on
which each coordinate sequence converges, hence the subsequence itself is convergent.
Suppose A is a subset of 𝕌n with the property that every sequence in A has a subsequence
converging to an element of A. Then A must be bounded, since otherwise there exists a
sequence {xm} in A with || xm || ≥ m for all m, and then every subsequence is unbounded and
therefore not convergent. Moreover A must be closed, since from a noninterior point x in the
complement of A one can build an A-valued sequence converging to x. Thus the subsets A of
𝕌n for which every sequence in A has a subsequence converging to an element of A, that is,
the subsets which are sequentially compact in the subspace topology, are precisely the closed
and bounded sets. 
This form of the theorem makes especially clear the analogy to the Heine-Borel theorem,
which asserts that a subset of 𝕌n is compact if and only if it is closed and bounded. In fact,
general topology tells us that a metrisable space is compact if and only if it is sequentially
compact, so that the Bolzano–Weierstrass and Heine–Borel theorems are essentially the
same.
Tenth session – ladder number transcendence
The countability allocation to numbers in 𝕌 has the consequence that transcendental
numbers like  are deemed countable in 𝕌.
Q: Are you abandoning transcendental numbers?
A: No. To describe in more detail what we have to say about real numbers, for the symbol 𝕌
representing the zeroth rung of countable real ladder numbers, a number will be called Unonterminating if it is represented as
u = all r  ℕ0 nrr ,
a
and it is not rational, or equivalently for every n there does not exist a representation of u in
which ar = 0 for every r satisfying for some m  ℕ, r > m.
Clearly, by a long proof   𝔸, and by these criterions is U-nonterminating, being given for
instance by the algorithm
4
 = all r  ℕ0(8𝑟+1
–
2
8𝑟+4
–
1
8𝑟+5
–
1
8𝑟+6).
A number which is U-nonterminating and not algebraic is U-transcendental.
However, an algorithm is normally thought of as implemented on a Turing machine, that is, it
halts, but here the algorithm is generated over ℕ and is countable. Thus in generating ar by
this type of algorithm the number of constructible U-transcendental numbers is countable.
The limit of the number of possible representations of such real ladder numbers in 𝕌 is n,
and the statement that 𝝮  n is an axiom that can be more tightly defined by specifying the
𝝮 algebra for exponentiation, and we adopt it.
A set S is distinct from a set T if there exists an s  S such that s  T or a t  T such that t 
S. The sets ℕ and ℚ for example are countable but distinct. ℚ and 𝕌 are also distinct. Then
we say there exist distinct sets, the first two being respectively ℕ and 𝕌, allocated to the
following sequence:
𝝮 < n < 𝝮,
all of which are countable.
To develop the idea mentioned for algebraic numbers in the second session, the ladder
number   𝕌 where rungs outside the zeroth are null has at most one representative,
otherwise it has members on rungs < 0.
If a substitute for 𝕌 were to have countable hyperinfinitesimals, so it would in fact be LU, and
say  belonged to this LU,  would be a set {} of U-transcendental numbers satisfying the
a
formula giving nrr , for  over zeroth rung places, where ar  {0, ... n} can be specified by an
inductive algorithm over ℕ.
Thus the algebra we have introduced is not dissimilar to non-standard analysis, exceptions
being the cardinality of 𝕌 and that the structure allows multiple types of hyperinfinitesimals.
If instead a 𝕌 substitute were to be uncountable, i.e. LR with its hyperinfinitesimals, the
number of places of a real ladder number t  LR would in general be uncountable. Further,
we have seen uncountable sets cannot be generated from ℕ alone.
A specification for  may be thought of as specifying a sequence of places in ℕ, but beyond
this it may be considered to be unspecified, so that in the uncountable case an uncountable
collection of members {} of  maps surjectively onto its sequence of places in ℕ.
Eleventh session – Galois theory and infinite ladder automorphisms
For polynomials, degree decrementing transformations are available by differentiating them.
Thus the approach via Newton’s method can be formulated in which tangents to the
polynomial approach the zeros U-countably by approximation algorithms. This is formulated
in the language of antitone Galois connections, the idea initially being that two structures are
compared, a lattice and a group. The fact that Newton’s method is available means there
exists a U-transcendental group defining the solutions.
Q: Are there further consequences, say for lattice theory?
A: To generalise the above considerations, a partial order defines an arrow between elements
of a sequence. A partial order where all finite sets have a least upper bound (l.u.b.) and a
greatest lower bound (g.l.b.) is called a lattice. For example, the l.u.b. of a subset S of (ℤ+, |),
where | denotes ‘divides’, is the least common multiple of the elements of S, and the g.l.b. is
the highest common factor.
Galois theory is a theory of finite automorphisms for polynomial rings. In the theory of the
zeros of polynomial equations, approximation techniques – ‘Newton’s method’ are available
for solutions in 𝕌 beyond the quartic, where by Galois theory there is no formula involving
finite radicals. We indicate why techniques involving infinitesimals are available here, in
which LU or LR are not well-ordered, and therefore have no l.u.b. and are not lattices, but for
which on setting all values outside 𝕌 to zero, we do have a lattice, so that for infinite
automorphisms, corresponding to infinite groups, finite Galois theory fails and there are
exact solutions. These solutions for polynomials of degree > 4 may be U-nonterminating.
Newton’s method depends on the existence of a tangent vector to a polynomial f(x) = 0 of
finite integer degree n; in 𝕌 the tangent is a good local approximation, so that taking
infinitesimals about the zero of the polynomial, the tangent can be obtained as a polynomial
number from calculus plus a polynomial infinitesimal. On setting all values outside 𝕌 to zero,
for a specified c  𝕌 in the domain there exists a Eudoxus interval in which the tangent does
not differ from the polynomial in the codomain by more than f(c). Moreover, the difference is
monotonically decreasing from f(c) to the zero of the polynomial, so that by the monotone
convergence theorem its limit is f(x) = 0.
There exist functions which do not correspond to such polynomials, for example, where a
finite open interval is of degree n and another finite open interval is of degree (n + 1),
which motivates the introduction of sheaves.
For polynomials with coefficients that are complex without infinitesimals or infinities, that is,
f(x)  ℂ, then a differentiable function from ℂ to ℂ is certainly differentiable as a function
from 𝕌2 to 𝕌2 (in the sense that its partial derivatives all exist). A local finite interval satisfies
the Eudoxus axiom with magnitudes belonging to positive components of the Cartesian
product, in ℂ+  𝕌+  𝕌+i. A polynomial with negative integer degree values may be
transformed by a polynomial factor to one with degree values  ℕ0.
When there are no duplicate roots and the zero is not at a maximum or minimum, there will
exist a c such that a selection of a point m  x will generate Eudoxus intervals including
precisely n intervals where [f(m + c), f(m + c + c)[ contains zero. If a zero is at a
maximum or minimum this is detectible.
Thus a finite polynomial of degree n without duplicates may be covered by Eudoxus intervals
over all its zeros. These local intervals are algorithmically specifiable over ℕ and the
algorithm halts, say by selecting an interval in the domain of f and partitioning it inwards and
expanding it outwards, with both in an extra 2k  ℕ segments.
Let (A, ≤) and (B, ≤) be two partially ordered sets. A monotone Galois connection between
these partially ordered sets consists of two monotone functions: F: A → B and G: B → A,
such that for all a in A and b in B, we have
F(a) ≤ b if and only if a ≤ G(b).
These considerations can be developed in category theory. In this situation, F is called the
lower adjoint of G and G is called the upper adjoint of F. Mnemonically, the upper/lower
terminology refers to where the function application appears relative to ≤; the term ‘adjoint’
refers to the fact that monotone Galois connections are special cases of pairs of adjoint
functors in category theory.
Every property has its dual, obtained by inverting the order-dependent definitions in the given
statement, that is, by reversing arrows. By definition an antitone Galois connection between
A and B is just a monotone Galois connection between A and the order dual Bop of B. All
statements on Galois connections can thus easily be converted into statements about antitone
Galois connections.
An essential property of a Galois connection is that an upper/lower adjoint of a Galois
connection uniquely determines the other:
F(a) is the least element b with a ≤ G(b), and
G(b) is the largest element a with F(a) ≤ b.
Given a Galois connection with lower adjoint F and upper adjoint G, we consider the
compositions GF: A → A, known as the associated closure operator, and FG: B → B, known
as the associated kernel operator. Both are monotone and idempotent, and we have a ≤ GF(a)
for all a in A and FG(b) ≤ b for all b in B. A Galois insertion of A into B is a Galois
connection in which the closure operator GF is the identity on A.
The above definition is common in many applications today, and prominent in lattice and
domain theory. However the original notion in Galois theory is slightly different. In this
alternative definition, a Galois connection is a pair of antitone, that is, order-reversing,
functions F: A → B and G: B → A between two partially ordered sets A and B, such that b ≤
F(a) if and only if a ≤ G(b). The symmetry of F and G in this version erases the distinction
between upper and lower, and the two functions are then called polarities rather than adjoints.
Each polarity uniquely determines the other, since
F(a) is the largest element b with a ≤ G(b), and
G(b) is the largest element a with b ≤ F(a).
The compositions GF: A → A and FG: B → B are the associated closure operators; they are
monotone idempotent maps with the property a ≤ GF(a) for all a in A and b ≤ FG(b) for all b
in B.
The motivating example comes from Galois theory: suppose J/K is a field extension. Let A be
the set of all subfields of J that contain K, ordered by inclusion, . If E is such a subfield,
write Gal(J/E) for the group of field automorphisms of J that hold E fixed. Let B be the set of
subgroups of Gal(J/K), ordered by inclusion. For such a subgroup G, define Fix(G) to be the
field consisting of all elements of J that are held fixed by all elements of G. Then the maps E
↦ Gal(J/E) and G ↦ Fix(G) form an antitone Galois connection.
Now define Fix(G) to be the field consisting of all elements of J that differ by at most an
infinitesimal from all elements of G. We map LC or LRRi  ℂ in using the Newton algorithm
by forgetting its components not in ℂ, simultaneously mapping the Fix(G) infinitesimals to
zero. Thus we obtain the theorem that there exists an antitone Galois connection between
infinite groups and their C-rational or C-nonterminating field extensions.
Trivially, there exist ascending Jordan-Hölder series on groups for polynomials of
successively higher degree corresponding to which the Galois group of a lower degree
polynomial has a symmetric group which has a normal group for a polynomial of higher
degree – to form the group for the lower degree polynomial set non-matching instances of
roots in the higher degree polynomial to the identity transformation for the group.
In algebraic geometry, the relation between sets of polynomials, that is, zero sets, and
varieties is an antitone Galois connection.
Fix a natural number n and a field K and let A be the set of all subsets of the polynomial ring
K[X1, ..., Xn] ordered by inclusion , and let B be the set of all subsets of Kn ordered by
inclusion . If S is a set of polynomials, define the variety of zeros as
V(S) = {x  Kn: f  S, f(x) = 0},
the set of common zeros of the polynomials in S. If W is a subset of Kn, define I(W) as an
ideal of polynomials vanishing on W, that is
I(S) = {f  K[X1, ..., Xn]: x  W, f(x) = 0},
then V and I form an antitone Galois connection.
We can form a (covariant) functor from an antitone Galois connection defined on
polynomials and their varieties – equate all variables to one instance – to an antitone Galois
connection defined on infinite groups and transcendental field extensions.
The method may be extended to functors between other antitone Galois connections. We give
some examples.
Given a path-connected topological space X, there is an antitone Galois connection between
subgroups of the fundamental group 1(X) and path connected covering spaces of X. In
particular, if X is semi-locally simply connected, then for every subgroup G of 1(X), there is
a covering space with G as its fundamental group [Sz08].
Likewise for an inner product space V, we can form the orthogonal complement F(X) of any
subspace X of V. This yields an antitone Galois connection between the set of subspaces of V
and itself, ordered by inclusion; both polarities are equal to F.
Given a vector space V and a subset X of V we can define its annihilator F(X), consisting of
all elements of the dual space V* of V that vanish on X. Similarly, given a subset Y of V*, we
define its annihilator G(Y) = {x ∈ V: φ(x) = 0 for all φ ∈ Y}. This gives an antitone Galois
connection between the subsets of V and the subsets of V*.
The closure on Kn is the closure in the Zariski topology, and if the field K is algebraically
closed, then the closure on the polynomial ring is the radical of the ideal generated by S. Mac
Lane and Moerdijk, [MM00] pages 116-121, describe the Zariski site for Grothendieck
topologies [Gr10] on sheaves.
More generally, given a commutative ring R (not necessarily a polynomial ring), there is an
antitone Galois connection between radical ideals in the ring and subvarieties of the affine
variety Spec R (namely Spec of the ring).
Also, there is an antitone Galois connection between ideals in the ring and subschemes of the
corresponding affine variety.
Twelfth session – ladder calculus
We discuss how the ladder number approach may be applied to directional derivatives, total
derivatives and jets.
Q: What is the approach here to differentiation and integration?
A: The derivative of a function at a chosen input value describes the best linear
approximation of the function near that input value. For an infinitesimal as we have defined
it, the derivative is the ratio of the infinitesimal change of the output over the infinitesimal
change of the input producing that change of output. For a U-valued function of a single 𝕌
(Eudoxus) variable, the derivative at a point is obtained by setting after its computation as a
ladder number its non-Eudoxus values to zero, and equals the slope, m, of the tangent line to
the graph of the function at that point:
m=
𝚫𝐟(𝐚)
𝚫𝐚
=
𝐟(𝐚+𝐡) − 𝐟(𝐚)
(𝐚+𝐡) − 𝐚
=
𝐟(𝐚+𝐡) − 𝐟(𝐚)
𝐡
The tangent line to f at a gives the best linear approximation
f(a + h)  f(a) +
𝛛𝐟(𝐚)
h
𝛛𝐚
to f near a (that is, for small h).
In higher dimensions, the derivative of a function at a point is a linear transformation called
the linearisation. A closely related notion is the differential of a function.
At a discontinuity of a function f(a) between two intervals connected at .. a[ and [a ..
or .. a] and ]a .. we define the derivative at a to belong to that for the closed interval at a
when it includes an open component  , that is, it corresponds in the closed interval to the
derivative on the right in the first case, alternatively on the left.
We apply the transfer principle for ladder numbers by stipulating that if a function may be
expanded out as a Taylor series in hyperinfinitesimals, infinitesimals and numbers, then its
derivative is obtained by a binomial expansion of these variables with respect to other ladder
numbers by the same means. However, when a ladder number is a vector space, the
directional derivative is available.
In some cases it may be easier to compute or estimate the directional derivative after
changing the length of the vector. Often this is done to turn the problem into the computation
of a directional derivative in the direction of a unit vector. To see how this works, suppose
that v = λu. If we substitute h = k/λ into the difference quotient, the difference quotient
becomes:
f(𝐱 + (k/)(𝐮)) – f(𝐱)
k/
=
f(𝐱 + (k𝐮)) – f(𝐱)
k
.
This is λ times the difference quotient for the directional derivative of f with respect to u.
Furthermore, taking the limit as h tends to zero is the same as taking the limit as k tends to
zero because h and k are multiples of each other. Therefore Dv(f) = λDu(f). Because of this
rescaling property, directional derivatives are frequently considered only for unit vectors.
If all the partial derivatives of f exist and are continuous at x, then they determine the
directional derivative of f in the direction v by the formula:
Dvf(x) = ∑nj=1 vj(f/xj).
It follows that the directional derivative is linear in v, meaning that
Dv + w(f) = Dv(f) + Dw(f).
However, the usual difference quotient does not make sense in higher dimensions because it
is not usually possible to divide vectors. In particular, the numerator and denominator of the
difference quotient are not even in the same vector space: the numerator lies in the codomain
𝕌m while the denominator lies in the domain 𝕌n.
We introduce the total derivative at a, so all the partial derivatives and directional derivatives
of f exist at a, and for all v, f ′(a)v is the directional derivative of f in the direction v. If we
write f using coordinate functions, so that f = (f1, f2, ..., fm), then the total derivative can be
expressed using the partial derivatives as a Jacobean matrix of f at a:
f (a) = Jaca = (fi/xj)ij.
The existence of the total derivative f ′(a) is strictly stronger than the existence of all the
partial derivatives, but if the partial derivatives exist and are continuous, then the total
derivative exists, is given by the Jacobian, and depends continuously on a. The total
derivative of a multivariable function has to record much more information than for a singlevariable function and gives a function from the tangent bundle of the source to the tangent
bundle of the target.
The analog of higher-order total derivatives, called a jet, reflects geometric information, such
as concavity, which cannot be described by linear data such as vectors and is not a function
on the tangent bundle. Jets take as arguments coordinates representing higher-order changes
in direction. The space determined by these is called the jet bundle.
We have introduced in [Ad14a] the hyperintricate D1 exponential algebra for matrices, which
is developed further in [Ad14b] to encompass differentiation and integration. We have
introduced the complex D1 algebra for ladder numbers in session two, and extensions of the
Cauchy-Riemann equations for hyperintricate D1 are available in the work quoted.
The reverse process of finding a derivative is antidifferentiation. The fundamental theorem of
calculus states that antidifferentiation is the same as integration.
Thirteenth session – further mathematical applications
Further applications are: automated theorem proving, matrix infinitesimals and K theory.
Q: Are there other applications of this theory?
A: Real number analysis is used in floating-point verification. There is also work by Bledsoe
[Ad15a], [BB77] on automated proofs in nonstandard analysis.
Consider an example, the prime number theorem, stating that (n), the number of primes < n,
has the limiting property
(x)
x/ln(x)
 1.
All known proofs of this result use analysis, and even finding a proof using just real analysis
was a major accomplishment. Some deep results about the distribution of primes are used in
polynomial-time primality testing algorithms.
Since any operation on ladder numbers can be carried out on infinitesimals, this can be
extended to linear algebra, and it is possible to consider matrix infinitesimals. Noncommutative infinitesimals are developed from a different point of view in [MR91].
A further application is to K theory, which deals with linear algebra over a ring R, and
associates a sequence of abelian groups, Ki(R), to R [Ros94]: “K0 underlies the EulerPoincaré characteristic in topology and Grothendieck’s approach to the Riemann-Roch
theorem [Ad15b]. K1(Z), the Whitehead torsion [Wh78], is a determinant. There is evidence
that the higher K-groups are related to special values of L-functions.
Finitely generated projective modules are used to define the Grothendieck module K0. An Rmodule is projective if and only if it is isomorphic to a direct summand in a free R-module.
The isomorphism classes of projective modules over R form a semigroup, and may not have
the cancellation property s + t = u + t implies s = u. To define an equivalence class, put (p, q)
~ (r, s) when there exists a t such that p + s + t = r + q + t. A group can be built out of these
equivalence classes. Denote by [(p, q)] the equivalence class of (p, q). Then addition is
defined by the law
[(p, q)] + [(r, s)] = [(p + r, q + s)],
so that associativity holds. The identity is given by
[(x, x)] = [(y, y)],
also this is a group because
[(x, y)] + [(y, x)] = [(x + y, x + y)] = 0.”
The monoid of countably generated modules is usually set isomorphic to the extended natural
numbers ℕ  {}, with n +  =  for any n. Under the commonly adopted algebra, this is
not a monoid with cancellation, since two elements become isomorphic after adding  to
each one, so K0 cannot be defined in this way. However, under an algebra for ladder
numbers, we are dealing with ℕ2 = ℕ  ℕN, and we can now obtain a monoid with
cancellation, thereby extending K0 to countable and also uncountable modules.
Fourteenth session – the special zero algebra
The infinitesimals developed in this session are smaller in magnitude than all those discussed
in earlier sessions, and the infinities are larger. We develop an algebra in which 0 is not
multiplied out with other terms, but in which the memory of 0 is retained. The algebra
developed is in general multiplicatively non-commutative in Ж =
0
0
(zheh), which we treat as
1
a set, and the infinity Ч = 0 (cheh).
Q: What is your intention in this session?
A: The objective of this session is to demonstrate that it is possible to develop a mathematics
which differs from ideas in set theory circulating currently. The operations will be on sets
(and their generalisations), similar to the extended elements considered in previous sessions,
and we develop an algebra combining numbers and sets.
0
An algebra is investigated in which the intuitive model we will use is 0 is a set, called a zheh,
denoted by Ж, the set of special real numbers, and is not itself a number. Corresponding to Ж
1
there is also a type of infinity, denoted by Ч or cheh, with value 0. In order to maintain
consistency, the algebra for 0, Ж and Ч is different from the algebra for c  0, Ж or Ч. It may
be thought of as an algebra for which the memory of instances of 0 is retained.
The algebra for infinite sets developed shares features with that commonly adopted, and is
extended to the special zero algebra, which for sets and infinities mirrors the algebra for
instances of numbers. Infinitesimal sets are introduced in this approach, but the idea is
different from what has been discussed previously, which introduced positive infinitesimals 𝝴
greater than the infinitesimals studied in this session.
We extend the special zero algebra to higher order sets and infinities. Natural number
subscripts label this extension. The range of these subscripts is then extended to other
numbers. For instance, for subscript -1 an antizheh is defined as satisfying an antizheh
divided by itself equals zero.
Q: What, precisely, is this algebra?
A: We will start off this mathematics by considering just two elements, c and 0, and two
operations, + and . For c  0 define the following states and operations to be valid
-0 = 0
0+0=0
00=0
c+0=0+c
and
c  0 = 0  c.
There is a special evaluation operator =0, which forgets structure, so that
c =0 0 + c
0 =0 0  c.
We then define operations that generate an extended algebra from the above, by considering a
number of relations that are used to generate new variables, both instances and sets, from
these simple conceptual building blocks. We itemise these next. The first we will consider is
c + (-c) = 0,
the second would be the existence of a c = 1 where
1  1 = 1,
a third would be to generate a value (2c) satisfying
c + c = (2c),
1
a fourth and a fifth to generate values c2 and c with
c  c = (c2),
1
c  c = 1,
also for square roots of c, and finally perhaps a value i in which
i  i = -1.
1
Note once again that all these values satisfy c, -c, 1, 2c, c2, 𝑐 , √c and i  0.
If x, y and z are such values  0 as are generated from c and 0 above, with variables we
will subsequently introduce, then we define that the following x, y, z rules hold
(x + y) + z = x + (y + z)
(x  y)  z = x  (y  z)
x+y=y+x
xy=yx
and the distributive law
x  (y + z) = (x  y) + (x  z) = (y + z)  x.
Q: What is the relation of these ideas to the set theory of Cantor?
0
A: We can extend our algebra further, having already introduced Ж = 0.
For sets and infinities, the characteristics we will discuss first will closely correspond to
mathematics as developed historically by the set theorist Cantor. Later and more explicitly,
we will consider a modification of the axioms and a departure from them which we feel is
more natural (and introduces more structure).
Essential to the set theory as developed by Cantor is the assertion that the power set (the set
of all subsets) satisfies
2 > ,
(1)
which characterises the uncountability of the real numbers. One aspect of (1) is that it is an
assertion about countability, the second is that it involves the > relation. Initially, the
algebraic relations we will develop between number systems and Ж or Ч are independent of
these countability considerations. Later, we develop a characterisation of > that these algebras
satisfy, and the question then arises whether (1) is deducible from this.
Cantorian set theory, in terms of the algebra adopted, is distinguished by an algebra for rings
which differs in its implementation from its algebra for exponentials.
In order to develop a Cantorian type of theory, we will put all positive numbers c  0 in the
same equivalence class, so that when we consider the relation
c + (-c) = 0
to hold, we will define
c
= Ч,
0
the interpretation being that Ч (infinity) is also a type of set, and define
−c
= (-Ч).
0
Since in the Cantorian algebra relations on sets and infinities are not transmitted downwards
to relations between numbers, we are safe if we isolate the number c to c > 0 equivalent to c =
1, c = 0, and c < 0 equivalent to c = -1.
We adopt the following relation between Ж, Ч and -Ч.
Ж=
c + (−c)
0
c
=0+
−c
0
= Ч + (-Ч).
We now define the Cantor algebra for infinities. This satisfies the axioms
Ч+Ч=Ч
and
Ч  Ч = Ч.
Continuing with this, we have
Ч  (-Ч) = (-Ч) = (-Ч)  Ч,
(-Ч)  (-Ч) = Ч.
We also have
0  Ч = 0  (c/0) = c  Ж = Ж,
so that in conformity with the idea of Ж as a set, and -0 = 0, we obtain
-Ж = Ж
and
Ж + Ж = Ж.
Consequently
0  Ж = c  Ж + (-c)  Ж = Ж + (-Ж) = Ж + Ж = Ж,
leading to
Ж  Ж = (Ч + (-Ч))  ((-Ч) + Ч) = Ж + Ж = Ж.
A possible double axiom states
1/Ж  Ж  Ч.
A further assumption, indirectly derved from Cantor’s diagonal argument for real numbers,
since this reasoning intends to make available more than one type of infinite set, is
Ж/Ж  Ж.
This enables us to give a non-constructive definition of countability, dependent on the
bijections available between countable infinite sets. Ж will be called Ж/ Ж countable if and
only if
Ж/Ж = Ж.
We are now using Ж as a signifier of an uncountable or a countable set.
Continuing in similar manner, we proclaim two possibilities for the value 1/Ч, namely
1/Ч  0,
in which case we are considering for objects derivable from sets an analogue of
infinitesimals, as discussed in previous sessions and
1/Ч = 0,
in which case according to the above reference we are choosing an analysis which is that of
the rational numbers. These situations are those for which a set is converted back to an
instance. 1/Ч  0 is this sort of reversion, but such infinitesimals cannot be boosted to
standard real numbers, although they may be added to them, and multiplying by a standard
real number changes the value of the infinitesimal.
We define the ‘belongs to’ relation by the mapping c  Ж as being c  Ж. There are two
extremal maps here, in which either c maps to c in Ж, which is isomorphic to the domain, and
the map is isomorphic to a singleton inclusion map, or c maps to the whole of Ж, isomorphic
to the codomain, and isomorphic to the Hom sets in which the maps of c range over the
whole of Ж. There are two types of modification to this map, where the instance c is
extended to a collection of c’s, and the set Ж is reduced by the removal of subsets of Ж.
Sets of which c is a member may be subject to union, , intersection, , and inclusion, ,
and under the mapping c  Ж, this is mirrored in the codomain.
We define further the relation < as that conforming to
(-c) < c,
(-c) < 0
and 0 < c,
(-Ч) < Ч,
(-Ч) < Ж
and Ж < .
The < relation can be extended to include real numbers in characteristic 0 generated from c,
and as we will see later, there exists a bijection between the first line of relations above and
the second.
Q: What is the special zero algebra?
A: To introduce the special zero algebra for  and Ж, we must incorporate the following
principle on the order of evaluation, where x, y, z  0 rules may be applied before this is done
0
1
determine 0 first, 0 second and instances of 0 last.
We have previously defined (2c) as
(2c) = c + c,
with implicitly (2c)  c or 0. In similar manner we change the rule for addition of Ч to
(2Ч) = Ч + Ч,
If we wished to use the variable c instead of 1, then we would suggest using the symbol (cЧ)
for this infinity. Then if we had
c=1+1+1=3
then
Ч + Ч + Ч = (3Ч),
in which it is understood that (cЧ)  Ч. We will say the natural algebra for this type of
behaviour is in ‘characteristic Ж’. If (cЧ) = Ч in a lowest value c > 0, we say the natural
algebra is in ‘characteristic (cЧ)’. The Cantor algebra is a special zero algebra in
characteristic Ч.
A consequence is that
(1 + 1) + (−1)
0
=
21
0
+
−1
0
= (2Ч) + (-Ч) = Ч,
and this above expression is
1 + (1 + (−1))
0
= Ч + Ж.
Indeed, under the mapping 0  Ж and 1  Ч it is now possible to translate any statement
concerning numbers under x, y z rules to the corresponding statement in the special zero
algebra of sets.
For each mapping 0  Ж and 1  Ч there is a dual mapping which reverses all arrows.
However, if we wish to maintain
2Ч > Ч,
which is a natural aspiration, then we cannot use the map 1  Ч. For example
(Ч2) = Ч  Ч = Ч,
(ЧЧ) = (Ч)Ч = Ч
and
e(i  Ч) = [(cos )  Ч] + [(i sin )  Ч]
may not, and we would prefer in the last two instances to say do not, hold.
We therefore adopt an algebra in which Ч acts like an irreducible number. For instance
eЧ = 1 + Ч + [Ч(Ч – 1)/2]Ч2 + [Ч(Ч – 1)(Ч – 2)/3!]Ч3 + ...,
whilst maintaining the mapping 0  Ж.
Q: Can the algebra be extended further?
A: Nonabelian and nonassociative algebras also become available to us via these means.
Indeed, in what follows we shall assume non-commutativity for multiplication between
instances of Ж and Ч.
What is Ж/Ж in the special zero algebra? We claim this is an example of a hyperzheh, given
0
through the use of the symbol Ж with subscripts. We define Ж1 = Ж as being the zheh 0, Ж2
as Ж1/Ж1, and generally Жn as the value Жn-1/Жn-1. For n > 1 these are called hyperzhehs. It
is convenient to put Ж0 = 0.
We also need symbols corresponding to Ч. Having defined cЧ0 as the cheh c/0, then cЧn is a
hypercheh cЧn-1/0.
We can extend these subscripts introduced above to negative values. We provide a
nomenclature for negatively subscripted zhehs; Ж -1 will be called an antizheh, so an antizheh
divided by itself equals zero. For n < -1 we use the terminology antihyperzheh. The question
arises as to whether the subscripts need be restricted to the integers. We also allow the
variable c to be of type (bЧm) or (bЖm).
A variation on this theme is that the collection of hyperzhehs forms a vector space. Addition
and multiplication are performed componentwise, and some components may have zero as a
coefficient, remembering 0  Жm = Жm, which is the default. We can choose a standard
arrangement, where say a coefficient c = (bЧj) is used in (cЧk) for the mth component of the
extended vector space, with j + k = m.
More generally, defining addition on components t and u to give component (t + u) and
multiplication of components t and u to give component t  u, the aggregated sum adds each
component of the first vector with those in sequence of the second to give additively the
result, and similarly for multiplication to give the result for the aggregated product. Since the
component of the aggregate is derived from each set of components, the distributive law
holds on components, and each component t, u, v or w contains as its value a function f on
hyperzhehs satisfying
f(t + u) = f(t) + f(u)
and
f(u  v) = f(u)  f(v),
this is a well behaved combination of operations on hyperzhehs.
For exponentiation, we might wish to consider whether Ж-1 = Ж-1, in which case
Ч-1nЧn = Жn.
The special zero algebra can be extended to incorporate matrices [Жm]ij and [Чn]ij in which
the place of zero is subsumed by determinant zero, that is, for singular matrices. We can
relate the determinant to the hypervolume of a vector space and highlight the well-known
relationship to linear independence in this context.
This case may be redeveloped further using hyperintricate number coefficients [Ad14a]
(these numbers are a representation of matrices), and the case by definition outside of
complex analysis where the Jacobean determinant is zero. Then an intricate coefficient
representation could be
(a1 + bi + c + d)Жm or (a1 + bi + c + d)Чn
These coefficients can also have ladder number components.
We note also that it is possible to define subscripts (mod p). The trivial case is (mod 1) when
0
1
zero is a zheh and 1 is cheh, in other words, 0 = 0 and 0 = 1.
We can define a combination of additive inverse subscripts, that is, an exponentiated
subtraction, for example when exponentiated antizhehs are combined to give an instance:
(aЖ -1)1/Ж-1 = aЖ0 = a0 = 1.
This algebra can be extended to superexponential operations, as we have already done in the
second session for the axiomatics of ladder algebra. For these higher-order operations, for
example we could explore the algebra
0
00 = 010-1 = 0 = Ж1,
where
Ж2 = 010-10-101  Ж1.
Q: Are there applications of this algebra?
A: Projective spaces, where the role of  is allocated to Ч.
Acknowledgements.
I would like to thank Tim Gibbs for asking stimulating and insightful questions, which
had the consequence of initiating this work, and Doly García and Paul Hammond for
their comments.
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Wikipedia. The following references have been used.
First session – prologue: First-order arithmetic.
Second session – ladder numbers and nonstandard analysis: Ultrafilters, Cauchy
sequence, Archimedean property.
Third session – the transfer principle: Leibniz law of continuity.
Fifth session – ordering and the axiom of choice: Well-ordering, axiom of choice,
completeness.
Seventh session – transfinite induction and analysis: Density.
Ninth session – limits and convergence: Bolzano-Weierstrass theorem, compactness,
Heine-Borel theorem.
Tenth session – ladder number transcendence: Formula for , lattice.
Eleventh session – Galois theory and infinite ladder automorphisms: Monotone and
antitone Galois connection.
Twelfth session – ladder calculus: Directional derivative, total derivative.