arXiv:2005.10080v12 [math.LO] 22 Sep 2021

APROOF

OF

P6=NP

RUPERT

McCALLUM

1

2 McCALLUM

Abstract. This work will show that P 6= NP is provable in PA

and indeed even in EFA. The opening section gives acknowledge-ments and the second section makes some introductory remarks.An outline of the overall proof-strategy is given in the third section.

A lemma stated in the third section and proved in the fourth sec-tion, serving as a lemma for P 6= NP , outlines a construction ofan arithmetically definable sequence φn : n ∈ ω of Π0

2-sentences,

which satisfy a number of properties and of which we can say,speaking roughly, that the growth in logical strength is “as fast aspossible”, manifested in the fact that the total general recursivefunctions whose totality is asserted by the true Π0

2-sentences in the

sequence are cofinal growth-rate-wise in the set of all total generalrecursive functions.

We then develop an argument which makes use of a sequence ofsentences constructed from the preceding sequence by means ofa certain somewhat involved construction, together with an ap-plication of the diagonal lemma. These sentences in this secondsequence are inspired by Hugh Woodin’s “Tower of Hanoi” con-struction as outlined in his essay [1]. The argument establishes theresult that it is provable in PA that P 6= NP (careful examinationof this proof seems to show that Σ0

3-induction arithmetic is suffi-

cient). The fifth section shows how to establish the stronger claimthat the result P 6= NP is actually provable in EFA.

In the sixth and final section of this work, we obtain a general-ization of the key lemma mentioned above to the context of thelanguage of set theory.

A PROOF OF P6=NP 3

1. Acknowledgements

I am very thankful to Henry Towsner for taking some care and patienceto critique flawed earlier versions of parts of this argument, from whichSections 3 and 4 benefitted substantially. I’m also very thankful toBenoit Monin who very kindly closely examined the argument andpointed out important corrections to the statement of requirement 3 inProposition 3.5. Scott Aaronson provided crucial assistance in pointingout that any attempted proof of P 6= NP must survive the basic “sanitycheck” of being consistent with the known fact that SAT is solvable inO((1.3)n). And the very great debt of this work to Hugh Woodin’s“Tower of Hanoi” construction will be apparent.

2. Introduction

In what follows, we will present an argument establishing that P 6=NP is provable in first-order PA, using strategies based on proof-theoretic reflection and a diagonal-lemma construction as clarified inthe abstract. The argument in the most natural form is highly non-constructive and, in its most natural form, seems to require Σ0

3-inductionarithmetic, but with some care it appears to be to possible to pull theargument down into EFA, as will be clarified in the fifth section. Thefollowing section outlines the general shape of the proof and the fourthsection presents the proof of the main result while the fifth section ex-plains how to get the argument to go through in EFA. The final sectiondiscusses a generalization of the first lemma of Section 3, as discussedin the abstract, to the context of the language of set theory.

3. Outline of the argument

Our main result of Section 4 shall be that PA proves P 6= NP . Wegive an outline of the overall shape of the proof strategy. We beginby stating a lemma to the effect that a certain class of decision prob-lems are all in NP . We shall be concerned with decision problemsabout formal provability in theories in a first-order language of finitesignature (except for possibly having infinitely many constant symbols)which belong to a certain class. Since we shall want to speak of suchdecision problems being in NP , we shall need to organize things sothat the alphabet of the language is finite, so therefore we shall saythat the variables are of the form x, x′, x′′, . . . where x and ′ are twosymbols of the language, and further that in the case where there areinfinitely many constant symbols, we shall require that each of thesebe specifiable from some fixed finite alphabet.

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Lemma 3.1. Consider the class of all theories T with a fixed recursiveaxiomatization in a fixed first-order language of finite signature – exceptfor possibly having infinitely many constant symbols – with the alphabetof the language being finite in the manner described above, and withthe possibility of constant symbols of length greater than one allowedfor as described above. We assume that the characteristic functionof the axiom set of the theory T is polynomial-time computable anda fixed presentation of the polynomial-time algorithm together with apolynomial bound on running time is given (however these items are notpart of the data of our decision problem, each special case of the decisionproblem being entirely relativized to T for the purpose of evaluatingits computational complexity). For all theories in the class described,consider the following decision problem. We are given a finite list ofaxioms to add to T and a positive integer n given in unary notation, anda particular sentence φ in the language of T , and the decision problemis to determine whether or not there exists a proof in T plus the givenlist of axioms, of length at most n, of the sentence φ. This decisionproblem, denoted in what follows by PROVABILITYT , is in NP .

Proof. It is sufficient to show that the decision problem for determiningwhether a given string of length at most n is a proof in T plus a fixedfinite list of axioms can be solved in a number of steps which is apolynomial function of n. Recall that we are assuming that the axiomset of T is polynomial-time computable. It easily follows that the axiomset of T with the fixed finite list of axioms adjoined is also polynomial-time computable. Now, it is well-known that it is polynomial-timedecidable whether or not a given string is proof in classical first-orderlogic from a fixed set of axioms which is polynomial-time decidable.

We shall need to make reference to a modified version of PRA whichwe shall now define.

In order to define our modified version of PRA, we shall need to citeBoolos’ work on quotational ambiguity in [7]. Let us remind ourselvesof the devices for quotation introduced in the language Lgood describedin that paper.

Definition 3.2. A q-mark is an expression j where j ≥ 0 and jis the expression consisting of j consecutive occurrences of ′. For anyexpression α, we define m(α), the q-mark of α, to be the shortest q-mark that is not a subexpression of α. The quotation of α, denoted byr(α), is defined to be m(α) ∗ α ∗m(α) where ∗ denotes concatenation.

A PROOF OF P6=NP 5

Definition 3.3. In what follows, we define a modified version of PRA,which by slight abuse of notation shall also hereafter be denoted byPRA, which shall be an example of a theory which can feature in theclass of decision problems described in Definition 3.1. We begin witha polynomial-time-computable version of PRA as a theory in the first-order language of arithmetic with signature (0, S,+, ∗, <) which is de-ductively closed in classical first-order logic, where as in Definition 3.1the language of the theory has a fixed finite alphabet. Let us denotethat language by L0 and the theory just described by T0, this is clearlyan example of a theory which can feature in the class of problems de-scribed in Definition 3.1. Now let us describe an extended theory T1 inan extended language L1. Given any sentence φ of L0 with length n

(containing no q-marks) there is a constant symbol cφ of L1 of lengthn+ 2 which appears as the expression r(φ), as given in Definition 3.2,and an axiom cφ = n is given for the theory T1, where n is a numeralfor the Godel number of φ under some fixed Godel coding scheme, withthe length of the numeral n being proportional to the logarithm of theGodel number for which it is a numeral. The theory T1 is also an ex-ample of a theory which can feature in the class of problems describedin Definition 3.1. Then we can construct an extended theory T2 in anextended language L2 in a similar way, except that now given a sen-tence φ the corresponding constant symbol will have length n+4, sincethe sentence φ may have q-marks of length at most 1, so that q-marksof length 2 will be needed to do the quotation. And so on. In thisway we construct an infinite ascending chain of theories Tn : n ∈ ω.The union of all such theories, which we denote by T∞, is also an ex-ample of a theory which can feature in the class of decision problemsdescribed in Definition 3.1. The language of all arithmetical theoriesshall hence-forward be assumed to be the underlying language of thistheory.

We must now state a lemma which will be proved at the start of Section4. In addition to giving the proof of this lemma, three more steps arerequired for the proof of the main result of Section 4, that P 6= NP

is provable in PA. We shall enumerate these three steps and indicatehow successful completion of these three steps would lead to the finalconclusion. Then Section 4 will be occupied with proving the firstlemma of this section and with completing the three steps in question.

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Lemma 3.4. Assume the axioms of PA. On those assumptions, it isprovable that there exists an arithmetically definable sequence of Π0

2-sentences φn : n ∈ ω in the language of PRA as defined in Definition3.3 with the property that

(1) No q-marks of length greater than n occur in φn (and we start withn = 0).(2) There exists a polynomial function with non-negative integer co-efficients such that the length in symbols of φn is eventually alwaysbounded above by the value of this polynomial function at n;(3) PRA+φn : n ∈ ω is Π0

2-sound and Π01-complete

(4) PRA+φn+1 always proves 1-consistency of PRA+φn, and the lengthof the shortest proof as a function of n is bounded above by a polyno-mial function with non-negative integer coefficients;(5) The provably total general recursive functions of PRA+φn : n ∈ ωhave growth rates cofinal in the set of growth rates of all the total gen-eral recursive functions.

That completes the statement of the first lemma of this section which isto be proved at the start of Section 4. We must now describe the threeremaining steps which are required for the proof of the main result ofSection 4.

Let us begin by describing Step 1. The completion of Step 1 amountsto proving a certain set of propositions, stated below.

Let us suppose that φn : n ∈ ω is the sequence of Π02-sentences of

Lemma 3.4, and let us consider a particular polynomial function p(n)with non-negative integer coefficients, and for each natural number n,let us define Sn to be the set of Σ0

1-formulas of length at most p(n),with exactly two free variables, which define PRA+φn+1-provably totalgeneral recursive functions. Let us suppose that we are entertainingthe hypothesis P = NP , with a view towards deriving a contradiction.Let us suppose that m is an integer such that m ≥ 2.

Proposition 3.5. Suppose that we are assuming P = NP for a contra-diction, and let us suppose that we are considering a particular polyno-mial function p(n) with non-negative integer coefficients. Suppose thatfor sufficiently large n the theory PRA + φn does prove a Π0

1-sentence(which can be chosen to work uniformly for all n, naturally) whichwitnesses P = NP , in the sense of identifying a particular polynomial-time solver for an NP -complete problem – we could take the NP -complete problem SAT as our example of an NP -complete problem,

A PROOF OF P6=NP 7

to be specific – which is always guaranteed to halt and give the correctanswer in a time bounded above by a particular polynomial bound withexponent m. This can naturally be transformed into a polynomial-timesolver of the decision problem PROVABILITYT as described in Lemma3.1 with T = PRA. Suppose that the set of formulas Sn be defined asimmediately above.

Then, when n is sufficiently large that the foregoing hypothesis onPRA+φn holds, there will also always exist at least one formula fn ∈ Sn

satisfying all of the following.

(1) It is PRA+φn+1-provable that the following holds for all k: there isa PRA+ φn-proof of the halting of a Turing machine Tn,k which is setup so that given empty input it will eventually halt and output fn(k)(as expressed in unary notation).(2) The following is PRA+ φn+1-provable. Denote by rn(k) the lengthof the shortest proof from PRA + φn of the halting of the Turing ma-chine Tn,k. We require it to be PRA + φn+1-provable that it holds forall sufficiently large k that the actual halting time of Tn,k lies between(rn(k))

k and (rn(k))2k.

(3) Denote by hn(k) the halting time of the Turing machine Tn,k andconsider the set of all Turing machines T ′ with halting time h′ ≤ hn(k)when given empty input, with the property that, if e denotes the in-dex of Tn,k and e′ denotes the index of T ′, then e′ ≤ Keβ, for fixedpositive integers K and β, to be chosen later on in Section 4 duringthe exposition of the main argument, and which will have to be cho-sen so as to enable the requirements of a model-theoretic argument forthe second of our three steps to go through. It is required to alwaysbe PRA + φn+1-provable that if T ′ is a Turing machine from this setthen a PRA + φn-proof of halting of T ′ given empty input does indeedexist, and in what follows we shall denote by t′ the shortest length insymbols of such a proof for a given T ′ in the set. We require it tobe PRA + φn+1-provable that, for all sufficiently large k, (the relevantlower bound for “sufficiently large” being the same as in (2)), on theassumption that T ′ is in the set described previously and t′ is definedas above, then (t′)2k ≥ h′, where h′ is the halting time of T ′ on emptyinput.

In the discussion above we were working with a sequence of sentencesφn : n ∈ ω which are assumed to be true in the standard model. Butthe above discussion is easy to generalise to any pair of sentences 〈φ, φ′〉with the property that PRA+ φ′ proves the 1-consistency of PRA + φ

and PRA + φ proves a Π01-sentence witnessing the truth of P = NP

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in the sense outlined in the statement of Proposition 3.5. Given anysuch pair, we want to claim that even then, without assuming that thesentences are true in the standard model or that the Turing machinesinvolved actually halt “in the real world”, an appropriate variant ofProposition 3.5 can still be proved, and that the defining formula forfn – where naturally we no longer make the assumption that there isa total general recursive function fn, only a defining formula which isclaimed by the appropriate extension of PRA to define a total generalrecursive function – will be a polynomial-time computable function ofφ, φ′, the shortest proof from φ of a Π0

1-sentence witnessing P = NP

in the foregoing sense, and the shortest proof from PRA + φ′ of 1-consistency of PRA+φ. Proposition 3.5 should be understood to coverthis case as well, and its proof, together with the proof of a Propositionto appear shortly below, is the first of the three required steps besidethe proof of Lemma 3.4 for the proof of the main result.

Proposition 3.6. The variant of Proposition 3.5 outlined just aboveis also true.

Suppose that Proposition 3.5 and its variant have both been estab-lished. One consequence will be that when we are dealing with thesequence of Π0

2-sentences φn : n ∈ ω which are all true in the stan-dard model – with some end-fragment thereof, φn : n ≥ N, beingsuch that each φn in the end-fragment entails, in conjunction with PRA,a Π0

1-sentence witnessing P = NP in the sense outlined in the state-ment of Proposition 3.5, with the same choice of Π0

1-sentence workinguniformly for all n ≥ N – then we will thereby obtain a sequencefn : n ≥ N of total general recursive functions, which will be one ofthe important objects in our proof and of which we shall speak morelater.

It is also important to note the following.

Proposition 3.7. Our proof of Proposition 3.5 can be relativized toany sequence of functions 〈gk : k ∈ ω〉 at all, where in the originalstatement of Proposition 3.5 our choice for 〈gk : k ∈ ω〉 was gk(n) = nk,but this can be replaced by any sequence of functions 〈gk : k ∈ ω〉, withthe proviso that the sequence of functions should have the followingset of properties: (1) The functions in the sequence should be closedunder functional composition (2) they should be dominated growth-rate-wise by the exponential function (3) we require that gk, for arbitraryk, be dominated growth-rate-wise by gk′ for some sufficiently large k′.

A PROOF OF P6=NP 9

Then, with the assumption that our sequence of functions satisfy (1)–(3), Proposition 3.5 suitably modified still holds, as long as we substituteevery application of a function n 7→ nk or n 7→ n2k in the statement ofProposition 3.5 with an application of gk or g2k.

In fact, the requirement (2) is not strictly necessary here. However, ifwe want it to turn out that our sequence of functions 〈fn : n ≥ N〉constructed from a sequence of sentences which are true in the standardmodel turns out to have the property that the functions fn are cofinalgrowth-rate-wise in the set of all total general recursive functions, whichis something that we require for later parts of our argument, thenwe will need to impose requirement (2). Then if we apply the aboveProposition in the case where 〈gk : k ∈ ω〉 is any sequence of totalgeneral recursive functions satisfying requirements (1)–(3), to obtain asequence of functions 〈fn : n ≥ N〉 which are cofinal growth-rate-wisein the set of all total general recursive functions, we will be able toconclude by means of arguments to be presented later that an NP -complete problem cannot be solved in time O(f(n)) for any f which isdominated growth-rate-wise by some gk. Thus in this way we indicatethe generality of our argument, notably indicating that our argumentwill imply as a final conclusion not only P 6= NP but also that thereare many sub-exponential functions f(n) such that an NP -completeproblem cannot be solved with run-time O(f(n)). However our proofwill be consistent with the known fact that the NP -complete problemSAT is solvable in O((1.3)n).

We should further note that our construction can indeed be relativizedto an oracle, but we will not end up proving the result PA 6= NPA

for arbitrary oracles A, which is known to be false, because when theconstruction is so relativized the requirement that the sequence of func-tions 〈fn : n ≥ N〉 be cofinal growth-rate-wise in the set of total generalrecursive functions will fail. We will return to this point briefly laterat the appropriate time.

We have now clarified, then, the propositions that must be proved inorder to complete the first of the three steps that are required for ourfinal conclusion, in addition to the proof of Lemma 3.4. We shall nowproceed to the description of the next two steps.

Suppose that Step 1 has been completed. We have then obtained asequence 〈fn : n ≥ N〉 of total general recursive functions, and for eachfunction fn in the sequence we can construct the “Tower of Hanoi”sentence ψn, which asserts “Either the Turing machine for fn(n) halts

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and I’m refutable from PRA+φn in at most fn(n) steps, or the Turingmachine for fn(n) does not halt and I’m refutable from PRA + φn+1”.We assume that n is represented by a closed term of length polynomialin log2 n. This construction can be achieved with the diagonal lemma,with the sentence ψn being a polynomial-time computable functionof the defining formula for fn, and we call it the “Tower of Hanoi”sentence, for reasons discussed below. We can also clearly constructa “Tower of Hanoi” sentence where φn, φn+1 are replaced by arbitraryφ, φ′ satisfying the usual set of requirements.

This construction was inspired by a similar construction of HughWoodin’sin an essay of his titled “The Tower of Hanoi”. Calling the construc-tion by this name is motivated by the fact that Hugh Woodin used asimilarly constructed sentence to give an instance of a polynomial-timecheckable property of a hypothetical string of bits of length 1024, suchthat there would be no feasible way to demonstrate that the stringof length at most 1024 with the required property does not exist from“anything in our mathematical experience”, but also such that if thestring were actually observed to exist then we would be able to inferthat a certain Turing machine involving 2011 iterations of the exponen-tial operator does not in fact halt, thereby having located an inconsis-tency in EFA. Naturally our mathematical intuitions lead us to believethat the string does not in fact exist, but this is not feasibly confirmablethrough “direct mathematical experience” but only by intuition. Thesentence would therefore reflect the “Tower of Hanoi” legend in thatsense, since the “Tower of Hanoi” legend holds that if the Tower ofHanoi game with 64 disks were ever completed then the gods wouldstep in and destroy the universe, analogous to the “destruction of themathematical universe” that would ensue from actually locating thebit-string. So for this reason we call it a “Tower of Hanoi” sentencebut will be using this construction for very different purposes.

Proposition 3.8. Given that the “Tower of Hanoi” sentences are con-structed as above, we construct a partial general recursive functiong in the following way. We search through every possible quadruple〈φ, φ′, P, P ′〉 such that P is a proof of 1-consistency of PRA + φ fromPRA + φ′ and P ′ is a proof of a Π0

1-sentence witnessing P = NP inthe appropriate sense from PRA+ φ. Given such a quadruple, this canbe used as a basis to compute in polynomial time a Σ0

1-formula withtwo free variables defining an appropriate analogue of the function fndescribed in the statement of Proposition 3.5 (but of course since we

A PROOF OF P6=NP 11

are remaining agnostic about whether φ and φ′ are true in the stan-dard model we must also remain agnostic about whether we are reallysucceeding in defining a total general recursive function here). Then,given the formula defining the appropriate analogue of fn, and givena closed term in the first-order language of arithmetic for the uniquepossible candidate for what n might be, where the closed term’s lengthis polynomial in log2 n, we may construct a “Tower of Hanoi” sen-tence ψ corresponding to this quadruple which may be obtained as apolynomial-time function of the quadruple.

We now define a partial general recursive function g as follows. Forevery quadruple as specified above with length bounded above by somepolynomial function of the argument k to the function g, we searcheither for a refutation from PRA + φ of the Tower of Hanoi sentenceψ, or else a proof in PRA + φ + ¬ψ of Σ0

r-unsoundness, for some r,of PRA + φ′ + ¬ψ. We claim that this search always terminates forevery tuple with the desired (polynomial-time-checkable) properties, nomatter what, provided that the constants K and β given in requirement(3) given earlier for the formula fn in the statement of Proposition3.5 are chosen appropriately. But, pending a demonstration of that,then of course we simply stipulate that if the search fails to terminatefor some tuple corresponding to a given argument k then that simplycounts as a value of k for which the function g is undefined. Then,having performed such a search and brought it to termination for everytuple with length bounded above by the appropriate polynomial bound,we take the maximum length of all proofs found as the outcome of thesearch over all the tuples, and that is the value of g(k) for this k. Inthis way we have clearly defined a partial general recursive function,computer code for which it is totally feasible to construct. The claimwe wish to make is that the function g is in fact provably total in Σ0

2-induction arithmetic, provided that the constants K and β given earlierare chosen appropriately.

Then suppose that we have completed this step, and suppose that weare also assuming P = NP for a contradiction, and suppose that weare also assuming Σ0

3-induction so that we can assume that predicatesdefined with reference to our sequence of sentences 〈φn : n ∈ ω〉 con-structed in Lemma 3.4 are such that if they are inductive then theirextension is the whole set of natural numbers (Σ0

3-induction appear-ing to be the amount of induction that is required for this on carefulinspection of the proof). Then our third step involves showing

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Proposition 3.9. Given the above assumptions and the previous step,the function g is in fact a total general recursive function growing fasterthan any fn in the sequence 〈fn : n ≥ N〉 discussed at the completionof Step 1, but this sequence of functions is cofinal growth-rate-wise inthe set of total general recursive functions, and that’s a contradiction.Thus we’ve derived a contradiction from P = NP in Σ0

3-inductionarithmetic, therefore we can now conclude that P 6= NP is provablein Σ0

3-induction arithmetic and in particular in PA, and this completesthe proof that PA proves P 6= NP .

This then, is the general sketch of how the proof goes. So to completethe proof we need to prove Lemma 3.4, Propositions 3.5 – 3.7 (Step 1),Proposition 3.8 (Step 2), and Proposition 3.9 (Step 3).

We also need to briefly note the reasons why if our argument for Propo-sition 3.5 were done in a context where we were relativizing to an oraclefor a set A known to have the property that PA = NPA then we wouldnot get the outcome that 〈fn : n ≥ N〉 is cofinal growth-rate-wise inthe set of total general recursive functions, thereby explaining how ourargument is not a “relativizing” argument. We also need to briefly notethat if requirement (2) were dropped from the statement of Proposition3.7 then we would fail to end up with a sequence 〈fn : n ≥ N〉 whichis cofinal growth-rate-wise in the set of total general recursive func-tions, thus showing that our argument survives the “sanity check” ofthe NP -complete problem SAT being solvable in O((1.3)n). Discussionof these issues will be given in the next section as well as completionof the three steps.

This then completes the general sketch of how the proof goes. Now weneed to fill in all the missing steps which will be the task of the nextsection.

4. Proof of the main theorem

We noted at the end of the previous section, that we’ve set ourselvesthe following tasks for this section: we must prove Lemma 3.4, Propo-sitions 3.5 – 3.7 (Step 1), Proposition 3.8 (Step 2), and Proposition3.9 (Step 3), and we must discuss the reasons why the argument isnot a “relativizing” argument and why the sequence of total generalrecursive functions 〈fn : n ≥ N〉 would fail to be cofinal growth-rate-wise in the set of all total general recursive functions in the event thatrequirement (2) were dropped from the statement of Proposition 3.7.So, those are the tasks of this section, and let us now proceed, startingwith the proof of Lemma 3.4.

A PROOF OF P6=NP 13

Proof of Lemma 3.4. Our background metatheory is PA. Choose a pos-itive integer C such that, given a Π0

2-sentence φ of length N symbolswith no q-marks of length greater than or equal to 2n, a Π0

2-sentence ψof length at most N+2n+C symbols exists which implies 1-consistencyof PRA+φ. The construction of the desired sequence φn : n ∈ ω isas follows.

Define p(n) to be a polynomial with constant term 100, degree at least2, and such that p(n) − p(n − 1) ≥ 3n + C for all n > 0. Select thesentence φn, starting from n = 0 and proceeding by induction on n,assuming at each step that all previous sentences have already beenconstructed, as follows. If we are in the case n > 0, then considerthe set S of all true Π0

2-sentences with at most p(n − 1) + 2n + C

symbols and no q-marks of length greater than n which (together withPRA) imply 1-consistency of PRA+φn−1, otherwise just consider theset S of all true Π0

2-sentences with at most 50 symbols and no q-marks;in either case this set is non-empty and finite. Each sentence in S

is true in the standard model, and so is associated with a particulargeneral recursive function, of which the sentence asserts totality. Thetotal general recursive functions are partially ordered by comparison ofgrowth rates.

Our sentence φn will be a conjunction of two sentences, the first one aΠ0

2-sentence chosen as follows. We choose our Π02-sentence by selecting a

maximal element of this partially ordered set at each stage, also strictlydominating growth-rate-wise, in the case n > 0, each total generalrecursive function whose totality is witnessed by a true Π0

2-sentence oflength at most q(n) (and with q-marks of arbitrary length allowed),where q(n) again is an appropriately chosen polynomial. With thepolynomial q(n) fixed, this is indeed possible for an appropriate choiceof the polynomial p(n).

Our second conjunct is chosen (where possible) from the set of trueΠ0

1-sentences with no q-marks, and of length bounded so as to makethe length of the conjunction at most p(n). For appropriate choices ofall the polynomial bounds involved, this gives rise to a sequence withthe desired set of properties. The sequence constructed in this way isclearly arithmetically definable.

We now need to complete all the steps outlined in the previous sec-tion in addition to the proof of Lemma 3.4.

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Proof of Propositions 3.5 and 3.6. The simplest way to proceed will beto prove Proposition 3.6.

We are assuming that we are given a quadruple 〈φ, φ′, P, P ′〉 whereφ and φ′ are Π0

2-sentences and P is a proof from PRA + φ′ of the1-consistency of PRA + φ and P ′ is a proof from PRA + φ of a Π0

1-sentence witnessing the existence of a polynomial-time-solver whichalways correctly solves the NP -complete problem SAT with run-timebounded above by a particular polynomial bound with exponent m.

We have to show how given such a quadruple we can construct inpolynomial time a Σ0

1 formula θ(a, b) with exactly two free variables aand b which PRA+φ′-provably defines a total general recursive functionwith a certain set of properties outlined in the statement of Proposition3.5.

We shall assume that there are fixed positive integers C and α, to bechosen later in such a way as to meet the requirements of our argument,and that p(n) := Cnα, and that the length in symbols of θ(a, b) isbounded above by p(n) where n is chosen to be the unique possiblecandidate for an n such that φ could possibly be equal to φn from thesequence 〈φn : n ∈ ω〉 of Lemma 3.4. The growth of length in symbolsof the φn is organized in such a way that there will be a natural choice,given a particular φ, for the unique possible candidate for such an n.We assume that n is chosen this way, and then we require that θ(a, b)be bounded above in length in symbols by p(n) as defined above.

We will let Sn be the set of all Σ01 formulas with exactly two free

variables of length at most p(n) which PRA + φ′-provably satisfy thefunctionality condition and also the totality condition. Although wecannot guarantee that θ(a, b) really does define a total general recursivefunction in the standard model, it does provably do so in PRA + φ′,so that when we are speaking in a context where we are assumingPRA+ φ′, we can use fn to denote the total general recursive functionwhich it defines and we will do so. Now we shall once again spell outthe properties that the formula θ(a, b) is required to have, quoting thesefrom the statement of Proposition 3.5.

(1) It is PRA+φ′-provable that the following holds for all k: there is aPRA+ φ-proof of the halting of a Turing machine Tn,k which is set upso that given empty input it will compute fn(k).(2) The following is PRA+ φ′-provable. Denote by rn(k) the length ofthe shortest proof from PRA+ φ of the halting of the Turing machine

A PROOF OF P6=NP 15

Tn,k. We require it to be PRA + φ′-provable that it holds for all suffi-ciently large k that the actual halting time of Tn,k lies between (rn(k))

k

and (rn(k))2k.

(3) Denote by hn(k) the halting time of the Turing machine Tn,k andconsider the set of all Turing machines T ′ with halting time h′ ≤ hn(k)when given empty input, with the property that, if e denotes the in-dex of Tn,k and e′ denotes the index of T ′, then e′ ≤ Keβ, for fixedpositive integers K and β, to be chosen later on in Section 4 duringthe exposition of the main argument, and which will have to be chosenso as to enable the requirements of a model-theoretic argument for thesecond of our three steps to go through. It is required to always bePRA+ φ′-provable that if T ′ is a Turing machine from this set then aPRA + φ-proof of halting of T ′ given empty input does indeed exist,and in what follows we shall denote by t′ the shortest length in symbolsof such a proof for a given T ′ in the set. We require it to be PRA+ φ′-provable that, for all sufficiently large k, (the relevant lower bound for“sufficiently large” being the same as in (2)), on the assumption thatT ′ is in the set described previously and t′ is defined as above, then(t′)2k ≥ h′, where h′ is the halting time of T ′ on empty input.

We are required, then, to show that given a quadruple with the statedproperties, and provided C and α are chosen appropriately, a for-mula θ(a, b) with all the required properties does indeed exist and isa polynomial-time computable function of the quadruple. We shall infact specify an algorithm for computing the unique b such that θ(a, b)as a function of a, where we are in a context where we are assumingPRA + φ′, and then knowledge of the formula θ(a, b) can of course berecovered from knowledge of this algorithm. We must first give a def-inition of a sequence of theories whose union is PRA. We shall definethis sequence of theories immediately below as a sequence of theories inL(0, S,+, ∗), but it should be understood that we take the obvious ex-tension of each of these theories to the language of PRA as was definedpreviously, which has additional constant symbols.

Definition 4.1. Let Γr : r ∈ ω be the following increasing sequenceof theories. Γ0 is defined to be the deductive closure in the predicate cal-culus for L(0, S,+, ∗) of the set of all consequences of the axioms of Qwhich can be formulated in L(0, S) together with bounded induction forthat language. Γ1 is defined to be the deductive closure in L(0, S,+, ∗)of Presburger arithmetic. Γ2 is Bounded Induction Arithmetic. ForΓ3, extend the language to include a symbol for exponentiation and in-clude the defining recursion equations for exponentation, and then add

16 McCALLUM

bounded induction axioms for that language, and take the set of conse-quences of this axiom set in L(0, S,+, ∗). Similarly with Γ4, except thatwe include a symbol for super-exponentiation as well as exponentiation.And with Γ5 we have symbols for exponentiation, super-exponentiation,and super-super-exponentiation. And so on. This completes the speci-fication of the hierarchy of theories Γr : r ∈ ω to which we will referin what follows.

In particular, with Γr : r ∈ ω as above, we have that the set ofprovably total general recursive functions of Γr is equal to the r-th setin the Grzegorczyk hierarchy of primitive recursive functions.

We shall work in a context where we are assuming PRA+φ′ and denoteby fn the unique total general recursive function (which does indeedprovably exist given PRA + φ′), giving the unique b such that θ(a, b)as a function of a, and we previously promised we would present analgorithm for computing this function fn. Our algorithm for comput-ing fn is as follows. For each natural number k which serves as theargument to the function fn, we search through the space of Turingmachines with a description of length bounded above by some polyno-mial function of k, to be chosen later. For each such Turing machineT , we then use the polynomial-time algorithm for solving the NP -complete problem PROVABILITYT for all theories T appearing in theclass of decision problems given in Lemma 3.1 – recalling that we areassuming for a contradiction that this polynomial-time algorithm doesindeed exist – to determine whether there exists a proof p1, with a givenpolynomial bound relative to n on its length, in PRA+φ+“Γr + φ is1-consistent”: r ∈ ω, that there exists a proof p2 from PRA+ φ of thehalting of the Turing machine T under consideration (with empty in-put). Note that the proof p1 is polynomially bounded in length relativeto n, but in the case where φ is true in the standard model so that theproof p2 is indeed guaranteed to exist, the length of the proof p2 is notin general polynomially bounded relative to n.

We then use the same algorithm to determine whether a proof p3 ofthe desired lower and upper bounds given earlier in requirement (2) ofthe statement of Proposition 3.5 – on the number of steps for Turingmachine T to halt, relative to the length of p2 – exists in PRA+φ′,with some appropriately chosen polynomial bound relative to n on thelength of p3.

A PROOF OF P6=NP 17

We now want to claim the following. There always exists some Turingmachine T with index e for which the proofs p1 and p3 are availablewith the appropriate bounds on their lengths, and for which there alsoexists a PRA + φ′-proof p4, with some appropriate polynomial boundrelative to n on the length of the proof, for the conclusion presentedin the following paragraph. We should note that for the purposes ofcalculating the length of the proof p4, it should be understood thatthe Turing machine T is referred to in the conclusion of the proof by adescription as being the unique Turing machine with a certain property,where that property shall be specified later on. There will also be aPRA + φ′-proof p5 of polynomially bounded length relative to n thatthere exists a PRA+φ′-proof p6 for the conclusion that this descriptionuniquely picks out the Turing machine T with index e, where e willbe specified by some actual closed term, but we are not assuming anypolynomial bound on the length of the proof p6 and will not have a usefor the proofs p5, p6 later in the argument except by way of noting thatour construction does require that the proof p5 exists.

Let us now describe what the conclusion of the proof p4 is required tobe, where it is understood that the Turing machine T shall be referredto by a description to be given later. The conclusion desired for theproof p4 is as follows: If we let T ′ be any other Turing machine whichhas halting time h′ when given empty input less than or equal to thehalting time h of T , and has an index less than or equal Keβ where eis the index of T , and as noted before the values of the positive integerconstants K and β are to be chosen later on during the justification ofProposition 3.8, and if we let t(T ′) be the shortest length of a proof fromPRA+φ of halting of T ′ when given empty input, and we let h(T ′) bethe halting time of T ′ when given empty input, then logt(T ′) h(T

′) ≤ 2kfor all such Turing machines T ′ on empty input. We have to showthat a description of T can be found so that the proof p4, using thisdescription, with polynomially bounded length relative to n, will exist,proving the preceding conclusion, and also such that there will also be aproof p5, from PRA+φ′ and with polynomially bounded length relativeto n, of the existence of a proof p6, from PRA+ φ′ and not necessarilywith polynomially bounded length relative to n, that this descriptionwill indeed uniquely refer to the Turing machine T with index e wheree is an appropriately chosen closed term. Let us now proceed to givethe description which satisfies these requirements.

Working in PRA+φ′, consider the finite set of Turing machines T ′′, forwhich the proofs p1 and p3 are available, with polynomially bounded

18 McCALLUM

length relative to n, and such that logt(T ′′,s)h(T′′, s) ≤ 2k is true for all

s, and also this universally quantified statement is PRA + φ′-provablevia a proof with appropriately polynomially bounded length relativeto n, and also the same is true of any other Turing machine whichhas a description of no greater length than T ′′. Let T be a member ofthis finite set with largest possible running time. This completes thedescription of T required for the purposes of the preceding paragraph,which we previously promised to give. The required proofs p4 and p5both clearly exist. Furthermore, as k tends towards infinity, the valueof t(T ) for such a Turing machine T , where t(T ) refers to the lengthof shortest proof from PRA + φ of halting of T on empty input, andwhere T is a Turing machine for which the proofs p1, p3 and p4 areavailable, may be chosen to be arbitrarily large (since the maximumvalue of logt(T )h(T ) over all Turing machines T with a given bound onthe length of their description, does go to infinity as that bound goes toinfinity). It is clear that the proof p3 does exist for the Turing machineT , although not necessarily with polynomially bounded length, whenit is referred to by a closed term for its actual index. We shall alsoneed to show that the proof p3 exists as a proof in PRA + φ′ withpolynomially bounded length, when the Turing machine T is referred toby the foregoing description, possibly with a somewhat more generouspolynomial bound on the length of the proof than was imposed on thelengths of the corresponding proofs for the Turing machines T ′′ in thefinite set. This point requires some discussion and we shall elaboratefurther on that point shortly.

Then the value of the function fn for the value of k under considerationwill be the maximum number of steps taken to halt for any Turingmachine in the set considered for which the following holds. We requireof the Turing machine that appropriate proofs p1, p3 both exist, and theproofs p4 and p5 can be found relative to some appropriate descriptionof the Turing machine in question, again with appropriate polynomialbounds on the length, and recalling that it will also be true, as wewill discuss later, that in that case a proof p3 will be available for theTuring machine under the description in question with a somewhatmore generous polynomial bound than the bounds used in the searchthrough the space of Turing machines.

We spoke of “searching through the space of Turing machines” in or-der to compute the maximum number, but actually the availability ofa polynomial-time solver for an NP -complete problem will make possi-ble a speed-up of the computation required to determine the maximum

A PROOF OF P6=NP 19

number, so that a complete search will not be necessary, although therun-time of the computation will still be approximately equal to itsfinal output. We will assume that some such speed-up is indeed used.We will be able to describe the algorithm for the computation as apolynomial-time-computable function of the quadruple 〈φ, φ′, P, P ′〉,and the fact that the algorithm will indeed eventually output the cor-rect maximum running time for all Turing machines in the set whileitself having a total run-time close to the final outcome of the compu-tation will be provable in PRA+ φ′.

With the assumption that some such “speed-up” is used, and still pend-ing the discussion promised earlier of why the proof p3 will be availablewith polynomially bounded length for the Turing machine T under theappropriate description, it can then be seen that a Σ0

1-formula for thefunction fn may be chosen so that the corresponding Turing machineT ′

n,k for fn(k) is such that the requisite proofs p1, p3 and p4 will be avail-able for that Turing machine, with polynomially bounds on the lengthof p1 and p4 relative to the n, when k is sufficiently large, possiblywith a slightly higher polynomial bound needed than the polynomialbounds that were imposed for the Turing machines T ′′ in the finiteset over which the search was done. We must give a discussion of thepoint that the proof p3 will indeed be available for the Turing machineT referred to two paragraphs previously, under the right description,with a somewhat more generous polynomial bound on its length thanthe bounds used in the search through the space of Turing machines,and also for the Turing machine T ′

n,k discussed just now, again witha somewhat more generous polynomial bound than the bounds usedin the search through the space of Turing machines, provided that wechoose the Σ0

1-formula appropriately. For confirming the second point,the fact that PRA+φ proves the correctness of a specific polynomial-time solver of any NP -complete problem with exponent m is crucial,and we will need to assume that k is much greater than m. We mustnow argue for this point.

Let us begin with consideration of the point that the proof p3 is avail-able for the Turing machine T under the right description with poly-nomially bounded length, only with a somewhat more generous poly-nomial bound than the bounds used in the search through the spaceof Turing machines. This really follows almost immediately from thedescription given of T , together with the fact that, given a certain up-per bound N on the lengths of proofs, one can show any given instance

20 McCALLUM

of proof-theoretic reflection as applied to a sentence which is a conclu-sion of a proof, with length constrained by the upper bound, is itselfprovable by means of a proof with length bounded above by a linearfunction of N . This is in fact true of any theory extending Q, but thepoint is even clearer regarding Σ0

1-sentences, which we are indeed deal-ing with, in the case of PRA, which actually proves the 1-consistency ofall of its finitely axiomatizable fragments. Still, the assumption of thetheory being reflexive is not needed and we shall need to avoid it lateron in the following section when we show that the argument can bepulled down into EFA. So, it is quite clear that the proof p3 is availablefor the Turing machine T under the right description with a somewhatmore generous polynomial bound on the length of the proof than thebounds used when searching through the space of Turing machines.But now when we turn to the Turing machine T ′

n,k, bearing in mindthat we have available a polynomial-time algorithm with exponent mfor solving NP -complete problems which PRA+φ-provably always ter-minates and gives the correct answer, and recalling the description ofhow the algorithm is actually constructed, we now see that when k islarge the run-time and shortest proof from PRA+ φ of halting for thisTuring machine T ′

n,k are (PRA+φ′-provably) going to be quite close tothe corresponding figures for the Turing machine T itself, such that ifwe choose our Σ0

1 formula appropriately, so that the run-time of T ′

n,k

is “padded out” a little if need be with only a modest correspondingincrease in the length of the shortest proof of halting, then the proofsp1, p3, and p4 will all exist for the Turing machine T ′

n,k, with slightlymore generous polynomial bound than the bounds used in the searchthrough the space of Turing machines.

We have shown how our Σ01-formula θ(a, b) can be constructed as a

polynomial-time-computable function of any quadruple 〈φ, φ′, P, P ′〉with all the required properties. But, returning to the case whereφ and φ′ are φn and φn+1 respectively from the sequence of sentences〈φn : n ∈ ω〉 of Lemma 3.4, where these sentences are all true in thestandard model, we see that we have now given a means of constructinga sequence 〈fn : n ≥ N〉 of total general recursive functions, where thelower bound N is determined by the least n such that PRA+φn provesa Π0

1-sentence, uniformly chosen for all n, witnessing P = NP in thesense given before.

Proof of Proposition 3.7. Because the length of the shortest proof fromPRA + φn of halting of T ′

n,k goes to infinity as k goes to infinity, this

A PROOF OF P6=NP 21

shows that each fn is provably total in PRA+φn+1 but not in PRA+φn.Thus the sequence of total general recursive functions 〈fn : n ≥ N〉 iscofinal growth-rate-wise in the set of all total general recursive func-tions, but the assumption P = NP is necessary for this. We canalso recall the discussion in the previous section of how this argumentcan be seen as invoking a sequence of total general recursive functions〈gk : k ∈ ω〉, with the case gk : n 7→ nk being the one under considera-tion here, and the argument will still work as long as any sequence offunctions 〈gk : k ∈ ω〉 are used which satisfy requirements (1)–(3) ofthe statement of Proposition 3.7, of which we give a quick reminder.

(1) The functions in the sequence should be closed under functionalcomposition(2) they should be dominated growth-rate-wise by the exponential func-tion(3) we require that gk be dominated growth-rate-wise by gk′ for somesufficiently large k′.

All points of the above discussion will still hold if the entire discus-sion is relativized to any sequence of functions 〈gk : k ∈ ω〉 satisfyingrequirements (1)–(3) above.

If requirement (2) is not imposed then the conclusion, that the lengthof the shortest proof from PRA + φn of the halting of T ′

n,k must go toinfinity as k goes to infinity, may fail. And then we will fail to get theresult that 〈fn : n ≥ N〉 is cofinal growth-rate-wise in the set of all totalgeneral recursive functions, which is needed for the later part of theproof. So our argument will eventually show that there cannot be analgorithm for solving an NP -complete problem with run-time O(f(n))where f is dominated growth-rate-wise by some gk′ from a sequence offunctions 〈gk : k ∈ ω〉 satisfying requirements (1)–(3) above, and theinclusion of requirement (2) is necessary for this. This clarifies that ourargument does show P 6= NP but is consistent with the known factthat there does exist an algorithm with run-time O((1.3)n) for solvingthe NP -complete problem SAT. We have now completed the proof ofProposition 3.7.

We also need to show why relativizing to an oracle in the case gk(n) =nk, or dropping requirement (2) from the constraints on 〈gk : k ∈ ω〉in the statement of Proposition 3.7 would give rise to 〈fn : n ∈ ω〉possibly not being cofinal growth-rate-wise in the set of all total generalrecursive functions.

22 McCALLUM

If we try to relativize to an oracle A while assuming PA = NPA,then the attempt to show that the proofs p1, p3 and p4 are alwaysavailable for the Turing machine T ′

n,k with polynomially bounded lengthwill fail. Because we will basically be speaking of proofs where we’reallowed to use any sentence we want from an axiom schema assertingmembership or failure of membership in the set A, and in order tooptimize running time over the entire set of Turing machines referredto in the description of the algorithm that T ′

n,k is running, we may notbe able to polynomially bound how many such axioms we need to usein order to construct our proofs. So this shows why our argument isnot a “relativizing” one.

To complete the argument that P 6= NP is provable in PA we just needto prove Propositions 3.8, and 3.9.

Proof of Proposition 3.8. We constructed a sequence of sentences 〈φn :n ∈ ω〉 in Lemma 3.1 and we showed how to construct a certain Σ0

1 for-mula with two free variables a and b as a polynomial-time-computablefunction of a quadruple 〈φ, φ′, P, P ′〉 satisfying certain properties. Webriefly indicate how to use these two constructions to construct thesequence of “Tower of Hanoi” sentences 〈ψn : n ∈ ω〉. Each sentencein the sequence of “Tower of Hanoi” sentences is constructed by meansof the diagonal lemma.

For each n ≥ N , where N is sufficiently large that a Π01-sentence wit-

nessing P = NP is provable from PRA + φn, the sentence ψn says“Either the Turing machine for fn(n) halts and I’m refutable fromPRA+φn in at most fn(n) steps, or the Turing machine for fn(n)does not halt and I’m refutable from PRA+φn+1”, where the sentencesφn, φn+1 are given by actually writing them out, and the function fn isgiven by means of the formula θn, and the Turing machine for fn(n) isagain determined by this formula θn, and n is given by a closed termof polynomially bounded length . Note that the growth of the lengthof ψn as a function of n is polynomially bounded.

Recall that we are assuming, for a contradiction, the existence of apolynomial-time solver for every NP -complete problem with an expo-nent of m, so there is always a refutation of ψn from φn in at mostC(fn(n))

m steps for some sufficiently large C, provided n ≥ N . It isalso apparent that the shortest refutation has more than fn(n) steps.

We are now going to obtain a contradiction by constructing a total gen-eral recursive function g, whose growth rate dominates every fn. Since

A PROOF OF P6=NP 23

the sequence of functions fn is cofinal growth-rate-wise in the set ofall total general recursive functions, by construction of the fn, includ-ing the previously mentioned fact that fn dominates any PRA+φn-provably total general recursive function growth-rate-wise, and theproperties of the sequence φn : n ∈ ω stated in the previous lemma,this will indeed be a contradiction and so our conclusion will be thatP 6= NP is indeed provable in PA, completing the proof.

Let us now describe the construction of g. We described how to con-struct the sentence ψn given the sentences φn and φn+1. More gener-ally, as described previously, given any quadruple 〈φ, φ′, P, P ′〉 whereφ and φ′ are Π0

2-sentences and P is a proof of 1-consistency of PRA+φfrom PRA+φ′, P ′ is a proof of a Π0

1-sentence witnessing P = NP fromPRA+φ, one may construct a corresponding sentence ψ in a similarway which is a polynomial-time-computable function of the quadruple〈φ, φ′, P, P ′〉. As noted earlier, the candidate for the n such that φ = φn

will be unique, so no ambiguity will arise from this issue.

So, given an n for which we wish to evaluate g(n), one now searchesthrough all such quadruples such that the length of each component ofthe quadruple is bounded above by an appropriate polynomial functionof n, and then, for each such quadruple in the resulting finite set, oneembarks on a search for either a proof in PRA+φ+¬ψ of arithmeticalunsoundness – a proof in PRA+φ+¬ψ of Σ0

r-unsoundness for some r –of PRA+φ′+¬ψ, or a refutation of ψ from PRA+φ, and the maximumlength of the shortest such proof or refutation found (the maximumbeing taken over all the quadruples whose components have lengthappropriately polynomially bounded) is the value of g(n). We mustprove that the search always terminates. If we can establish in Σ0

2-induction arithmetic that this search always terminates for any givenquadruple, using features of the construction of the “Tower of Hanoi”sentence ψ, and also choosing the positive integer constants K and βthat were mentioned in requirement (3) of the previous section in sucha way as to suit the needs of our argument, then it will follow fromthis that g is a Σ0

2-provably total general recursive function, computercode for which is perfectly feasible to construct. Then as discussedearlier P = NP plus Σ0

3-induction will give rise to the conclusion thatg grows faster than any total general recursive function thereby givinga contradiction, and then we will be finished. In any case we haveclearly at this point established the existence of g as a partial generalrecursive function.

24 McCALLUM

So our sole remaining task to complete the proof of P 6= NP in PA isto show that g is Σ0

2-provably total and also to show Proposition 3.9,given the way the “Tower of Hanoi” sentence ψ is constructed, and alsogiven appropriate choices uniformly made for the constants K and β

mentioned earlier. This is our final task, and then we will be done.

Suppose that the search never terminates. Then T := PRA+φ+¬ψ+“PRA+ φ′ + ¬ψ is Σ0

r-sound”:r ∈ ω is consistent and PRA + φ + ψ

is consistent. We are going to be doing model-theoretic reasoning inWKL0 which is adequate for the basic results of model theory includingthe proof of the completeness theorem and which is Π0

2-conservativeover PRA, as discussed in [5]. Let M be a model of T . M is a modelfor (schematic) arithmetical soundness of PRA+φ′+¬ψ, and PRA+φ′

predicts (with a proof of standard length) that PRA+φ will eventuallyrefute ψ, as can be seen from the way that ψ is constructed. The reasonfor this is that PRA+ φ′ predicts that the Turing machine referred toin the sentence ψ does indeed halt. And, if the Turing machine halts,then clearly one way or another a refutation of ψ from PRA + φ willeventually be found, and so PRA + φ′ makes this prediction. Thus inthe model M is a proof p in PRA+ φ of ¬ψ.

We are going to be speaking of “standard” and “non-standard” ele-ments of a model, so it will be desirable to clarify what is meant by“standard”. We shall be making use of the notion of the theory WKL0as defined in the previously mentioned work [5], and also making useof various Reverse Mathematics results from that same work. We willeventually want to claim that all of our reasoning can be formalisedin WKL0 which is Π0

2-conservative over PRA, but currently we are as-suming for a contradiction the consistency of T , and then given thatassumption WKL0 implies the existence of a model M of T . There isa cut of such a model – a Σ0

1-definable class relative to M , where weshould note that in WKL0 we do not have a set existence axiom forarbitrary Σ0

1-definable classes, but we can still talk about them meta-linguistically in the usual way – and this cut is the least cut that modelsPRA+“PRA is 1-consistent”. We can indeed assume that such a cutdoes exist as a Σ0

1-definable class, given that we are assuming that Tis consistent, and given the strength of T .

Elements of M will be said to be standard if they belong to this cut.Thus it will be okay in what follows to assume that the standard frag-ment ofM is closed under Ackermann’s function. So, with that under-standing of what “standard” means, could p be a standard element of

A PROOF OF P6=NP 25

the model M? If that were the case, then with the further assumptionthat Ackermann’s function is total (in the meta-theory, as an extra as-sumption added to WKL0) we could conclude that the search of whichwe spoke in the previous paragraph terminates. Our final conclusionwill be that WKL0, which is Π0

2-conservative over PRA, proves eitherthe termination of the search or the termination of the search on theassumption that Ackermann’s function is total. That will be enoughfor the function g to be PA-provably total (in fact provably total fromΣ0

2-induction arithmetic). Given this consideration, we will be ableto assume without loss of generality in what follows that p is a non-standard element ofM (since we already see that the desired conclusionis easy to obtain on the assumption that p is standard).

Denote by T ′ the theory PRA+φ+“Γr+φ is 1-consistent”: r ∈ ω. Theleast cut M ′ of M satisfying T ′ satisfies PRA+φ + ¬ψ since ¬ψ is Π0

1,and it can be seen – recalling that the functions fn were constructedso that a proof of halting of fn(k) could always be found in T ′ in anumber of steps bounded above by a polynomial function of k – thatp may be chosen so that p ∈ M ′, and we can assume without loss ofgenerality that p is non-standard by the foregoing. Denote by A be the(standard) Turing machine which plays in ψ the part that was playedby the Turing machine for fn(n) in ψn. That is, we are speaking ofthe Turing machine which is appropriately the analogue of a Turingmachine which computes fn(n) when given empty input.

Since M ′ is a cut of a model of T , and p is a non-standard element ofM ′, the halting time of A in M ′ is non-standard. Then the shortestproof of halting of A from PRA+φ has non-standard length, and whenraised to the power of a sufficiently large standard natural number,it exceeds the halting time for A (as we see when we recall the wayin which A was constructed). One can then also conclude that foreach standard natural number k, the length of the shortest proof fromPRA+φ of “PRA+φ proves PRA+φ proves PRA+φ proves... A halts”,going k levels deep, is also non-standard. Let us discuss how to arguefor this point.

We note that the shortest proof in PRA + φ of “A halts” is “close to”the actual halting time of A in the sense that if a is the length of theshortest proof and h is the halting time then a2r ≥ h for a standard r.Further for any h′ ≤ h each Turing machine T ′ of halting time h′ – andalso with its index appropriately bounded via constants K and β whichwill be chosen at the end – is such that the length t′ of the shortest

26 McCALLUM

proof in PRA+φ of halting of T ′ is such that (t′)2r ≥ h′ for a standardr, as we can see when we recall the construction of the algorithm forfn earlier, and also recall that we are working in a cut of a model forthe theory T .

It follows that the length of the shortest proof in PRA+φ of “PRA+φ

proves A halts” must also be non-standard. We see this as follows.Let T ′ be the Turing machine which searches for the shortest proofin PRA+φ of “PRA + φ proves PRA + φ proves... A halts” for somefinite number of levels deep. (We will therefore later have to choose Kand β so that T ′ can perform this function and at the same time theindex of T ′ can be chosen so as to be bounded appropriately by K andβ.) The halting time h′ of T ′ is such that a ≤ h′ ≤ h where a2r ≥ h

for a standard r. So the shortest proof of halting of T ′ is such thatraising the length of the proof to the power of a standard integer yieldsa result that dominates the non-standard halting time h′, so thereforethe length of the shortest proof of halting of T ′ is itself non-standard.As a corollary, the length of the shortest proof in PRA+φ of “A halts”is non-standard, and going two levels deep, the length of the shortestproof in PRA + φ of “PRA + φ proves A halts” is non-standard, andso on for any standard number of iterations of this. Thus we can nowsee as claimed before that for any standard k the length of the shortestproof from PRA+φ of “PRA+φ proves PRA+φ proves PRA+φ proves...A halts”, going k levels deep, is non-standard. We want to argue thatgiven the way the Turing machine A is constructed, this gives rise toa contradiction. The nature of the argument for this point is basicallyproof-theoretic.

One can introduce function symbols for every primitive recursive func-tion and also for the following total general recursive functions (relativeto our model M ′ which is a model for T ′ with no proper cuts whichmodel T ′): the function h0 whose totality is asserted by φ and the eachof the functions hr for r > 0 whose totality is asserted by the sentence“Γr+φ is 1-consistent”. Then one can construct a “quantifier-free”system in a language with these function symbols, with open formulasas axioms, with the first-order theory T ′ being Π0

2-conservative overthis quantifier-free system in an appropriate sense (where here we aredealing entirely with proofs of standard length).

We can now consider that there will be a closed term of standardlength in this language which denotes the halting time of A in everymodel of T ′. If we recall the construction of A, we see that such a

A PROOF OF P6=NP 27

term of standard length can be obtained computationally by searchingthrough the space of proofs in T ′ of standard length bounded above bya certain fixed standard upper bound. Furthermore, it can be provedin PRA+“PRA is 1-consistent” – which we are allowed to use here giventhe previous discussion of how we are using “standard” in such a waythat it is okay to assume that the standard fragment of any model ofT or T ′ models PRA+“PRA is 1-consistent” – that every such termis bounded above (in every model of T ′) by another term of standardlength, where nested occurrences of hr in the first term are replaced bya single occurrence of hs for a larger s, such that for every occurrenceof hs with s > 0 in the term, the argument to hs is a term with theproperty that it denotes a standard natural number in every model ofT ′. Having noted this, we then see that this term of standard lengthcan in turn be proved in PRA+“PRA is 1-consistent” to be boundedabove (in every model of T ′) by another term of standard length inwhich only the primitive recursive functions and h0 occur. From theexistence of such a term of standard length one can now obtain theresult that there exists some standard natural number k such thatPRA+φ proves PRA+φ proves... A halts, where we only go k levelsdeep. But this contradicts the observation previously made.

We are now at the point where we have derived a contradiction (inWKL0+“Ackermann’s function is total”) from supposing that the the-ory T has a model and also that PRA+φ + ψ is consistent, but thissupposition was seen to be a consequence of the assumption that thesearch never terminates. Our conclusion is that the search does prov-ably terminate in Σ0

2-induction arithmetic. Hence the function g isprovably total in Σ0

2-induction arithmetic as claimed.

Proof of Proposition 3.9. So, we do indeed have a provably total gen-eral recursive function here, and computer code for this function isperfectly feasible to construct. But the assumption that P = NP ,together with Σ0

3-induction, can now be seen to give rise to a conclu-sion which leads to a contradiction, namely that this general recursivefunction grows faster than any fn, but on the other hand the sequence〈fn : n ∈ ω〉 is cofinal growth-rate-wise in the set of all total generalrecursive functions. This follows from the construction of the “Tower ofHanoi” sentence ψn for n ≥ N whose length is polynomially boundedrelative to n and whose shortest refutation from PRA+ φn has lengthat least fn(n). In this way, from P = NP we can derive a contradictionin PA, and in fact in Σ0

3-induction arithmetic.

28 McCALLUM

From all of the foregoing, discussion we finally have

Theorem 4.2. It is provable in PA that P 6= NP .

5. Showing that the argument works in EFA

We have completed the proof in PA that P 6= NP . We shall de-scribe how to adapt this proof to show that Γ5 proves P 6= NP , andthat the total general recursive function witnessing the truth of theΠ0

2-statement P 6= NP is elementary recursive. We will conclude byshowing how the argument can in fact be adjusted to show that Γ3,that is EFA, proves P 6= NP .

We begin by observing that when we used the theories Γr in our con-struction of fn we could have replaced them by I∆0 +Ωr. These theo-ries are all provably 1-consistent in Γ4 (and the universally quantifiedstatement asserting that all of them are 1-consistent is provable in Γ4).From considerations of the speed at which proofs can be found of theform searched for in the algorithm for the function g, as can be ob-served from examining the proof of Proposition 3.8, we can in fact seethat the version of the function g defined this way would be elementaryrecursive.

We could also have replaced every occurrence of PRA with EFA (in asuitably extended language with all the appropriate constant symbols,that is), while still having g come out elementary recursive. There isa weak version of WKL0 called WWKL0 which is Π0

2-conservative overEFA, which is presented in [6]. We could have have used WWKL0 + Γ5

which is Π02-conservative over Γ5 to construct a model of EFA which

would serve for the construction of the sequence 〈φn : n ∈ ω〉, (thesesentences being required to be true in this model, which would be nec-essarily non-standard), and the principle of induction for sets definedfrom this sequence using set existence axioms allowable in WWKL0would be permitted in WWKL0 + Γ5, which again is Π0

2-conservativeover Γ5. We require adding Γ5 to WWKL0 in order to be able to prove1-consistency of EFA. Thus we end up with the conclusion that theΠ0

2-sentence P 6= NP is provable in Γ5 and the total general recursivefunction witnessing its truth is elementary recursive, as claimed.

We can in fact pull the argument all the way down into EFA. To achievethis, substitute EFA everywhere with I∆0 + Ωr : r ∈ ω, howeverwe use a quantifier-free version, so that the 1-consistency claim forthis theory is provable in EFA. Now re-work the argument so thatall object theories use quantifier-free logic rather than full first-order

A PROOF OF P6=NP 29

classical logic, and do the model-theoretic reasoning in WWKL0. Inthis way we can dispense with the assumption of Γ5. Thus our finalconclusion is that EFA proves P 6= NP .

6. The language of set theory

In what follows, when we speak of the first-order language of set theory,we shall be speaking of a language extended by a infinite family ofconstant symbols, some of them of length greater than one relative toa fixed finite alphabet, as in Section 3. All theories will be in thislanguage.

The argument in the previous section was inspired by the initial obser-vation that it is provable in ZFC+“V=Ultimate-L” that there existsa certain set-theoretically definable sequence φn : n ∈ ω of Σ2-sentences in the first-order language of set theory with a certain setof properties. Here by “set-theoretically definable” we mean, definablein the first-order language of set theory without parameters. Let usdiscuss this result.

Recall the statement of the axiom “V=Ultimate-L”, formulated byHugh Woodin in [4].

Definition 6.1. The axiom V=Ultimate-L is defined to be the asser-tion that

(1) There is a proper class of Woodin cardinals.(2) Given any Σ2-sentence φ which is true in V , there exists a univer-sally Baire set of reals A, such that, if ΘL(A,R) is defined to be the leastordinal Θ such that there is no surjection from R onto Θ in L(A,R),

then the sentence φ is true in HODL(A,R) ∩ VΘL(A,R).

Now we state the main lemma of this section.

Lemma 6.2. Assume ZFC+“V=Ultimate-L”. On those assumptions,it is provable that there exists a set-theoretically definable sequence ofΣ2-sentences φn : n ∈ ω with the property that

(1) There exists a polynomial function with integer coefficients and pos-itive leading coefficient such that the length in symbols of φn is even-tually always bounded above by the value of this polynomial function atn;(2) ZFC+φn : n ∈ ω is Σ2-sound and Σ2-complete;(3) ZFC+φn+1 always proves Σ2-soundness of ZFC+φn.

30 McCALLUM

Here, by saying that the sequence φn : n ∈ ω is set-theoreticallydefinable, we mean that the set of ordered pairs 〈n,m〉 such that m isthe Godel code of φn is a set which is definable by a formula in thefirst-order language of set theory without parameters.

Proof. Choose a positive integer C such that, given a Σ2-sentence φof length N symbols, and with no q-marks of length greater than orequal to 2n, a Σ2-sentence ψ of length at most N + 2n + C symbolsexists which implies Σ2-soundness of ZFC+φ. The construction of thedesired sequence φn : n ∈ ω is as follows.

Use a similar construction to as in the proof of Lemma 3.4, with thefollowing modifications. Select the sentence φn, starting from n =0 and proceeding by induction on n, assuming at each step that allprevious sentences have already been constructed, as follows. If weare in the case n > 0, then consider the set S of all true Σ2-sentenceswith at most p(n − 1) + 2n + C symbols which (together with ZFC)imply Σ2-soundness of ZFC+φn, otherwise just consider the set S of alltrue Σ2-sentences with at most 50 symbols; this set is non-empty andfinite. Each sentence in S is true in V , and recall that we are assumingV=Ultimate-L. Recall also that the Borel degrees of universally Bairesets are well-ordered, as shown in [3]. For each sentence φ in S, wecan select a minimal Borel degree corresponding to φ such that thereis a universally Baire set A of that Borel degree witnessing condition(2) of Definition 6.1 for the sentence φ. Now consider the set T ⊆ S

consisting of all sentences φ from S for which the associated Boreldegree is as large as possible. Choose a universally Baire set A fromthe Borel degree corresponding to all sentences T ⊆ S, and then foreach sentence φ ∈ T , select the least possible α < ΘL(A,R) such thatthe sentence φ is true in (HOD)L(A,R) ∩ Vα. In this way an ordinalis associated to each sentence in T ; then choose a sentence in T withthe largest possible associated ordinal. Let r be the maximum integersuch that the conjunction of a sentence of length p(n − 1) + 2n + C

and a sentence of length r has length at most p(n). Now form theconjunction of the previously mentioned sentence chosen from T withthe Σ2-sentence of length at most r least in the lexicographical orderingwhich has not occurred as a conjunct in any sentence so far, if thereis any such sentence (otherwise simply leave the chosen sentence fromT as is). This completes our account of how we select the sentenceφn. This construction will produce a sequence φn : n ∈ ω with thedesired list of properties. It is clear from our construction that thesequence is set-theoretically definable.

A PROOF OF P6=NP 31

One could of course note that the same lemma is provable from as-sumptions much weaker than ZFC+“V=Ultimate-L” without makingany essential use of the machinery about universally Baire sets; thepoint of interest about this particular construction is that in somesense the Σ2-sentences grow in logical strength as fast as possible whilestill remaining only polynomially bounded in their growth in length. Iam not currently aware of any interesting application of this particularconstruction of a sequence of sentences, but it may be a topic thatwould repay further thought.

References

[1] Hugh Woodin. The Tower of Hanoi, Chapter 18 of [2].[2] Truth in Mathematics, eds. H. G. Dales and G. Oliveri, Oxford Science Publi-

cations 1998.[3] Set Theory of the Continuum. eds. H. Judah, Winfried Just, Hugh Woodin.

Springer (22 October 1992).[4] In Search of Ultimate-L: The 19th Midrasha Mathematicae Lectures. W. Hugh

Woodin. Bulletin of Symbolic Logic, Volume 23, Issue 1, March 2017, pp. 1–109.

[5] Subsystems of Second Order Arithmetic. Perspectives in Mathematical Logic;Springer-Verlag 1999.

[6] Measure theory and weak Konig’s lemma. Xiaokang Yu and Stephen G. Simp-son. Arch. Math. Logic, Volume 30, Issue 3, pp. 171–180.

[7] Quotational ambiguity. George Boolos. In Paolo Leonardi and Marco San-tambrogio (eds.), On Quine: New Essays, Cambridge University Press, pp.283–295 (1996).