On the Fixed-Point Theorem of GL
Todd W. Schmid
University of Victoria
0. Abstract
Metamathematics is, to state the obvious, abstract and mysterious. And though it usually has many
applications to philosophy, it seldom has much to offer the working mathematician. This seems to be
especially true in the case of modal logic. That being said, a construction of a particularly interesting theory
of provability can be found within the scope of modal logic, to be presented in part in the following pages.
It is particularly interesting because of a theorem proved by Solovay in 1976, namely that the proof theory
we will present is capable of giving a full picture of provability in the famous formalization of arithmetic
offered by Russel and Whitehead. This is somewhat unexpected, as the modal provability logic happens to
be model-complete, which cannot be said about arithmetic. The material presented below does not focus
on this theorem, but instead aims to develop the necessary machinery to state and prove possibly the most
dazzling theorem from modal provability logic, the Fixed-Point Theorem. The Fixed-Point Theorem, when
equipped with the mentioned theorem of Solovay, has as corollaries Gödel’s Incompleteness theorems from
arithmetic, and at the same time raises many interesting questions about the nature of contradiction and
its relationship with incompleteness.
1. Informal Discussion of Formal Arithmetic, Background
In Principia Mathematica, an important series of books written by Russel and Whitehead, a famous formalization of arithmetic is given. Like many other installments of Hilbert’s Program, it allows us to express
most things we might like to say about a mathematical doctrine in a purely symbolic fashion – in the
case of Principia Mathematica, the theory of arithmetic. Russel and Whitehead’s formalization transforms
arithmetic deduction into symbolic manipulation, which has since become its own formal theory. As is only
natural when confronted with any sort of formalism, mathematicians have done much work on this ‘theory
of proofs’ that emerged. And in particular, about the existence of proofs for arithmetical results.
It was Kurt Gödel who pointed out that Russel and Whitehead’s arithmetic, which will from now on simply
be refered to as ‘arithmetic,’ can be turned on itself to formalize its own proof theory: every arithmetical
formula could be assigned a unique natural number, and each natural number assigned a unique formula
[1]. Note that the symbols we will take as primitive to our logic are ‘¬’ for negation, ‘→’ for implication,
and ‘⊥’ for contradiction. The other logical symbols, ‘∧,’ ‘∨,’ ‘↔,’ have their usual definitions.
Definition 1.1: If A is a formal sentence of arithmetic, we denote its associated integer with [A]. Similarly,
if n is a natural number, n̄ will be used to denote its arithmetical formula.
To reassure the reader, it is indeed the case that [n̄] = n. To perhaps scare the reader, this also insinuates that
proofs of arithmetical sentences correspond to series of symbolic manipulations, each of which representable
with series of arithmetical operations. Arithmetic, we have learned from Gödel, contains all the necessary
ingredients to encode and manipulate statements about itself!
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But let us not forget the original reason for our intrigue: the provability, or disporovability, of arithmetical
statements. We will from now on denote ‘the sentence A is provable in arthmetic’ with ‘`P M A’ (P M for
‘Principia Mathematica’). The question at hand is essentially the following: since we have determined that
proofs in arithmetic correspond to series of arithmetical operations, can we be assured that there exists
a unary predicate symbol with the free-variable n whose truth value agrees with the provability of n̄ and
whose provability is determined for each n ∈ N? The answer to this question is very clearly ‘no,’ as we will
soon see, but let that not discourage us from defining such a predicate and exploring the expressive power
it carries. Doing so will illuminate several very subtle facts about the nature of proofs in arithmetic and,
eventually, in other fundamental mathematics as well.
Definition 1.2: Let n be a natural number. We define a unary predicate symbol, Bew(n), to be read, ‘n̄
is provable in arithmetic.’
The problem that arises with definitively giving this predicate expressive power over any natural number is
due to arithmetic’s inherent ability to self-reference. Consider the following famous example (named after
Gödel) of a self-referential arithmetical sentence A such that `P M A ↔ ¬Bew([A]). If `P M A ↔ ¬Bew([A]),
what would that make the truth-value of `P M A? If you find yourself in a loop, you have done something
right!
Gödel’s sentence is, without a doubt, confounding. The proof of it, found in his 1931 paper, ‘On Formally
Undecidable Propositions in Pricipia Mathematica and Related Systems,’ is even more so. But as it turns
out, it is inherent to formal arithmetic (even with the addition of our predicate Bew) that any unary
predicate P (n) has at least one sentence A for which `P M A ↔ P ([A]). Carnap ws the first to make this
observation in Logical Syntax of Language [2].
For each unary predicate P (n) we will have a special name for such a sentence A, as it is the intention of
this document to make a similar observation to that of Carnap’s within an extension of the minimal normal
modal logic we will call Provability Logic. This new logic will include as axioms the analogues of some
important properties of Bew in the hope that they will form a believable theory of provability in a general
context.
2. Provability Logic from K
What we will call the minimal normal modal logic, or K, takes as its primitive modality the symbol ‘,’
and extends the formulae of propositional logic in the obvious way. K has as its valid sentences
a) the tautologies from propositional logic (wherever does not interfere it is included here),
b) any sentence of the form (A → B) when (A → B) is also a tautology (distribution),
c) any sentence of the form A when A is also a tautology (necessitation),
and any sentence which is inferred from the above via modus ponens. Traditionally in K, p reads, ‘it is
necessary that p,’ though we are about to abandon this interpretation of in favour of another.
Our preferred interpretation is ‘it is provable that p.’ Now, there are three properties of Bew noticed by
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20th century mathematicians that we will need to enjoy should it provide a model of provability (in
arithmetic, at the very least). Letting A be any arithmetical sentence, these properties go as follows:
a) If ` A, then ` Bew([A]).
b) If ` Bew([A → B]), then ` Bew([A]) → Bew([B]).
c) (Lös’ Theorem) If ` Bew([A]) → A, then ` A.
The first two properties already have their analogues in K, so the desired extension needs only the third
property.
Definition 2.1: The modal Provability Logic, abbreviated GL, extends K by including A as a tautology
whenever A → A is, and closing itself under modus ponens.
The act of showing that a sentence is a tautology in a given logic much resembles that of any other proof.
For this reason, to denote that A is a tautology of GL we write `GL A, or ` A whenever the context is
clear. For our purposes, tautologies will simply be called ‘valid.’
An auxilliary concept we will need for this discussion is that of a model. Models are intended to be somewhat
concrete examples of the structure suggested by the inner workings of a logic. As one might expect, an
example of a model of GL that has already been discussed is that of provability within arithmetic. The
specifics of the modelling process for arithmetic are cumbersome, and won’t be discussed at length here.
However, it has been clear since the 1970’s that, at the very least, each statement of arithmetic may be
realized as a statement of GL, and provability is sound under such a realization [5]. Solovay proved the
much more dubious converse statement in 1976 [3]. More precisely, there is a function f from the sentences
of GL to the sentences of arithmetic such that the following proposition is true.
Proposition 2.2: Let A be a sentence in GL. `GL A if and only if `P A f (A).
The concept of realization will be made more precise in the concluding section, and its philosophical relevance
hinted at simultaneously.
It is to be expected at this point that the axiom from definition 2.1 (named after Martin Löb) gives us
exactly what we should want out of any model of our Provability Logic*: no statement is a proof of itself;
proofs are always finite in length; and if A is a viable proof of B, and B a viable proof of C, then A is
also viable proof of C (though in practice less detailed). In order to show that GL has only models of this
variety, we will need the following technical results, and to make precise the conception of a model of GL.
Proposition 2.3:
a) `GL (A ∧ B) ↔ (A ∧ B) (conjunctive distribution).
b) `GL A → A (the K4 axiom).
Proof. For the first direction of part ‘a),’ notice that ` A ∧ B → A and ` A ∧ B → B. Necessitation yields
` (A ∧ B → A) and ` (A ∧ B → B), and distribution ` (A ∧ B) → A and ` (A ∧ B) → B. Hence
` (A ∧ B) → A ∧ B.
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The opposite direction of ‘a)’ is similar. Notice first that ` B → (A → (A ∧ B)), and that necessitation and
distribution give ` B → (A → (A ∧ B)). Distributing again, ` B → (A → (A ∧ B)), and hence
` A ∧ B → (A ∧ B).
The deduction of part ‘b)’ goes as follows:
(tautology) ` (A ∧ A ∧ A) → (A ∧ A)
(logical equivalence) ` A → ((A ∧ A) → (A ∧ A))
(1)
(part ‘a)’) ` A → ((A ∧ A) → (A ∧ A))
(necessitation & distribution) ` A → ((A ∧ A) → (A ∧ A)).
Recall that Löb’s axiom states that ` (B → B) → B. Hence, in replacing replacing B with A ∧ A,
we learn that
(Löb’s axiom) ` ((A ∧ A) → (A ∧ A)) → (A ∧ A)
(2)
(part ‘a)’) ` ((A ∧ A) → (A ∧ A)) → A ∧ A
By modes ponens, combining (1) and (2) gives us ` A → (A ∧ A). This provides us with the desired
result.
Definition 2.4: A frame consists of a set W called its universe, and a binary relation R ⊆ W × W called
its accessibility relation. When a frame is accompanied by a function V from the class of sentence letters
to the powerset of W , called its valuation, it is called a model. Formally, a frame is denoted by an ordered
pair, F = (W, R). The same goes for a model, M = (F, V ).
Given a particular point w ∈ W , we say that v is accessible to w if (w, v) ∈ R, more conveniently written
wRv. Defining the set acc(w) to be the set of worlds accessible to w provides some notational convenience,
as well as writing
A instead of A ∧ A. A point w ∈ W is called an endpoint if acc(w) = ∅.
Definition 2.5: Let M = ((W, R), V ), p be a sentence letter, and w ∈ W . The symbolic phrase ‘M, w |=_’
is used to denote ‘in the model M, w believes _ to be true.’ We formally define M, w |=_ inductively as
follows:
M, w |= p iff w ∈ V (p)
M, w |= A ∧ B iff M, w |= A and M, w |= B
M, w |= ¬A iff it is not the case that M, w |= A
M, w |= A iff for any v ∈ acc(w), M, v |= A
A model of GL is one which considers each sentence which is valid in GL to be true, regardless of perspective
or valuation. For convenience’s sake, the sentence (x0 Rx1 ∧ x1 Rx2 ∧ ... ∧ xn−1 Rxn ) is replaced with the
simpler x0 Rx1 Rx2 R...Rxn and called a chain. Infinite chains are denoted without a tailing ‘Rxn .’ We are
now ready to verify that each model of GL indeed satisfies the desired conditions*.
Theorem 2.6: Let F := (W, R) be a frame. Then for any model M = (F, V ) of GL, R is irreflexive and
transitive, and there are no infinite chains x0 Rx1 Rx2 R..., where {xi : i ∈ N} ⊆ W .
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Proof. The transitivity of R follows directly from the K4 axiom (Proposition 2.2b): if we let x, y, z ∈ W , p
be a sentence letter, V be a valuation for M such that V (p) = W \ {z}, xRyRz, and ¬xRz, it is true that
M, x |= p, but not that M, x |= p: there is a world accessible to y which does not believe p. This
contradicts the mentioned proposition, and thus cannot be the case.
We will also show that there are no infinite chains via a contradiction. Let xi ∈ W for each i ∈ N,
V (p) = W \ {x0 , x1 , ...}, and x0 Rx1 Rx2 R.... This chain guarantees us that there are no endpoints amongst
the xi , so given our valuation of p, M, xi |= ¬p for each i ∈ N. Therefore, by vaccuity, M, xi |= p → p
for each i ∈ N. Now, for each y ∈ W \ {x0 , x1 , ...}, M, y |= p, so we know that M, x0 |= (p → p). But
M, x0 |= ¬p, which contradicts Löb’s axiom (Definition 2.1).
Again, to show the irreflexivity of R, we produce a contradiction. Let x ∈ W , wRw, and V (p) := W − {x}.
Since x is not itself in V (p), if follows not only that M, x |= ¬p, but also that M, x |= ¬p. Thus,
M, x |= p → p. As well, since for each w ∈ W − {x} we have M, w |= p, M, v |= p → p for every v ∈ W .
Hence, M, x |= (p → p). But M, x |= ¬p, which contradicts Löb’s Axiom.
The irreflexivity of R assures us that in no model of GL does a sentence provide a proof for itself, the
transitivity of R permits less detailed proofs, and the lack of infinite chains allows for only finite proofs.
Corollary 2.6.1: Let M = ((W, R), V ) be a model of GL, and w ∈ W . Then if acc(w) 6= ∅, acc(w)
contains an endpoint.
Proof.
If there were no endpoints in a given acc(v), our lack of infinite chains would tell us there were a
cycle x0 Rx1 R...Rxn Rx0 in acc(w). However, no such cycle can exist since transitivity would imply x0 Rx0 .
Given the above details, there are two last things we will need to know about models of GL.
Proposition 2.7: Let M = (W, R, V ) be a model of GL, w ∈ W , and A be a sentence. If M, w |= A,
then for any x ∈ acc(w), M, x |=
A.
This I welcome the reader to verify for themself, as it follows directly from the involved definitions.
Proposition 2.8: Let M = (W, R, V ) be a model of GL. For any sentence A, and w ∈ W for which
M, w |= ¬A, there is an x ∈ acc(w) such that M, x |= A ∧ ¬A.
Proof.
First note that there is a y ∈ acc(w) for which M, y |= ¬A, and that this means acc(w) 6= ∅. By
Corollary 2.6.1, acc(w) contains an endpoint, v. Since acc(v) = ∅, M, v |= A vaccuously. Thus, there is
a maximal chain x0 Rx1 R...Rxn in acc(w), where xn = v, such that for 0 ≤ i ≤ n, M, xi |= A. x0 is the
desired choice for x by maximality.
We end this section with the mention of a theorem whose proof the reader can find in [2]. It is a necessary
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fact to keep in mind when deriving theorems of GL from arbitrary models.
Theorem 2.9:(The Completeness Theorem of GL) For any sentence A, `GL A if and only for any model
M = (W, R, V ) of GL and w ∈ W , M, w |= A.
3. The Fixed-Point Theorem and a Syntactic Proof
As we saw within arithmetic, the ability to denote formulae with unique natural numbers allows for arithmetical sentences to reference other arithmetical sentences – including themselves. In our newly developed
proof logic, we will fortunately be able to avoid transfering the burden of representation to an external
collection of objects and instead focus our attention on the syntax of the sentences themselves. We will
need a short discussion of substitution, and then a few final definitions to keep in mind, in order to conclude.
Definition 3.1: Let A be a sentence containing the sentence letter p, and B, C and D be sentences. We
define the sentence Ap (D) inductively as follows:
if A is p, then Ap (D) is D
if A is B ∧ C, then Ap (D) is Bp (D) ∧ Cp (D)
(3)
if A is ¬B, then Ap (D) is ¬Bp (D)
if A is B, then Ap (D) is Bp (D)
It is advised to read Ap (D) as ‘the sentence A where each instance of p is replaced with D.’ Beware that I
will sometimes write (Ai )p (D), where i is an indexing element. For example, (C1 )p (D) is the sentence C1
with each instance of p replaced by the sentence D.
We will not be using this definition very often in the next few pages, but instead using a similar definition to
do simultaneous substitutions. This is denoted with a sequence of predicates p1 , ..., pn found in A, alongside
a sequence of sentences D1 , ..., Dn , and written Ap1 ...pn (D1 , ...Dn ). The inductive definition of simultaneous
substitution one might imagine is nearly identical. The following three comments about the given notation
are worth keeping in mind: Ap (D) is well-defined; (¬A)p (D) is ¬(A)p (D), (A∧B)p (D) is (A)p (D)∧(B)p (D),
and (A)p (D) is (A)p (D); (Ap1 (D1 ))p2 (D2 ) is not always the same as Ap1 p2 (D1 , D2 ).
The following result is what one would expect of substitution.
Proposition 3.2: (The Substitution Lemma) Let A and B be sentences containing the predicate p, and C
any sentence. Then `
(A ↔ B) → (Ap (C) ↔ Bp (C)).
The proof of this is done via an induction on the complexity of A, the case being the only case of genuine
interest. Setting up an arbitrary model, appealing to proposition 2.6, and appealing to completeness should
do the trick.
The term subsentence found below has the obvious inductive definition: B is a subsentence of A if it is
well-formed sentence appearing within A. We now direct our attention to the aforementioned definitions
necessary to continue this discussion.
Definition 3.3: Let A be a sentence containing the sentence letter p. We say that A is modalized in p,
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or p-modalized, if for some subsentence B of A also containing p, there is a subsentence C of A which
contains B as a subsentence. In other words, every instance of p in A appears within the scope of a .
Definition 3.4: We say that A is n-decomposable if for some sentence B containing the predicates q1 , .., qn
distinct from each other and another predicate p, there are sentences C1 , ..., Cn containing p such that A is
Bq1 ...qn (C1 , ..., Cn ).
We are now ready to state the fixed point theorem, and make our way through its proof.
Theorem 3.5: (The Fixed-Point Theorem) Let A be a sentence modalized in p. There exists a sentence F
which contains only the predicates distinct from p which are found in A such that
(p ↔ F ) ↔
(p ↔ A).
Such a sentence F we call a fixed-point of A, for reasons made more clear in the proof. The fixed point
theorem applies only to p-modalized sentences, and clearly, every p-modalized sentence is n-decomposable
for some n. This fact plays directly into the proof of the fixed point theorem: essentially, we will construct
a fixed point for an arbitrary n-decomposable sentence via induction, and then show it is in fact a fixed
point.
Proof.
Our base case is rather simple. Suppose that A is a 0-decomposable sentence. Then A does not
contain p whatesoever, and hence is a suitable F .
For the inductive hypothesis, suppose that n-decomposable sentences have fixed-points, that A is an (n + 1)decomposable sentence, and A in its decomposed form is
Bq1 ...qn+1 (C1 , ..., Cn+1 ).
Note that each Ci contains the sentence letter p. We define for each i ∈ {1, ..., n + 1} the sentence Ai , which
takes the form
Bq1 ...qn (C1 , ..., Ci−1 , ¬ ⊥, Ci+1 , ..., Cn+1 ).
Clearly, since our decomposition of each Ai was explicit, Ai is n-decomposable. This assures us of a fixedpoint for each Ai , which I’ll denote with Fi . Now, we define a new sentence F which is
Bq1 ...qn+1 ((C1 )p (F1 ), ...(Cn+1 )p (Fn+1 )).
Talk about meta! The thing to notice here about our choice of each Di is the seemingly out-of-place ‘¬ ⊥,’
which is a tautology in K. This substitution hands us the following fact, which is necessary in completing
our proof of the theorem.
Lemma 3.5.1: For each i,
Proof.
(p ↔ A) →
(Ci ↔ (Ci )p (Fi )).
Let M = (W, R, V ) be a model, and w ∈ W . Suppose that M, w |=
(p ↔ A). Of course, it
suffices to show that for any y ∈ {w} ∪ acc(w), M, y |= Ci ↔ (Ci )p (Fi ).
To prove that M, y |= Ci → (Ci )p (Fi ), we suppose M, y |= Ci . Then for any x ∈ acc(y), M, x |= Ci
as well. Pedantically, M, x and M, y |= Ci ↔ (¬ ⊥), so that M, w |=
(Ci ↔ (¬ ⊥)). But a little
pedanticism every now and then doesn’t hurt – especially if you’re a logician and have the substitution
lemma on your side! In such a case, we have
M, y |=
(Ci ↔ (¬ ⊥)) → (Bq1 ...qn+1 (C1 , ..., Ci , ..., Cn+1 )) ↔ (Bq1 ...qn+1 (C1 , ..., (¬ ⊥), ..., Cn+1 ),
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from which we may conclude that M, y |= Ai ↔ A. Since y was chosen arbitrarily, M, w |= (A ↔ Ai ),
and
M, w |=
Going back to our initial supposition that M, y |=
(A ↔ Ai ).
(p ↔ A), and combining it with the above expression,
M, y |=
(p ↔ Ai ).
The inductive hypothesis yielded us a fixed-point of Ai , recall – namely Fi . Combining this fact with the
above expression, M, y |= p ↔ Fi , and hence M, w |=
(p ↔ Fi ). Using the substitution lemma once more,
M, w |= Ci ↔ (Ci )p (Fi ), and so by necessitation M, w |= Ci ↔ (Ci )p (Fi )). This implies the desired
result.
To show that M, y |= Ci ← (Ci )p (Fi ), we assume M, y |= ¬Ci and show the contrapositive. In such a
case, proposition 2.7 guaruntees us that for some x ∈ acc(y), M, x |= Ci even though M, x |= ¬Ci . Since
x ∈ {w} ∪ acc(w), we know now from the first direction of the proof that M, x |= Ci ↔ (Ci )p (Fi ), and
so it must be the case that M, x |= ¬(Ci )p (Fi ). This gives us M, y |= ¬(Ci )p (Fi ), which completes our
proof of the other direction.
Only a small bit of discussion within an arbitrary model is needed to finish this proof. Theorem 2.9 is
also used throughout this paragraph. Let M = (W, R, V ) be a model of GL, and w ∈ W such that
M, w |=
(p ↔ A). From our lemma 3.5.1, M, w |= (Ci ↔ (Ci )p (Fi )). The substitution lemma, used
to its full extent, now tells us that
M, w |= Bq1 ...qn+1 (C1 , ..., Cn+1 ) ↔ Bq1 ...qn+1 ((C1 )p (F1 ), ..., (Cn+1 )p (Fn+1 )).
That is to say that M, w |= A ↔ F . Now, since M, w |= p ↔ A, we have M, w |= p ↔ F . Due to our
initial assumption, M, w |=
(p ↔ A) → (p ↔ F ). By the arbitrary choice of M and w, and Theorem 2.9,
(p ↔ A) → (p ↔ F ) is valid in GL.
4. Conclusion
There are numerous proofs of the fixed-point theorem, each with its own merits. However, there is something
rather mystifying about the proof given, due in part to the metalogical nature of the algorithm we used
to compute fixed-points: if A is a sentence modalized in p, and D is the fixed-point of A generated by the
algorithm above, D will ‘closely resemble’ A. What’s meant by ‘closely resemble’ is merely that they will
be similar looking sentences.
As an example, let us consider the p-modalized sentence A, which we will take to be ¬p. In the notation of
the algorithm given, we have C1 = ¬p and B = q, since A = Bq ((C1 )) = ¬p. Now, A1 = Bq (¬ ⊥) = ¬ ⊥,
which is 0-modalized in p, so the fixed-point D1 of A1 is simply ¬ ⊥. We now have the fixed-point of A:
D = Bq ((C1 )p (D1 )) = Bq ((C1 )p (¬ ⊥)) = Bq (¬¬ ⊥) = ¬(¬ ⊥). Parentheses have been placed to
emphasize the similarity between A and D. Nevertheless, the resemblance is clear.
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On a separate note, let us consider a similar example of a p-modalized sentence which is relevant not only
to the initial discussion, but also to the brief philosophical discussion below. Let A = ¬p. Keeping with
the notation of the proof, we have C1 = p and B = ¬q, so that A = Bq (C1 ) = ¬p. This gives us
A1 = Bq (¬ ⊥) = ¬¬ ⊥, which is logically equivalent to ⊥. D1 may be ⊥, for A1 is 0-modalized in p.
D = Bq ((C1 )p (D1 )) = ¬(C1 )p (⊥) = ¬ ⊥. Thus, by the rest of the fixed-point theorem,
`GL ¬ ⊥↔ ¬¬ ⊥ .
(4)
In order to clarify the relationship with the initial discussion, we would like to make more precise the concept
of representation mentioned in the first section.
Definition 4.1: A map f from the sentences of GL to those of arithmetic is called a representation if
a) f (⊥) =⊥
b) where A and B are sentences of GL, f (A → B) = f (A) → f (B)
c) f (¬A) = ¬f (A)
d) f (A) = Bew([f (A)]).
Intuitively, a representation simply amounts to a translation from the language of GL into the language of
arithmetic. Now Theorem 2.2 tells us something rather special about (4), for it provides an explicit solution
to the logical equation A ↔ ¬Bew([A]) - a Gödel sentence!
Theorem 4.2: (Gödel’s Second Incompleteness Theorem) `P M ¬ ⊥↔ ¬¬ ⊥.
Proof. We have already shown `GL ¬ ⊥↔ ¬¬ ⊥ . Let f be a representation. Then from
`GL ¬ ⊥↔ ¬¬ ⊥ iff `P M f (¬ ⊥↔ ¬¬ ⊥)
iff `P M ¬f ( ⊥) ↔ ¬f (¬ ⊥)
(5)
iff `P M ¬Bew([f (⊥)]) ↔ ¬Bew([f (¬ ⊥)])
iff `P M ¬Bew([⊥]) ↔ ¬Bew([¬Bew([⊥])]),
the result follows from equivalence.
The fastidious reader will notice that the Gödel sentence above is of a particular character. The theorem
reads ‘one cannot prove a contradiction within arithmetic if and only if one cannot prove that they cannot
prove a contradiction within arithmetic.’ We have have found a concrete example of a sentence within
arithmetic which cannot be proven or disproven, which is not to mention it is at the very same time a hefty
statement about the consistency of fundamental mathematics. Essentially, arithmetic cannot prove its own
consistency, for this would require you to violate the lack of ability, found in Theorem 4.2, to prove that
your proof is correct.
Which brings us to our final observation. Theorem 2.8 insinuates that GL is, in fact, a decidable formal
theory: for each GL sentence A, either `GL A or else `GL ¬A. This is due to our ability to delegate the
determination of provability to the model-theoretic tools found in section 2. This is a stark difference between
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GL and arithmetic. So stark, even, that it seems almost paradoxical that on the one hand arithmetic contains
examples of undecidable sentences, and on the other, by Theorem 2.2, GL appears to be a completely
accurate model of decision within arithmetic. Thusly, GL provides us with an impeccable example of the
power of educated simplification – the act of breaking an unmanageable bit of machinery down to the
workings of a simple set of subroutines to learn about the behaviour of the system as a whole.
I would like to thank Dr. Audrey Yap from the University of Victoria for her insight and direction, Adrian
Yee for inspiration, and David Douglas, to whom this piece was originally aimed to inform and entertain.
References
[1] K. Gödel, On Formally Undecidable Propositions of Principia Mathematica And Related Systems, Monatshefte für Mathematik, 1931
[2] G. Boolos, The Logic of Provability, Cambridge University Press, New York, 1993
[3] P. Henk, A new perspective on the arithmetical completeness of GL, University of Amsterdam, 2011
Link: https://www.illc.uva.nl/Research/Publications/Reports/X-2011-06.text.pdf
[4] L. Reidhaar-Olson, A New Proof of the Fixed-Point Theorem of Provability Logic, Notre Dame Journal
of Formal Logic, 1990
Link: http://projecteuclid.org/download/pdf1 /euclid.ndjf l/1093635331
[5] Verbrugge, Rineke (L.C.), Provability Logic, Stanford Encyclopedia of Philosophy, 2016
Link: http://plato.stanford.edu/entries/logic-provability/
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