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Entanglement in Peano Arithmetic
Giulio Filippi
December 2021
1 Introduction
In 1931 Gödel showed the incompleteness of Peano Arithmetic (PA), crushing Hilbert’s dream of
a proof of the consistency of mathematics. The proof works by showing that statements about
numbers can also be statements about statements of PA (through Gödel numbering). In particular
Gödel showed there is a statement that asserts it’s own unprovability, resulting in the first unprov-
able truth. This shows that there are causal tangles in PA. At first mathematicians assumed they
were pathological cases and could be ignored, however Matiyasevich’s theorem ([2]) shows us this
is not the case by displaying undecidable Diophantine equations (that have solutions but cannot
prove it in PA). In this paper I generalise Gödel’s theorem, showing that there are causal tangles
involving links between arbitrary amounts of sentences.
2 Gödel’s Theorem
Note that a lot of the following exposition of Gödel’s theorem is inspired by the book Gödel Escher
Bach by Douglas Hofstadter [1].
The main idea needed to prove Gödel’s theorem is encodings. We devise a way of encoding
statements of PA into numbers (there are many ways). Similarly we can choose a way of encoding
proofs in PA into numbers (again there are many ways). Hence when we are speaking about
statements of PA, we might also be speaking about numbers, this will help us achieve circularity.
Let us define function PROOF(m, n) as true iff mis an encoding of a proof of statement n.
Lemma 2.1 (PROOF(m, n) is computable).
Proof. PROOF(m, n) is computable because a computer can decode m, decode nand check that
every line in the proof is correct and that the last line corresponds to the statement with Gödel
number n.
The next idea is the one that enables us to make a self referential statement. In literature, the
Quine of a sentence A is the new sentence where A is preceded by it’s quotation. For instance
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Quine(hello how are you?) is
"hello how are you?" hello how are you?
Consider the following : Quine(Is false when preceded by it’s quotation)
"Is false when preceded by it’s quotation"
Is false when preceded by it’s quotation
Here we have a sentence that speaks about itself without use of cheating such as using "I" or "this
sentence", etc. We will now devise a PA analogue of quining.
The number theory analogue for quining is as follows, take a sentence Lwith at least one free
variable in Peano arithmetic, let ube it’s Gödel number, insert uin the first free variable available
in L, get new sentence K. We call Kthe quine of L. We define the following QUINE(a, b)=True
iff aand bboth code for sentences in PA and the sentence of bis the quine of the sentence of a.
Then
Lemma 2.2 (QUINE(a, b) is computable).
Proof. We can imagine a computer decoding a, b and checking that the sentence of bis indeed the
quine of the sentence of a.
The next result allows us to tie the notion of computability to the notion of definability in PA.
We have shown PROOF and QUINE are computable but what does that have to do with PA?
Lemma 2.3 (Computable function is definable in PA).Any function Fon any number of inputs
that is computable by an algorithm (Turing machine or any other model of computation), will be
definable in PA through some statement AF. That is
PA ⊢ AF(x, y)↔F(x) = y
Proof. We omit the proof, it can be found in any sufficiently complete exposition on Gödel’s
theorem.
Now since PROOF and QUINE are computable, by Lemma 2.3 they are definable in PA. We
will take the sentences in PA that define PROOF and QUINE to have the same names as the
programs for simplicity. Now consider the following sentence of PA
U=¬ ∃ p, b PROOF(p, b)∧QUINE(a, b)
Now suppose this sentence has Gödel number u, quine this sentence (that is insert uin the place
of a) to produce
G=¬ ∃ p, b PROOF(p, b)∧QUINE(u, b)
Let gbe the Gödel number of G. Let us reason about the truth or falsehood of G. Suppose Gis
false. Since QUINE(u, b)is satisfiable (uniquely when b=g), we conclude ∃pPROOF(p, g).
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Then Ghas a proof, so under any reasonable truth predicate it must be true, contradicting our
claim. Hence we must conclude that Gis true, again QUINE(u, b)is satisfiable (uniquely when
b=g), so to have Gbe true we would need ¬ ∃ pPROOF(p, g)to be true. Hence Gis true but has
no proof. In a way that is precisely what Gis saying (when interpreted on a meta level) : "I am not
provable!".
3 Generalising Gödel’s theorem
The main result of this section is that we can "entangle" any number nof statements of PA into
a causal network where each sentence is able to specify definable properties of itself or others.
Gödel’s proof will then be a special case of this result. We start by defining an ENTANGLER
program on 2ninputs that does the following
ENTANGLER(u1, ..., un, a1, ..., an)=true iff
•u1, ..., uncode for sentences of PA each with nfree variables, call them U1, ..., Un
• If we insert u1, ..., unin the nfree variables of U1, ..., Un, we obtain new sentences of PA
A1, ..., An
• The codes for A1, ..., Anare exactly a1, ..., an
Lemma 3.1. ENTANGLER is computable
Proof. It is a tedious procedure but it is clear that every step of ENTANGLER can be executed by a
computer program. Hence by Lemma 2.1, ENTANGLER is definable in PA. To avoid introducing
more notation we will once again use ENTANGLER for the sentence in Peano arithmetic.
Now suppose we have nstatements of number theory that specify definable properties of each
other. Call them ψ1(a1, ..., an), ..., ψn(a1, ..., an). These could be anything as long as the properties
are definable! Consider the new statements
U1=∃a1, ..., anψ1(a1, ..., an)∧ENTANGLER(x1, ..., xn, a1, ..., an)
...
Un=∃a1, ..., anψn(a1, ..., an)∧ENTANGLER(x1, ..., xn, a1, ..., an)
Now let u1, ..., unbe the Gödel numbers of U1, ..., Unrespectively. Now consider the new sentences
A1, ..., Anproduced by inserting u1, ..., unfor the free variables x1, ..., xnin each of U1, ..., Un.
A1=∃a1, ..., anψ1(a1, ..., an)∧ENTANGLER(u1, ..., un, a1, ..., an)
...
An=∃a1, ..., anψn(a1, ..., an)∧ENTANGLER(u1, ..., un, a1, ..., an)
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What is the Gödel number of each Ai? It is precisely the value of aithat makes the ENTANGLER
true by construction. So each of A1, ..., Anis able to speak about itself and every other of the
sentences. What are they able to say about each other? They can say whatever they want as long
as they are using definable (or equivalently computable) properties through the use of the Gödel
numbers.
4 Examples
4.1 Gödel’s sentence
If we take n= 1 and ψ1(a1) = ¬ ∃ pPROOF(p, a1), then we recover Gödel’s sentence,
A1=∃a1¬ ∃ pPROOF(p, a1)∧QUINE(u, a1)
4.2 Two sentence liar’s paradox
Suppose we have Alice and Bob, Alice claims ’Bob is telling the truth’, Bob claims ’Alice is ly-
ing’. Then we get the two sentence liar’s paradox. We can find an analogue of this in PA.
Take n= 2,ψ1(a1, a2) = ∃pPROOF(p, a2),ψ2(a1, a2) = ¬ ∃ pPROOF(p, a1), then
A1=∃a1, a2∃pPROOF(p, a2)∧ENTANGLER(u1, u2, a1, a2)
A2=∃a1, a2¬ ∃ pPROOF(p, a1)∧ENTANGLER(u1, u2, a1, a2)
If A1is true, then A2has a proof so A2is true, so A1has no proof. So A1is true but has no proof.
If on the other hand A1is false, then A2has no proof, but also A1has no proof (otherwise it would
be true), so A2is true. So A2is true without proof. Hence in both cases we find a non-provable
true statement.
4.3 n sentence truth teller paradox
Take ψ1(a1, ..., an), ..., ψn(a1, ..., an)to all be "Nobody has a proof". If one of them is false, then
one of them must have a proof, but then he can prove that he has no proof, a contradiction. So all
of them must be true, but unprovable!
4.4 More complicated structures
We can specify more than just the provability of other sentences. Any definable property can be
used, and we can even use definable properties that link up other sentences together
ψ1(a1, ..., an) = PROPERTY(a1, ..., an)
For instance PROPERTY could be something like :
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1. A2, ..., Anare provable from A1and the traditional axioms of PA
2. A1, ..., Anall have different numbers of symbols
3. 0 = 1 (an obviously false statement can be useful in some networks)
4. A2, ..., Ancan be obtained from A1by rearranging the order of the symbols
The possibilities are endless. It is important to note that we are not interested it building true
statements here, it is the connections between these statements that are more interesting to study.
In particular the numerous paradoxes we can create in PA. Since every individual sentence (the
part) has access to specifications of the whole network (the whole), could this lead to interesting
implications regarding locality/non-locality?
5 Conclusion
We have gone over the proof of Gödel’s theorem and generalised it to tangle nstatements of PA.
My hope is that I can motivate the need for more metamathematical investigations. It is like we
have been closing one eye, focusing only on the parts of mathematics that behave classically in
a reductionist way, and have thus ignored the beauty that lies beyond the horizon of provability.
These tangled sentences are as much a part of PA as any other statements. Maybe long standing
conjectures like the Goldbach conjecture or the Riemann hypothesis are tangled in some ways and
might be unprovably true.
References
[1] Douglas R. Hofstadter. Godel, Escher, Bach: An Eternal Golden Braid. Basic Books, Inc.,
USA, 1979.
[2] Yu. V. Matiyasevich. Martin Davis and Hilbert’s tenth problem. In Eugenio G. Omodeo and Al-
berto Policriti, editors, Martin Davis on Computability, Computational Logic, and Mathemat-
ical Foundations, volume 10 of Outstanding Contributions to Logic, pages 35–54. Springer,
2016.
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