Symbolic Logic: Gödel’s Incompleteness Theorem
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Course textbook: Peter Smith – An Introduction to Gödel’s Theorems
Main things to focus on: videos, chapters, exercises, seminar
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Presentation: maybe on unreasonable effectiveness of maths? But how to relate to Gödel’s
theorem? Or: Putnam/Quine, Is Logic Empirical? – Quine/Putnam: not sure how this would be
related to Gödel. Other topic is good. Alternatively, explain in detail some stuff from chapter 21
Significance of different interpretations of Gödel’s theorem, sufficiently strong etc.
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Course evaluation
https://duke.evaluationkit.com/MyEval/LoginShib.aspx?lt=jNDe7LmUcdkFO%2bRnnILd%2bsjN3xDM
oPoQ84X3vlpGzFg%3d
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Then, reading: ch. 17-22
Look at link Ben sent you
Go over class notes & reading notes. Rehearse stuff. Write down stuff in Q and fact sheet
Reread bits of book/rewatch some videos etc.
Redo ex. Set 3, 4, 5, 6. In particular 5, 6 and 7
Do last set of exercises at some point
Watch Veritasium video
Go over all notes and rehearse new concepts – logic and maths
Make logic fact sheet and hang it up. Print out all axioms/order adequacy conditions etc. Make a list
of ALL theorems in the book. Make a comprehensive summary
Go over logic basics (propositional and predicate)
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Will you do exercise on explaining all concepts for last week? Ben: maybe yes
Does presentation need to be on a topic that explicitly relates to Gödel’s theorem? Ben: yes
Does paper need to be on a topic that explicitly relates somehow to Gödel’s theorem? Ben: yes
Paper – how does it work? Ben will propose paper topics, but you can also choose own topic to be
approved by Ben
Topics like implications of Gödel’s theorem for Platonism
 Week 1 – Smith, foreword
 ‘In 1931, the young Kurt Gödel published his First Incompleteness Theorem, which tells us that, for
any sufficiently rich theory of arithmetic, there are some arithmetical truths the theory cannot
prove’
 ‘The formal explanations are interwoven with discussions of the wider significance of the two
Theorems’
 ‘These Incompleteness Theorems settled -- or at least, seemed to settle -- some of the crucial
questions of the day concerning the foundations of mathematics. They remain of the greatest
significance for the philosophy of mathematics, though just what that significance is continues to be
debated’
 ‘For those who want to fill in more details and test their understanding there are exercises at
www.logicmatters.net, where there are also other supplementary materials’
1. Peter Smith, Chapter 1&2: ‘What Gödel’s Theorems Say’ and ‘Functions and Enumerations’
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Iff: if and only if
Goldbach’s conjecture: every even number greater than two is the sum of two primes
A set of axioms for arithmetic makes every proposition about arithmetic either true or false
‘This so-called `logicist' view would then give us a very neat explanation of the special certainty and
the necessary truth of correct claims of basic arithmetic’
‘Gödel’s First Incompleteness Theorem shows that the entirely natural idea that we can give a
complete theory of basic arithmetic with a tidy set of axioms is wrong’
‘What he then did is find a general method that enabled him to take any theory T strong enough to
capture a modest amount of basic arithmetic and construct a corresponding arithmetical sentence
GT which encodes the claim `The sentence GT itself is unprovable in theory T'. So GT is true if and
only if T can't prove it’
A sound theory: everything that the theory proves is true
Conclusion: theory T cannot be negation-complete
‘Suppose then that T is sound. If T were to prove GT , GT would be false, and T would then prove a
falsehood, which it can't do. Hence, if T is sound, GT is unprovable in T. Which makes GT true. Hence
GT is false. And so that too can't be proved by T, because T only proves truths’
‘In sum, still assuming T is sound, neither GT nor its negation will be provable in T’ sure but that is
because GT is a highly unusual sentence – it is a self-referential sentence! We have defined this
sentence as unprovable, so of course it is unprovable!! If it wasn’t, it wouldn’t be that sentence!
‘But doesn't this make GT rather uncomfortably reminiscent of the Liar sentence `This very sentence
is false' (which is false if it is true, and true if it is false)?’  couldn’t we argue that we cannot
meaningfully give a true value to these kinds of sentences? They are neither true nor false?
‘As we will see, there really is nothing at all suspect or paradoxical about Gödel’s First Theorem as a
technical result about formal axiomatized systems (a result which in any case can be proved without
appeal to `self-referential' sentences).’  okay so which types of sentences? And if this is the case,
then why did you not just explain this stuff with a non-referential sentence, so we didn’t have to
deal with that problem?
‘Thus T is not only incomplete but, in a quite crucial sense, is incompletable’  i.e. T is not just
incomplete, but it is impossible for T to be complete. I’m super interested in understanding more
about what we mean when we say ‘not only x, but impossibly not-x’
‘What gives Gödel’s First Theorem its real bite is that it shows that any nicely axiomatized and sound
theory of basic arithmetic must remain incomplete, however many new true axioms we give it’
 why is it known as the Incompleteness Theorem? Should rather have been called the
Incompletability Theorem
‘At this point it can begin to seem that we must have a rule-transcending cognitive grasp of the
numbers which underlies our ability to recognize certain `Gödel sentences' as correct arithmetical
propositions’ (bottom of p. 5)  I don’t understand (the significance of) what is being said here.
What does it mean to have a rule-transcending cognitive grasp of numbers? And why does that
imply we cannot be machines? The definition here refers to rules governing the machine’s
behaviour, not rules governing whatever topic the machine is engaging in (like arithmetic)
 See Hilbert’s programme. Look up what this is
A total one-place function maps each and every element of its domain to some unique
corresponding value in its codomain
When he talks about function, in this book, he always means total function
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Important terms: numerical functions, properties and relations
 Can you give an example of the characteristic function of the two-place relation R?
Convention in this book: 0 stands for true, 1 stands for false
Cantor’s theorem is an indenumerable set
Cantor’s theorem: there are infinite sets that are not enumerable  enumerable the same as
countable?
Indenumerable: the opposite of enumerable
‘Georg Cantor first established this key result in Cantor (1874), using the completeness of the reals.
The neater `diagonal argument' first appears in Cantor (1891)’
 I’m not saying that there’s no difference between what are called ‘analytical/a posteriori’ and
‘synthetic/a priori’ concepts; but I am saying that the difference is not that the one is not empirical
whereas the other is; they are BOTH empirical, but analytical… (this is the tricky part) is ‘less’
empirical, relies less on empirical knowledge, more basic knowledge, knowledge of the empirical
world that is true everywhere we know (??) whereas a posteriori/synthetic statements do not. They
rely more directly on empirical knowledge, even if all statements rely on empirical knowledge
 Week 2
1. Peter Smith, Chapter 3: Effective Computability
 Effectively computable function: ‘A one-place total function f : Δ  Γ is effectively computable iff
there is an algorithm which can be used to calculate, in a finite number of steps, the value of the
function for any given input from the domain Δ.
 Effectively enumerable set: a set that can be enumerated by an effectively computable function
 Effective computation involves (1) executing an algorithm which (2) successfully terminates.
 An algorithm might compute only a partial function. But an effective algorithm always computes a
total function since it must give an output for every possible input
 Algorithmic computability is architecture-independent
 Well-formed formula: wff
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Effective decidability: a property/relation is effectively decidable iff there is an algorithmic
procedure that a suitably programmed computer could use to decide, in a finite number of steps,
whether the property/relation applies to any appropriate given item(s).
Effective computability: a numerical property/relation is effectively decidable iff its characteristic
function is effectively computable
Explanation: ‘suppose there is an algorithm which decides whether n is P, delivering a `yes'/`no'
verdict: just add the instruction `if the answer is yes, output 0: otherwise output 1' and you get an
algorithm for computing cP (n). 5 Conversely, suppose you have an algorithm for computing cP (n).
Then you can add the line `if the value is 0 output yes: otherwise output no' to get an algorithm to
decide whether n is P’
Remember that numerical properties can be represented by the set of numbers that have that
property. i.e. a set is ED iff its CF is EC
Properties/sets are effectively decidable
Functions are effectively computable
Two theorems:
Any finite set of natural numbers is ED
If Σ is an ED set of numbers, so is its completement Σ’
Since the complement of a finite set is infinite, it follows that there are infinite sets of numbers
which are ED  that’s what I don’t understand. Completement of a finite set has infinitely many
elements, and so we cannot just take the finite number of outputs from the finite set and swap the
0 and 1, because there will be infinite numbers of output right? What am I misunderstanding here?
Effective enumerability
Enumerability: a non-empty set Σ is enumerable so long as there is some surjective function f: ℕ  Σ
which enumerates it (but need not be a EC function)
E.E: The set Σ is effectively enumerable iff either Σ is empty or there is an effectively computable
function that enumerates it, i.e. and EC surjective function f: ℕ  Σ
So: any finite set of numbers is EE
Theorem: if Σ is an ED set of numbers, it is EE
Theorem: if Σ and also its completement Σ’ are both EE sets of numbers, then Σ is ED
If every EE set of numbers had an EE complement, then every EE set would be ED. However, there
are some EE sets that have complements which are not EE. And there are EE sets that are not ED
Numerical domain of an algorithm Π: the set of natural numbers n such that when Π is applied to an
element of n as input, the algorithm will compute the output in a finite number of steps
‘e (i), choose any algorithm that never produces any output and we are done.’  I don’t get this.
Surely what we want for the empty set W is an algorihm that outputs 1 for any input, since no
number will be a member of the empty set??
Theorems about EE sets of numbers
Theorem 3.5: W is an EE set of numbers iff it is the numerical domain of some algorithm Π
Theorem 3.6: the set W of all EE sets of natural numbers is itself enumerable
Theorem 3.7: some sets of numbers are not EE, and hence not ED
Theorem 3.8: there is an EE set of numbers K such that its complement K’ is not EE
Theorem 3.9: Some EE sets of numbers are not ED
 Week 3
1. Chapter 4: Effectively Axiomatized Theories
 Gödel’s Incompleteness Theorems tell us about the limits of effectively axiomatized formal theories
of arithmetic
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‘in assessing complex arguments, it helps to regiment them into a suitable artificial language which
is expressly designed to be free from obscurities, and where surface form reveals logical structure’
‘Of course the benefits may come at a price; for it is not always clear how faithful the regimentation
is to the intentions of the original.’  Yes!! So many problems with translating natural language
sentences into formal languages – of course the ambiguities are gone, because you’ve just changed
the entire meanings of the sentences!
The point of axiomatizing theories: The aim is to discipline a body of knowledge by showing how
everything follows from a handful of basic assumptions p. 26
Axiomatized formal theory: ‘a theory built in a formalized language, with a set of formulae from the
language which are treated as axioms for the theory, and a deductive system for proof-building, so
that we can derive theorems from the axioms’
Formalized language L: < ℒ, 𝐼 > where ℒ is a syntactically defined system of expressions, and I gives
the interpretation of these expressions. I.e. ℒ is the syntactic component and I is the semantic
component
L’s syntactic component ℒ:
Need to specify ℒ’s non-logical vocabulary: constants (names), predicates, and function-signs
Need to specify ℒ’s logical vocabulary: variables, identity sign, parentheses, and symbols for
connectives and quantifiers
Need to specify ℒ’s terms: terms are constructed by applying (perhaps repeatedly) functions to
constants and/or variables
Rules: to determined which sequences of symbols constitute ℒ’s well-formed formulae (wffs). Open
wff: wff with free variables. Closed wff: wff without any free variables (only bound variables)
Sentence in ℒ: closed wff
L’s semantic component I:
Fix a domain of quantification for the languages variables to range over
Fix the semantic significance of all the non-logical vocab: assign values to the individual constants,
give satisfaction conditions for predicates, and assign functions to the function symbols
‘Then we explain how to assign truth-conditions to the atomic wffs. For example, in the simplest
case, the wff ‘var phi a’ is true iff the value of a satisfies the predicate’
‘Given the aims of formalization, a compositional semantics needs to yield an unambiguous truthcondition for each sentence, and moreover to do this in an effective way’
‘The semantics effectively tells us the conditions under which a given sentence is true’
Axioms: fundamental non-logical assumptions of the theory
‘for a usefully axiomatized formal theory, we must be able to effectively decide whether a given
sentence is an axiom or not’
Proof derivation: finite arrays of wffs that proves theorems from the theory’s axioms
It should be effectively decidable whether a given array of symbols counts as a well-formed proof
derivation according to the theory’s proof system (deductive apparatus)
‘For a nicely axiomatized formal theory T, then, we want it to be effectively decidable which wffs are
its logical or non-logical axioms and also want it to be effectively decidable which arrays of wffs
conform to the derivation rules of T's proof system’
Summary: T is an (interpreted) effectively axiomatized theory just if (i) T is couched in an
(interpreted) formalized language < 𝓛, 𝑰 >, such that it is effectively decidable what is a
wff/sentence of 𝓛 , and what the unique truth-condition of any sentence is, etc., (ii) it is
effectively decidable which 𝓛-wffs are axioms of T, (iii) T has a proof system such that it is
effectively decidable whether an array of 𝓛-wffs conforms to the proof-building rules, and hence
(iv) it is effectively decidable whether an array of 𝓛-wffs constitutes a proof from T's axioms.
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 what is the difference between 3 and 5? Don’t they say pretty much exactly the same thing?
‘However, note that although T1 is a decidable theory that doesn't mean T1 decides every wff ; it
doesn't decide e.g. the wff `(\sansq \wedge \sansr )', since T1's sole axiom doesn't entail either
`(\sansq \wedge \sansr )' or `\neg (\sansq \wedge \sansr )'. To stress the point: it is one thing to
have an algorithm for deciding what is a theorem; it is another thing for  WHAT??? This makes me
furious. That is EXACTLY what a decidable theory must be able to do! It must, for any putative wff in
the language, tell us whether or not it is actually a wff/whether or not the wff is true. That is exactly
what makes a theory effectively decidable!
 what? So what’s the difference between effectively decidable and ‘decides’? p. 33
The EE of theorems
Theorem 4.1: If T is an effectively axiomatized theory then (i) the set of wffs of T, (I’) the set of
sentences of T, (ii) the set of proofs constructible in T, and (iii) the set of theorems of T, can each be
effectively enumerated
Theorem 4.2: any consistent, effectively axiomatized, negation-complete theory T is effectively
decidable
Chapter 5: Capturing Numerical Properties
Effectively axiomatized theories for formal arithmetic
Need to be able to ‘express’ and ‘capture’ numerical properties
‘Gödel's First Incompleteness Theorem is about the limitations of axiomatized formal theories of
arithmetic: if T satisfies some minimal constraints, we can find arithmetical truths that can't be
derived in T. Evidently, in discussing Gödel's result, it will be very important to be clear about when
we are working `inside' a given formal theory T and when we are talking informally `outside'
that particular theory (e.g. in order to establish truths that T can't prove)’  yes, interesting
‘We'll standardly use Greek letters for this kind of `metalinguistic' duty. So note that Greek letters
will never belong to our formal languages themselves: these symbols belong to logicians'
augmented English’
Language LA: syntax
LA: the language of basic arithmetic
LA = ⟨𝐿𝐴 , 𝐼𝐴 ⟩ has a standard first-order syntax, with one two-place predicate (the =) and three
function expressions (S, + and x)
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Logical vocab: not, and, or, if, if and only if symbols, an infinitely countable number of variables,
quantifiers (‘for all’ and ‘there is’), identity symbol =, and brackets
Non-logical vocab: 0, S, + and x where 0 is a constant, S is a one-place function-expression, and +
and x are two-place function expressions
 why is = part of logical vocab whereas + and x part of non-logical? In general I don’t really
understand the division, here what makes the non-logical vocab non-logical and the logical v logical
Term: an expression that is constructed from ‘0’ and/or variables using any of the functionexpressions S, addition and multiplication. 1. 0 is a term and all variables are terms 2. If a and b are
terms, then so are S(a), (a + b), (a x b)
Numeral: a term built up from the single constant in our language, ‘0’, using only the S function. E.g.
S0 = 1, SS0 = 2, SSS0 = 3 etc.
Atomic wffs: since the only predicate in the language is =, all atomic wffs have the form a = b, where
a and b are any terms.
Wffs: atomic wffs plus those wffs formed by using connectives and quantifiers
‘With details completed in a standard way, it will be effectively decidable what is a term, what is a
wff, and what is a sentence of LA‘
Language LA: semantics
Interpretation is pretty straightforward. See p. 39
 don’t understand no. 4 p 39, why not just put ‘n’ under ‘there is an n’?
‘n has the property of being prime iff it is greater than 1, and its only factors are 1 and itself’
Definition of ‘expressing’ numerical properties: a numerical property P is expressed by the open wff
φ(x) with one free variable iff, for every n,
If n has the property P then φ(n) is true
If n does not have P then ¬φ(n) is true
‘True' of course continues to mean true on the given interpretation built into L’
Expressing just means: getting the extension right
Theorem 5.1: L can express a property P iff it can express P’s characteristic function C P
Definition of ‘capturing’ numerical properties: the theory T captures the property P by the open wff
ϕ(x) iff, for any n, if n has the property P, then T proves ϕ(n)
if n doesn’t have the property P, then T proves ¬ϕ(n)
You can also ‘capture’ a function: a theory T captures a function iff it captures the corresponding
functional relation
Distinction between expressing and capturing
A theory captures or expresses numerical properties
Capture: Case-by-case-prove. Remember it this way!
1. Whether P is expressible in a given theory just depends on the richness of that theory’s
language
2. Whether P can be captured by the theory depends on the richness of its axioms and proof
systems
Expressibility does not imply capturability!
Other direction: if T is a sound theory of arithmetic (one whose theorems are all true on the given
interpretation of the language), then if ϕ(x) captures P in T, then ϕ(x) expresses P
d
 Week 4
1. Chapter 6: The Truths of Arithmetic
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Chapter 4: we proved that the theorems of any effectively axiomatized theory can be effectively
enumerated. In short: theorems of any EA theory can be EE
This chapter: we prove by contrast that the set of truths of any language which is sufficiently
expressive of arithmetic can't be effectively enumerated
‘it immediately follows that a sound axiomatized theory with a sufficiently expressive language can't
be negation-complete.’
‘An interpreted formal language L is sufficiently expressive iff (i) it can express every effectively
computable one-place numerical function, and (ii) it can form wffs which quantify over numbers’
We will show that sound effectively axiomatized theories with sufficiently expressive languages
cannot be negation-complete
 why not just create a language that is not sufficiently expressive? Then maybe we won’t have the
negation-incompleteness problems
Theorem 6.1: The set of truths of a sufficiently expressive language L is not effectively enumerable.
Definition: ‘A set of wffs Σ is effectively axiomatizable iff there is an effectively axiomatized formal
theory T such that, for any wff ϕ, ϕ ϵ Σ if and only if T proves ϕ (i.e. Σ is the set of T-theorems)’ 
here it’s talking about a set being effectively axiomatizable, you talked about a theory being that –
difference? How does that work?
Theorem 6.2: the set T of true sentences of a sufficiently expressive language L is not effectively
axiomatizable.
Working up to the incompleteness theorem: Suppose we have an EA theory T in a sufficiently
expressive language L. Since T is EA, its theorems can be EE. Since T’s language L is sufficiently
expressive, the truths expressible in L cannot be EE. So, there is a mismatch between T’s theorems
and the truths expressible in L: these two sets cannot be identical, since one is EE and the other is
not. Suppose that T is also a sound theory (all its theorems are true). The mismatch between T
theorems and the truths in L must be due to there being truths which T cannot prove. Suppose ϕ is
one of the truths that T cannot prove. Then T doesn’t prove ϕ, and since not- ϕ is false, T also
doesn’t prove not-ϕ (since T is sound). So, there are truths expressible in T’s language L that T
cannot prove.
Incompleteness theorem 1: if T is a sound, effectively axiomatized theory whose language is
sufficiently expressive, then T is not negation-complete.
 in what way are there incompleteness theorems different/less profound/less consequential than
Gödel’s theorems? – specifically because we assume the theory T to be sound here?
Even the minimal language LA is already sufficiently expressive, so a vast number of sound theories
of arithmetic must be negation-incomplete
2. Chapter 7: Sufficiently Strong Arithmetics
 A sufficiently strong theory: one that by definition can prove what we’d like any decent theory of
arithmetic to be able to prove (about ED properties of numbers)
 We are interested in theories that can capture ED properties: a theory T which 1) has an open wff
ϕ(x) which expresses property P, and 2) is such that if n has property P, T proves ϕ(n) and if n does
not have property P, T proves not-ϕ(n). In short, we want T to capture P (to get it right which n have
the property P and which do not)
 Definition: a formal theory of arithmetic T is sufficiently strong iff it captures all effectively
decidable numerical properties
 Undecidability theorem: there is no consistent, sufficiently strong, effectively axiomatized theory of
arithmetic that is effectively decidable
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See proof: yeah, sure, but you’ve defined it such that there was necessarily going to be a
contradiction; you assumed the contradiction in order to then derive the contradiction. ?! You
assumed that n has property D iff T proves not-ϕ(n), and that n has property D iff T proves ϕ(n). Of
course then you’ll get an inconsistent theory!
‘In sum, the supposition that T is a consistent and sufficiently strong axiomatized formal theory of
arithmetic and decidable leads to contradiction’  how do we know that the problem then is
effective decidability, and not one of the other conditions (or one of the other combinations)?
‘What we've just shown is that this is a false hope: as soon as a theory is strong enough to capture
the results of boringly mechanical reasoning about decidable properties of individual numbers, it
must itself cease to be decidable’
Another incompleteness theorem: put together theorem 4.2 and theorem 7.1. If T is a consistent,
effectively axiomatized, sufficiently strong theory of arithmetic, then T is not negation-complete
E
Week 5 – Mathematical Induction
Chapter 8 – Interlude: Taking Stock
8.1 Comparing incompleteness arguments
‘However, we announced right back in Section 1.2 that Gödel’s own arguments rule out complete
theories even of the truths of basic arithmetic. Hence, if our easy Theorems are to have the full
reach of Gödel’s work, we'll really have to show (for starters) that the language of basic arithmetic is
already sufficiently expressive, and that a theory built in that language can be sufficiently strong’ 
don’t understand what is being said here. Ask Ben. P. 53
‘a general treatment of the class of computable functions,’  what do you mean, a ‘general
treatment’? No idea what you’re talking about. P. 53
In what way Gödel’s theorem is stronger: actually constructs a true but unprovable-in-T sentence (‘I
am unprovable in T’) and doesn’t need as many conditions to be in place for the proof to work
Gödel’s arguments also come in two flavours, like 6.3 and 7.2 (sound and sufficiently expressive vs
consistent and sufficiently strong)
8.2 A roadmap
PA: Peano Arithmetic
Q: Robinson Arithmetic, a subsystem of PA
‘G is true just if it is unprovable in PA. Given PA is sound and only proves truths, G can't be provable;
hence G is true; hence not-G is false, and so is also unprovable in PA . In sum, given PA is sound, it
cannot decide G’  exactly – you have DEFINED the sentence as being unprovable, so of course it is
unprovable!!! This is ridiculous! And if we don’t need this self-referential sentence, then why do we
keep on using it! Confusing at best, and thoroughly unconvincing at worst
 Chapter is saying that the incompleteness proofs we’ve just gone through are ‘informal’ as
opposed to Gödel’s proofs which are formal. I don’t get this – in what ways were the proofs we’ve
done informal? Seemed pretty formal to me. Drew on all this formal logical apparatus and
formalized arithmetical theories
‘Only after we have travelled the original Gödelian route (which doesn't presuppose a general
account of computability) will we return to consider how to formalize the arguments of Chapters 6
and 7 (a task which does presuppose such a general account).’  what do you mean by a general
account of computability?
Answer: General account of computability: so far our treatment has been in places somewhat
informal. Formal would be showing Turing machines
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2. Chapter 9 – Induction
 ‘A standard method (from ordinary, informal mathematics) for establishing general truths about
numbers’  so confused by this strange dividing line of formal and informal. How is this a method
from informal mathematics?
 Main subject of chapter: proof by induction
 9.1 The basic principle
- Principle of arithmetical induction
- Base case: ϕ(0), i.e. 0 has property P
- Φ: the expression attributing some property P to numbers
- Induction step: for all n(ϕ(n)  ϕ(n+1)
- ‘Arithmetical induction takes us from a universal premiss, the induction step (ii), to a universal
conclusion. That's why it can be deductively valid, and is of course not to be confused with an
empirical induction from a restricted sample to a general conclusion!’  so then why is it called
induction?
 9.2 Another version of the induction principle
- Course-of-values induction (strong or complete induction) principle: slightly different variant of the
simple induction principle  I don’t get the difference between these two
- ‘Every wff of the propositional calculus is balanced, i.e. has the same number of left-hand and righthand brackets’
- ‘We proceed by (course-of-values) induction on the number of connectives in the wff. This time, let
ϕ(n) hold iff an n-connective wff is balanced. Once more, the base case (i), i.e. ϕ(0), holds trivially’
- ‘So let \varphi (n) hold when the result of an n line proof is a tautology. Trivially, (i) \varphi (0) -since a zero line proof has no conclusion, so -- vacuously -- any conclusion is a tautology’  that
doesn’t seem right!
- ‘Soundness proofs for fancier logical systems will work in the same way -- by a course-of-values
induction on the length of the proof.’
 9.3 Induction and relations
- ‘Note that in the middle of the reasoning, at the point where we applied induction, `m' was acting as
a `parameter', a temporary name not a bound variable’
 9.4 Rule, schema or axiom?
- ‘We will finish this chapter by thinking just a little more about how the basic induction principle
should be framed’
- Three different presentations of the basic induction principle
- Induction rule of inference: same as the principle
- Family of true conditionals: an induction schema
- ‘Induction Axiom: for all numerical properties X, if 0 has property X and also any number n is such
that, if it has X, then its successor also has X, then every number has property X’
 Week 6 – Baby Arithmetic and Robinson Arithmetic
1. Chapter 10 – Two Formalized Arithmetics
 BA: a theory which `knows' about the addition and multiplication of particular numbers, but doesn't
`know' any arithmetical generalizations at all (for it lacks the whole apparatus of quantification)
 As with any formal theory, we need to characterize (a) its language, (b) its logical apparatus, and (c)-(e) its non-logical axioms.
  Theorem 10.4 proof sketches (p. 65): don’t understand at all. We are assuming what needs to be
proved!
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Recall, to say that \sansB \sansA correctly decides \varphi is to say that if \varphi is true \sansB
\sansA \vdash \varphi , and if \varphi is false \sansB \sansA \vdash \neg \varphi (see Section 4.4,
defn. 4).
Important theorems:
1. BA correctly evaluates every term of LB
2. BA correctly decides every atomic wff of LB
3. BA correctly decides every wff of LB
4. BA is negation-complete
BA is sound, effectively axiomatized, and negation-complete; by theorem 6.3, this entails that it
cannot be sufficiently expressive
Q: Robinson Arithmetic
‘however, that while these Axioms tell us that zero isn't a successor, they leave it open whether
there are other objects that aren't successors cluttering up the domain of quantification (there
could be `pseudo-zeros')’  don’t understand why this is the case
‘There is a wide variety of formal deductive systems for first-order logic, systems which are
equivalent in the sense of proving the same sentences as conclusions from given sentences as
premisses’
Two types of formal deductive systems: Hilbert-style axiomatic systems and natural deduction
systems
 Section 10.4 which logic? (Hilbery vs natural deduction): why is this significant/interesting? Okay,
inference rules and axioms are more or less the same, so what? Why do we even make a distinction
if they are indeed more or less the same? (explain difference between these two with examples)
‘So which style of logical system should we adopt in developing Q and other arithmetics with a firstorder logic? Well, that will depend on whether we are more concerned with the ease of proving
certain metalogical results about formal arithmetics or with the ease of proving results inside the
theories. Hilbertian systems are very amenable to metalogical treatment but are horrible to use in
practice. Natural deduction systems are indeed natural in use; but it takes more effort to theorize
about arboriform proof structures.’
‘Like \sansB \sansA , \sansQ is evidently a sound theory (and so consistent). Its axioms are all true;
its logic is truth-preserving; its derivations are therefore proper proofs in the intuitive sense of
demonstrations of truth. In sum, every \sansQ -theorem is a true LA sentence. But just which LAtruths are theorems?’
We know that all Q-theorems are truths in LA, but we don’t know whether all truths in LA are Qtheorems. So question is: are all truths in Q-theorems? If P is a truth in LA, is P a theorem of Q?
Answer: no! Q is not negation-complete. Not all truths in all are theorems of Q
 10.8: just add commutativity to Q and then it can prove this!! Also: break down exactly how to
understand this theorem
If there is one interpretation of the theory’s language on which a given wff is false, then we know
that the theory does not prove this wff (‘proving’ is a syntactic notion, so if T proves wff A then
wff A is true on any interpretation of T’s language)
 this seems like cheating. Couldn’t you always find some rogue interpretation, an interpretation
which seems intrinsically wrong or false, on which any theorem of T could be shown to be false? You
just have to engineer it right! I have a hard time believing that you couldn’t just show that any
theorem of Q is false on some interpretation, if you are allowed to change the interpretation
radically in the sense of changing the domain and the meaning of the symbols!
‘But not so. Despite its great shortcomings, and perhaps rather unexpectedly, we'll later be able to
show that \sansQ is `sufficiently strong' in the sense of Chapter 7. For `sufficient strength' is a matter
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of being able to case-by-case prove enough wffs about decidable properties of individual numbers.
And it turns out that Q's hopeless weakness at proving generalizations doesn't stop it from doing
that’
2.
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Chapter 11 – What Q Can Prove
Abbreviations
LL: Leibniz’s law
MP: modus ponens
RAA: reductio ad absurdum
UI: universal quantifier introduction
∃/∀: moving between quantifiers
Q can capture the less-than-or-equal relation
11.2: ‘But obviously that initial definition could get us into trouble. Suppose it happens that what we
put into the slot marked by `\xi' already has a free occurrence of `\sansv : we then get a nasty ‘clash
of variables’ p. 72  I don’t get what is the problem here
Bound variables, not the same as bounded quantifiers
Q is order-adequate
 Why do we have O2 and O3? Why not make a biconditional and have only one proposition?
Q can correctly decide all delta-0 sentences
Using course-of-values induction in order to
‘a \Delta 0 sentence has degree k iff it is built up from wffs of the form \sigma = \tau or \sigma \leq
\tau by k applications of connectives and/or bounded quantifiers. Then we can use a course-ofvalues induction on the degree of complexity k.’
 I don’t get the proof on top of p. 75
‘There is no induction rule inside Q we don't start considering formal arithmetics with an induction
principle until the next chapter. But that doesn't prevent us from using induction informally, outside
Q, to establish meta-theoretical results about what that theory can prove’  I TOTALLY DO NOT
UNDERSTAND THIS!!
 bound vs. bounded: are variables bound but quantifiers bounded? Or is bounded just a variation
in spelling on bound?
Q is Σ1-complete
Intriguing corollaries
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 Week 7 – Arithmetics with Induction and Peano Arithmetic
1. Chapter 12 – IΔ0, an Arithmetic with Induction (7 pp)
 When it comes to proving universal generalizations about numbers, Robinson Arithmetic Q is very
weak
 What do we need? An Induction principle!
 The formal Induction Schema
- Since LA lacks second-order quantifiers, we don’t have the option of going for a single second-order
induction axiom
- Induction schema for monadic properties: a wff
- Induction schema for relational properties (R) a wff
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Because the second axiom has a free variable, it is not a closed sentence with a fixed truth value on
an interpretation. Thus we cannot treat this wff as an axiom
Universal closure: closing some open wff by adding enough universal quantifiers to bind the free
variables. E.g. taking the universal closure of some wff: prefixing the wff with enough universal
quantifiers so that all its free variables become bound
So: (R) itself won’t be an axiom, but the universal closure of (R), (R’), will!
 so, why didn’t we just do this from the start? Why are we making a fuss out of this?
Introducing IΔ0
IΔ0 is the theory that adds a minimal amount of induction to Q
we want wffs which express genuine numerical properties (and relations), for these fall under the
intuitive induction principle
‘e \varphi (\sansx ) is a wff of LA which has just `\sansx ' free and which has at most bounded
quantifiers; in other words, suppose \varphi (\sansx ) is \Delta 0. Then such a wff surely does express
an entirely determinate monadic property.’  how do we know this? Can you explain?
Will restrict the application induction to Δ0 wffs
Definition of IΔ0: IΔ0 is the first-order theory whose language is LA, whose deductive logic is
classical, and whose axioms are those of Q plus the following extra axioms: (the universal closures
of) all instances of the Induction Schema where ϕ(x) is a Δ0 wff’
What IΔ0 can prove
The following are all theorems of IΔ0, meaning that IΔ0 can prove them:
‘Theorem 12.2 In any theory which is at least as strong as IΔ0, a wff starting with more than 1
unbounded existential quantifiers is provably equivalent to a wff starting with just a single
unbounded quantifier’  I’m not quite sure what this theorem is saying and why it is true. What
does it mean to be provably equivalent?
IΔ0 is not negation-complete
 don’t understand outline proof of the theory’s incompleteness at all
On to IΣ1
‘The most natural way of plugging the identified gap in \sansI \sansDelta 0 is, of course, to allow
more induction’
‘We of course can't swap the order of unbounded quantifiers of different flavours; but we can drag
an unbounded existential forward past a bounded universal (leaving behind a bounded existential)’
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 but WHY do we want to do this?? What is the point of doing this if they mean the same thing
anyway??
2. Chapter 13 – First-order Peano Arithmetic (8 pp)
 Peano Arithmetic: Q + (the universal closure of) all instances of the Induction Schema for any LA
predicate, not just Δ0 predicates
 The result is first-order Peano Arithmetic
 Being generous with induction
- ‘\varphi in the Schema by a suitable open wff which expresses a genuine property/relation.’  in
our last meeting you said that any wff expresses a property, but Smith keeps on specifying that we
need to look at wffs that express genuine properties only. ??
- ‘. Hence, if we are to philosophically motivate a restriction on induction, it must be grounded in the
thought that some predicates that we can syntactically construct (and which we thought our
classical semantics \scrI A gave sense to) in fact don't express kosher properties at all. But why
should we suppose that?’
 Summary overview of PA
- ‘Where ϕ(x) is an open LA wff that has ‘x’, and perhaps other variables, free’ P. 92  x is bound here
no?
- ‘It is still decidable whether any given wff has the right shape to be one of the induction axioms, so
PA is indeed an effectively formalized theory’
 Hoping for completeness
- Theorem 13.1: the theory P (which is PA minus the multiplication symbol) is a negation-complete
theory of successor and addition
-  what is model theory? Smith often talks about model-theoretic arguments
- Also: the theory A (which is PA minus successor and addition) is a negation-complete theory of
multiplication
- ‘Why then does putting multiplication together with addition and successor produce
incompleteness? The answer will emerge over the coming chapters?’
- ‘PA must indeed be incomplete. Starting in the next chapter we will begin to put together the
ingredients for a version of the original Gödelian argument for this conclusion (an argument that
doesn't require the idea of `sufficient strength')’
 Is PA consistent?
- ‘it appeals to our supposed grasp of the structure of the natural numbers and also to the idea that
all open wffs of LA express genuine numerical properties and relations’  how could this idea be
challenged, then? This appears to be a recurring theme in these chapters
- ‘an argument for a theory's consistency which appeals to our alleged grasp of some supposedly
intuitive truths (however appealing) can lead us badly astray’  YES!!
- Majorly problematic assumption: ‘Frege takes as his fifth Basic Law the assumption, in effect, that
for every well-constructed open wff ϕ(x) of his language, there is a class (possibly empty) of exactly
those things that satisfy this wff’ class = set
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Week 8 – Primitive Recursive Functions
Chapter 14 – Primitive Recursive functions
Introducing the primitive recursive functions
‘In each definition, the second clause fixes the value of a function for argument Sn by invoking the
value of the same function for argument n. This kind of procedure is standardly termed `recursive'
-- or more precisely, `primitive recursive'.
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And our two-clause definitions are examples of definition by primitive recursion.
‘Addition is defined in terms of the successor function; multiplication is then defined in terms of
successor and addition; then the factorial (or, on the other chain, exponentiation) is defined in
terms of multiplication and successor’
Primitive recursive function: a function that can be characterized using a chain of definitions by
recursion and composition
They are functions that can be defined by a chain of definitions by recursion and composition
Summary on p. 101
An aside about extensionality
Note on identity conditions for functions:
‘a p.r. function is by definition one that can be specified by a certain sort of chain of definitions. And
an ideal way of presenting such a function will be by indicating a definitional chain for it (in a way
which makes it transparent that the function is p.r.). But the same function can be presented in
other ways; and some modes of presentation can completely disguise the fact that the given
function is recursive’
‘primitive recursiveness is a feature of a function itself, irrespective of how it happens to be
presented to us.’
‘We count numerical functions as being the same or different, depending on whether their
extensions are the same.’
‘a numerical property, in particular, is a function that maps a number to the truth-value true (if the
number has the property) or false (otherwise). Which comes very close to identifying a property
with its characteristic function’  what do you mean, ‘comes very close’? it just IS that, no?
The PR functions are effectively computable
1. the initial functions S, Z, and I k i are effectively computable
2. The composition of two effectively computable functions g and h is also computable
3. if g and h are effectively computable, and f is defined by primitive recursion from g and h, then f is
effectively computable too
Theorem 14.1: Primitive recursive functions are EC, and computable by a series of (possibly
nested) ‘for’ loops
This gives us a quick-and-dirty (but reliable!) way of convincing ourselves that a new function is p.r.:
sketch out a routine for computing it and check that it can all be done with a succession of (possibly
nested) `for' loops which only invoke already known p.r. functions; then the new function will be
primitive recursive.
Not all EC functions are PR
‘We have just seen that the values of a given primitive recursive function can be computed by a
program involving `for' loops as its main programming structure, where each such loop goes through
a specified number of iterations. However, back in Section 3.1 we allowed procedures to count as
computational even when they don't have nice upper bounds on the number of steps involved. In
other words, we allowed computations to involve open-ended searches, with no prior bound on the
length of search’
See diagonalization argument
Defining PR properties and relations
‘We have defined the class of PR functions. Next, without further ado, we extend the scope of the
idea of primitive recursiveness and introduce the ideas of PR (numerical) properties and relations’
‘A PR property is a property with a PR characteristic function, and likewise a PR relation is a
relation with a PR characteristic function’
‘Given that any PR function is effectively computable, PR properties and relations are among the
effectively decidable ones’
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Building more PR functions and relations
 I don’t understand the definitions from point b) (bottom of p. 107 and then 108)
Theorem 14.3a
Theorem 14.3b
Theorem 14.3c
Theorem 14.3d
Theorem 14.3e
 what is the sg(n) about on p. 109? Why all of a sudden an ‘s’ in front?
Theorem 14.4: the properties, relations and functions listed in 1-6 are all PR
‘By the Fundamental Theorem of Arithmetic, which says that numbers have a unique factorization
into primes, this function is well-defined’  ?? What is the fundamental theorem of arithmetic?
When have we seen this? What theory is it a theorem of – PA?
 Week 9 – Expressing and Capturing Primitive Recursive Functions
1. Chapter 15: LA can express every PR function
 ‘We now radically extend our list of examples by proving that LA can in fact express any primitive
recursive function’
 Theorem 15.1: Every PR function can be expressed in LA
 We want to prove this theorem
 The most interesting part to prove – part (3): if the function f is defined by recursion from functions
g and h which are already expressible in LA, then f is also expressible in LA.
  Proof for (1): I don’t understand the definitions of the functions and wff p. 113
 ‘There is a sequence of numbers k0, k1, . . . , kx such that: k0 = 1, and if u < x then kSu = ku times Su,
and kx = y’  you said in your video lecture that k0 = 0, but here it says k0 = 1. Which one is right? P.
114
 ‘Can we construct a \beta - function just out of the successor, addition and multiplication functions
which are built into LA? Then we can use this \beta -function to prove claim (3)’
 Theorem 15.2 + 15.3: every PR function can be expressed by a Σ1 wff, and so also every PR
property and relation
2. Chapter 16: Capturing Functions
 ‘In the next chapter we will show that even the simple theory Q, despite its meagre proof-resources,
can in fact capture every PR function and thereby correctly evaluate it for arbitrary inputs’
 ‘This short intervening chapter does some necessary groundwork, pinning down the requisite notion
of `capturing a function'’
 This chapter explains what it means to capture a function
 Theorem 16.1: on minimal assumptions, theory T captures a property P iff T captures its
characteristic function CP. So, two directions:
- If T captures CP, then T captures P
- If T captures P, then T captures CP
 Strong vs. weak capturing
 ‘T may not even have quantifiers! But even if it does (e.g. because it contains Q), T's inference rules
might allow us to prove all the particular numerical instances of a certain generalization without
enabling us to prove the generalization itself’  I still don’t understand why this is the case. Are we
saying we that the generalization is more than the collection of all its numerical instances?
 Strong capturing  capturing  weak capturing
 If ϕ strongly captures f in T, then ϕ captures f in T
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If ϕ captures f in T, then ϕ weakly captures f in T
If ϕ weakly captures f in T, then there is a closely related wff, ϕ’, that captures f in T
If ϕ captures f in T, then there is another closely related wff, ϕ’’, that strongly captures f in T
‘once we are dealing with arithmetics as strong as \sansQ , if a function is weakly capturable at all it
is also capturable in both stronger senses’  yes exactly, even if it is not captured in the stronger
sense by the exact same wff that it is weakly captured by
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Week 10 – PR Adequacy and Russell’s Principia
Chapter 17: Q is PR Adequate
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2. Chapter 18 – Interlude: A Very Little about Principia
 Godel’s 1931 paper is the big important one: ‘On formally undecidable propositions of Principia
Mathematica and related systems I’
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Chapter 19: The Arithmetization of syntax
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Chapter 21: PA is Incomplete
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2. Chapter 22: Gödel’s First Theorem
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4. The Lucas-Penrose Argument about Gödel’s Theorem – Internet Encyclopedia of Philosophy
 ‘In 1961, J.R. Lucas published “Minds, Machines and Gödel,” in which he formulated a controversial
anti-mechanism argument. The argument claims that Gödel’s first incompleteness theorem shows
that the human mind is not a Turing machine, that is, a computer’
 ‘if we cannot establish that human minds are consistent, or if we can establish that they are in fact
inconsistent…’  what do you mean by ‘if the human mind is consistent’??
 ‘Penrose argues that the Gödelian argument implies a number of claims concerning consciousness
and quantum physics; for example, consciousness must arise from quantum processes and it might
take a revolution in physics for us to obtain a scientific explanation of consciousness’
 This argument called the anti-mechanism argument
  in what way is Godel’s sentence different the liar paradox?
 Go back to 1. Lucas’ original version of the argument
 The point is, there are arithmetical truths we can have/understand/?? That machines cannot.
 It’s not the case that we can in some more intuitive sense of the word prove, things that a machine
cannot. It is that the machine cannot prove something we which intuitively can see is true. 1. These
are two different notions, two different demands! 2. Lucas is begging the question when he says
that . A machine cannot ‘see’ intuitively whether some statement is true. Of course then he’ll arrive
at the conclusion that machines and humans are different
 Also, something strange
 Spell out his argument deductively – formalize it!
 We have to be really careful here. As with many philosophical arguments, there is a risk that we
take seemingly perfectly plausible/reasonable/true premises and arrive at a highly surprising or
just generally significant conclusion; but before accepting that conclusion, let us analyse the
argument, as history shows there are all kinds of ways that arguments can on the face of it seem
impeccable (deductive, plausible premises etc.) after all have a gap
 I’m wondering if the significance of Godel’s theorem to these .. like has been overstated. If I had to
make a bet, I would say that I’m not sure Godel’s theorem is hugely significant beyond the realm of
mathematical and philosophy of mathematics.
 I’m not really sure that I think that Gödel’s theorem has any bearing at all upon the question
whether the mind is (like) a computer. This just isn’t something that can be proven a priori.
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 Whilst this argument is very creative and imaginative, I cannot say I am particularly compelled by it.
 1.
 2. Circularity objection
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 Explain the Lucas-Penrose argument. Note the differences between Lucas’ and Penrose’s views.
 At least at first glance, I can’t say that I am very impressed with the argument. (jumping the gun). I
happen to agree to some significant extent that the human mind is not a Turing machine, but not
because of Gödel’s theorem. I don’t think the theorem is so strong as to be able to show this
 It could still be the case that for any one arithmetical truth, there is some formal theory which
proves it, and so all arithmetical truths are actually proven, just not all by the same formal theory
 Week 13 –
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