The Consistency of Arithmetic
Storrs McCall
The paper presents a proof of the consistency of Peano Arithmentic (PA) that
does not lie in deducing its consistency as a theorem in an axiomatic system. PA's
consistency cannot be proved in PA, and to deduce its consistency in some stronger
system PA+ is self-defeating. Instead, a semantic proof is constructed which
demonstrates consistency not relative to the consistency of some other system but in an
absolute sense. Some philosophical consequences of consistency for the thesis which
identifies minds with machines are explored.
Is Peano arithmetic (PA) consistent? This paper contains a proof that it is:- a
proof moreover that does not lie in deducing its consistency as a theorem in a system with
axioms and rules of inference. Gödel's second incompleteness theorem states that the
consistency of PA cannot be proved in PA, and to deduce its consistency in some
stronger system PA+ that includes PA is self-defeating, since if PA+ is itself inconsistent
the proof of PA's consistency is worthless. In an inconsistent system everything is
deducible, and therefore nothing is provable. If there is to be a genuine proof of PA's
consistency, it cannot be a proof relative to the consistency of some other stronger
system, but an absolute proof, such as the proof of consistency of two-valued
propositional logic using truth-tables. In this paper I construct a proof of PA's
consistency based not on manipulation of symbols, i.e. the syntactical derivation of a
consistency-theorem from axioms, but on a formal semantics for PA. An essential
element of any semantics is a nonlinguistic component, a domain of objects that is
necessarily absent from pure syntax. To my knowledge, no semantic consistency proof
of Peano arithmetic has yet appeared in the literature. In the last part of the paper I
briefly explore some of the philosophical consequences of the argument.
1. The system PA
1.1 Primitive symbols
Logical symbols: &, ~, =, plus brackets
Arithmetic symbols: 0, S, +, ×
Variables: x, y, z, ...
1.2 Rules of formation
1.2.1 For terms:
(i) All variables, plus 0, are terms
(ii) If X and Y are terms, then so are SX, X+Y, and X×Y
(iii) Nothing else is a term.
1.2.2 For wffs:
(i) If X and Y are terms, then X = Y is an atomic wff.
(ii) If A is a wff and x is a variable, then (x)A is a wff.
(iii) If A and B are wffs, then so are ~A and A&B.
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(iv) Nothing else is a wff.
1.3 Definitions.
A => B =df ~(A&~B)
A v B =df ~(~A & ~B)
(Ex)A =df ~(x)~A
1 =df S0, 2 =df SS0, etc.
1.4 Axioms
Axioms and rules for first-order logic plus the following (Mendelson
(1964), p. 103; Goodstein (1965), p. 46):
A1.
A2.
A3.
A4.
A5.
A6.
A7.
A8.
(x=y) => (x=z => y=z)
~(Sx = 0)
(x=y) => (Sx=Sy)
(Sx=Sy) => (x=y)
x+0 = x
x+Sy = S(x+y)
x×0 = 0
x×Sy = (x×y)+x
Induction rule. From |- F(0) and |- (x)(Fx => F(Sx)) infer |- (x)Fx.
2. Formal semantics for PA
2.1 A model M consists of a domain D, in this case the set of natural
numbers including 0, and an assignment function v, which assigns each term of PA a
member of D. The function v is defined inductively.
2.1.1 (Basis) v(0) = 0. Also, where x is a variable, v(x) is a member of D.
2.1.2 (Induction step) Where X and Y are terms, assume that v(X) and
v(Y) have already been defined. Then
(i) v(SX) = v(X) + 1
(ii) v(X+Y) = v(X) + v(Y)
(iii) v(X×Y) = v(X) × v(Y)
2.2 The valuation function vM with respect to a model M is a function
which takes wffs of PA into the set of truth-values{T, F}. It too is defined inductively.
2.2.1 (Basis) Where A is an atomic wff X=Y, vM(X=Y) = T iff
v(X) = v(Y).
2.2.2 (Induction step) Assume vM(A) and vM(B) have already been
defined. Then:
(i) vM(~A) = T iff vM(A) = F
(ii) vM(A&B) = T iff vM(A) = T and vM(B) = T
(iii) vM((x)A) = T iff for all models M' which differ from M at
most in assignment to the variable x, vM'(A) = T.
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2.3 Truth-in-a-model. We say that the formula A is true in a model M iff
vM(A) = T.
2.4 Validity. A valid formula is a formula which is true in all models M.
It is not difficult to show that the axioms A1-A8 are valid. For the induction rule,
if in every model F(0) is true, and if in every model in which Fx is true, F(Sx) is also true,
then in every model Fx is true. In the domain of natural numbers, either (i) all members
of the domain have a certain property F, or (ii) 0 lacks F, or (iii) there is a pair n, n+1
such that n has F and n+1 does not. There is no fourth alternative not covered by these
three. Hence the induction rule preserves validity. (By an element of the domain "having
a property F" is understood "having an arithmetic property expressible using only the
functions S, + and ×, the relation =, truth-functions and quantifiers". An example is the
property "being a prime number". Such properties do not include set-theoretical
properties like "being equal to the number of the set of all subsets of the natural
numbers". They do however include other properties that no numbers have, such as that
expressed by the open formula (y)y=x.)
The remaining axioms and rules of inference of PA belong to first-order predicate
logic, and it is known that these axioms are valid in the formal Tarski-style semantics
above, and that the rules preserve validity. Therefore all theorems of PA are valid. But
since no two formulae A and ~A can both be valid, it follows that PA is consistent.
3. Philosophical consequences of semantic consistency.
In a frequently cited paper written almost 50 years ago, Hilary Putnam says the
following in connection with Gödel's incompleteness theorem:
"It has sometimes been contended ... that a Turing machine cannot serve as a
model for the human mind, but this is simply a mistake.
"Let T be a Turing machine which 'represents' me in the sense that T can prove
just the mathematical statements I can prove. Then the argument ... is that by using
Gödel's technique I can discover a proposition that T cannot prove, and moreover I can
prove this proposition. This refutes the assumption that T 'represents' me, hence I am not
a Turing machine. The fallacy is a misapplication of Gödel's theorem, pure and simple.
Given an arbitrary machine T, all I can do is find a proposition U such that I can prove:
(3)
If T is consistent, U is true,
where U is undecidable by T if T is in fact consistent. However, T can perfectly well
prove (3) too! And the statement U, which T cannot prove (assuming consistency), I
cannot prove either (unless I can prove that T is consistent, which is unlikely if T is very
complicated)!" (Putnam (1960/1975), p. 366)
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If T is a Peano-arithmetic-proving-machine, capable of outputting all and only
theorems of PA, then Putnam's observation is unassailable. A human being can prove
nothing other than what can be proved in Peano arithmetic unless the human being can
prove that PA is consistent. Despite his warning, however, many attempts have been
made since 1960 to do what Putnam asserted cannot be done, namely use Gödel's
theorem to argue that human minds are not Turing machines and that mechanism is
false.1 I include myself in this list of what Putnam would classify as "failed attempts",
having argued that whether PA is consistent or not, the set of propositions a human being
can know to be true is different from the set of theorems a Turing machine can prove.2
The argument goes as follows:
Let G denote the Gödel sentence, which says of itself that it is unprovable in
Peano arithmetic. There are two alternatives. (1) If PA is consistent, then G is
unprovable but true. (2) If PA is inconsistent, then G is provable but false. Either way,
provability and truth part company, and human beings can make judgments of truth and
falsehood which elude the theorem-proving capacities of PA-machines.
In reply, a supporter of mechanism could maintain, with Putnam, that the
conditional PA sentences:
(1) If Cons(PA) then ~Bew[G]
(2) If ~Cons(PA) then Bew[G]
that we accept as true, a Turing machine can prove. ("Bew" is the PA predicate
"beweisbar", provable). The result for the mechanist is a draw, neither humans nor
machines being entitled to claim victory. But the picture changes if PA is shown to be
consistent.
Although PA cannot prove its own consistency using axioms and rules of
inference, as was seen there exists a semantic consistency proof which we humans can
accept but which PA-machines cannot reproduce. The reason why a Turing machine
cannot deal with a semantic proof is that the machine recognizes only symbols on its
input tape, and its response is confined to outputting more symbols. But a semantic proof
contains as an essential component a non-linguistic domain of elements which a Turing
machine cannot access except through the medium of symbols. To "fully formalize" a
semantic proof would require symbolizing the domain and axiomatizing its structure, that
is to say converting the semantic proof into a syntactic proof. But as Gödel showed, the
consistency of PA cannot be proved in PA, and if proved in a stronger axiomatic system
is contingent upon the consistency of that system. It follows that a conclusive, watertight
consistency proof of arithmetic must necessarily be a semantic one. A human being can
go through such a proof, detach the consequent of the conditional (1) above, and know
for certain that the Gödel sentence is true. A Turing machine can't do this. Conclusion:
human beings are not machines.
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Footnotes
1
See Lucas (1961), (1996); Penrose (1989), (1994); Redhead (2004); Wright (1994).
For criticism see Benacerraf (1967); Boolos (1990); Davis (1993); Dummett (1994);
Raatikainen (2005a), (2005b); Shapiro (1998); and numerous short comments in
Behavioral and Brain Sciences 13 (1990), pp. 655-92.
2 See McCall (1999), (2001). Critics of this approach include George and Velleman
(2000); Tennant (2001); Raatikainen (2002) and (2005a), pp. 523-4
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6
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