Absolute undecidability
- made as simple as possible
This presentation gives an example
based on a simplified version of
the formalism in my paper
On the existence of absolutely undecidable sentences of
number theory
Decimal numbers in base 2
Let r be a real number. We will consider r
written in base 2 notation.
Here is an example:
r = 0.10101110001111011001011010…
Winners and losers
If r is a real number written in base 2, we divide the decimal
bits of r into disjoint groups starting from the decimal point,
in such a way that the nth group consists of n bits. Below, the
bits of every second group of r is marked red:
r = 0.10101110001111101001011010010…
 We call the natural number n a winner with respect to r
if n = 0, or the nth group of bits from r consists of 1’s only.
 In the example above, n = 1 and n = 5 are winners.
 If n is not a winner, we call it a loser with respect to the
number r.
Numbers r definable in L(PA)
We assume that the real number r is definable in the
language L(PA) of Peano Arithmetic, in the following
sense:
 There is a formula A(x,y) of L(PA) with two free
variables x and y such that if n is a natural number,
then A(x/n,y/1) is true in N iff the nth base 2 decimal
of r is 1, and A(x/n,y/0) is true otherwise.
[The notation x/n means that the numerical term n corresponding to the
natural number n is substituted for the variable x in A. Similarly for y.]
Chaitin’s Omega
 In our simple example, we are going to take
r = Ω, where Ω is the number commonly known as
Chaitin’s constant.
 The number Ω can be interpreted as a halting
probability.
 In the following, a winner (loser) will mean a winner
(loser) with respect to Ω. Note, however, that the
theorem we will prove, trivially can be generalized to
hold for any given r which is definable in L(PA). Other
interesting choices for r include the Euler number e,
the number pi etc.
The maximal winner principle
(MWP)
 By the maximal winner principle (MWP) will be
meant the following statement: There exists a
maximal winner among the natural numbers.
 That is, the MWP states that there is a winner m
with the property that there is no winner n
greater than m.
 Since the bits of Ω are definable in the language
L(PA) of Peano Arithmetic, the MWP can be
expressed as a closed sentence in this language.
The probability that there are no
winners
 If the bits of Ω are assumed random, then the probability
that there are no winners n > 0, is the infinite product
p = (1/2)(3/4)(7/8)(15/16)(31/32)…
 This infinite product, in which the nth factor is
(2^n-1)/2^n, can be shown to converge to a nonzero
number. We have p ≈ 0.289.
The probability that there are no
winners beyond a certain point
Let n be a given natural number. If the bits of Ω are assumed
random, then the probability
p[n]
that there are no winning numbers j greater than n, is
nonzero. The probability
q[n] = 1 – p[n]
that there are winners j greater than n, is also nonzero.
Example: The probability that there
are no winners beyond n = 3
 The probability p[3] that there are no winners
greater than n = 3, is
p[3] = (15/16)(31/32)(63/64)…
 The probability that there are winners greater
than n = 3, is
q[3] = 1 - (15/16)(31/32)(63/64)…
The crucial question: What is the
truth value of the MWP?
 This is the simple question that opens the door
theoretically. Since both the probabilities p[n] and
q[n] are nonzero for all n, we will see that under the
assumption that all closed statements in the
language L(PA) of Peano Arithmetic have a well
defined truth value in the set N of natural numbers,
both of the alternatives (MWP true and MWP false)
lead to situations in which we can obtain nontrivial
knowledge about an infinite number of Ω decimals.
If the MWP is true…

If the MWP is true in N, then there exists a
natural number m such that the statement D
saying that m is the maximal winner, is true.

The statement D can be expressed as a closed
sentence of the language L(PA), with the
witness m represented by a numerical term m.
What happens if we
are able to prove D?

Note first that although we know by indirect reference that D is
true in N, we can not use this knowledge to infer the truth value of
D when given it. The reason is that we do not know if the value of
m in D is correct. In other words, we cannot use this knowledge to
‘recognize D if we see it’.

On the other hand, if the MWP is true, then the concrete formula
D is ‘out there somewhere’. If we can establish with direct
reference that the concrete formula D is true, then we will know
the value of m. In other words, we will know that there are no
winners n greater than m. Since in a random, equal probability bit
sequence the probability q[m] that there are winners greater than
m is nonzero, this gives us nontrivial information concerning an
infinite number of bits in Ω.
If the MWP is not true…




If the MWP is not true in N, then proving this very fact becomes the
problem.
If at some point in time we could establish that the MWP is false in N,
then we would know that a winning group occurs an infinite number of
times in the bit sequence of Ω.
In contrast, if the bits in Ω were random, then the probability p[n] that
there does not exist another winner greater than a given n is nonzero, in
fact greater than p.
Based on this, one may show (see my paper) that if we establish that the
MWP is false, then we will know that the Ω bit sequence deviates
strongly from an infinite random sequence in the following sense:
Let g be a natural number. Then there exists natural numbers q and d
such that the following holds for all integers d’>d: The number of
winners j between q and q+d’ is greater than gE, where E is the
expected number of winners between q and q+d’ if the Ω bit sequence
is considered random.
Now we may prove:
Theorem
Assume that every closed sentence of L(PA) has a well defined truth value
in N. Then there exists a closed sentence A in L(PA) with the property that
if at some point in time we are able to decide the truth value of A in N,
then at that point we will have nontrivial information (to be precise:
information of one of the two kinds described previously) concerning an
infinite number of bits in Ω.
Proof
If the MWP is true in N, then we take A = D. By indirect reference, we
know that D is true in N. If at some point we are able to prove the
concrete formula D, then we will know the value of the witness m in D.
Thus we will know that there are no winners n > m along the bits of Ω.
If the MWP is false in N, then we take A = MWP. If we are able to refute
the MWP, then we will know that the bit sequence of Ω deviates strongly
from a random sequence in the sense described on the previous slide.
But…
 Based on the definition of Ω, one may argue
convincingly that we cannot at any point in time
obtain information of the two kinds we have
encountered about the sequence of bits in Ω. The
reason is, as has been mentioned, that both types of
information gives us nontrivial knowledge
concerning an infinite number of Ω bits.
 Therefore, we are led to conclude that the truth
value of either the MWP or the formula D is
nonconstructible in the sense that we can never find
it. I express this by saying that either the MWP or D
is absolutely undecidable.
To sum up:
 Either the MWP is an absolutely undecidable
sentence of L(PA), or else it is true.
 It the MWP is true, so that the sentence D
exists, then D is an absolutely undecidable
sentence of L(PA).
 In either case, it follows that there exists a
closed sentence of L(PA) which is absolutely
undecidable.
The nature of the argument
Logically, the structure of the preceding undecidability argument is similar to
reduction ad absurdum. However, since we are dealing with underdetermination
instead of overdetermination, the absurd results encountered are not contradictions.
Instead, they describe absurd coincidences, or absurd situations.
The event that some day we obtain nontrivial information about an infinite number of
bits in Ω in one of the two ways we were led to by the preceding argument, indeed
would be an absurd situation. However, we probably cannot give a proof of the
impossibility of this which can be formalized in ZF. If we found such a mechanical
proof, then we could use it to prove inconsistency of ZF. Then we would be back at the
overdetermination side of the table. Still, the arguments that can be given for the
impossibility of the situations in question, based on the definition of Ω, to me are just
as convincing as a proof which can be formalized and checked by a computer. They are
just different in nature.
Further details concerning the arguments one may use, are included in my paper.
Roughly speaking, a guiding idea is that since no given axiomatized theory can
establish more than a finite number of bits in Ω, we will at no point in time be able to
obtain nontrivial information concerning an infinite number of Ω bits.
Relation to previous work I
As far as I can tell, previous examples of undecidable sentences in mathematics seem
to fall into one of the following three categories:
1. Sentences shown to be undecidable in a given axiomatizable theory. Each of
these may be proved in another more powerful theory with the same intended
model as the original one. This is not the case with the examples constructed here.
2. Sentences which seem difficult to prove/refute. Examples include the continuum
hypothesis and so forth. These examples differ from the examples constructed in
this paper because no specific arguments are given to support the conclusion that
they are absolutely undecidable in our sense.
3. Sentences whose complexities indicate that we will never have enough time or
space to write down a proof/refutation of them. Examples include statements
about concrete decimals in Ω. If there is a natural number m such that the mth
decimal of Ω is “absolutely undecidable”, then there is also a smallest natural
number n such that the nth decimal of Ω is absolutely undecidable. This gives no
meaning. In contrast, the kind of undecidability we have encountered here,
represents undecidability in principle, regardless of limitations in space or time.
Relation to previous work II
If N is a given natural number, then our construction concerning the base 2 decimal
expansion of real numbers r is very close to questions like the following:
Is there an infinite number of sequences of N consecutive 1’s
in the base two decimal expansion of the number r?
To see how our situation differs from this, consider the example r = Ω. Suppose we
defined n as a winner iff the nth bit in Ω is 1, and the next (N-1) bits are also 1. Let
MWP be the statement that there is a maximal winner. Then if the bits in Ω form a
random sequence, as is widely believed, it follows that the MWP is false. [The
probability of N consecutive 1’s is 2^(-N). The experiment is repeated an infinite
number of times beyond any given point in the bit sequence, so by the law of large
numbers there will be a winner beyond any given point.] However, we do not obtain
any nontrivial information about an infinite number of Ω bits from this. Hence, our
undecidability argument does not work in such cases.