One reason ZFC (and other similar systems) is interesting is that it is
a candidate for providing a basis for mathematics, and in particular
arithmetic. Now if ZFC should turn out to be inconsistent, this will
certainly make it unsuitable for such purposes. Everybody knows this.
What is not so commonly appreciated is that even if ZFC is consistent,
there might still be statements about the natural numbers that are
*theorems of ZFC* but nevertheless *false*. In this sense, consistency
is a "piddling condition" because it does not *in itself* give us any
guarantee that arithmetical theorems that we prove on the basis of ZFC
are necessarily true.
That's the basic point; the rest of this article is just a clarification
and explication of the above claim.
First of all, here is one point that confused me at first. How can
*theorems* be *false*? Well, the point here is that when one says that
a statement about the natural numbers is "false" one means that it is
false of the *standard* integers, i.e., the integers that we normally
work with. The statement could be true in a nonstandard model of
arithmetic. Thus for example, a statement of the form "there exists
an integer k that is P" could be a theorem of ZFC even if there is no
ordinary integer with property P. This phenomenon is often referred
to as "omega-inconsistency." We have the infinite family of theorems
"1 is not P," "2 is not P," etc., along with the theorem "there is some
integer that is P." So one can rephrase the claim above as, "the
consistency of ZFC does not imply the omega-consistency of ZFC."
For example, even if ZFC is consistent, it is possible for "ZFC is
inconsistent" (which I will abbreviate to ~Con(ZFC) from now on) to be
a theorem of ZFC. For ~Con(ZFC) really means "there is some integer k
that encodes a proof of a contradiction from the axioms of ZFC." As
indicated above, this does not necessarily mean that one can exhibit a
*standard* integer that encodes such a proof. In other words, even if
~Con(ZFC) is a theorem of ZFC, we needn't believe it's true (or more
precisely we need not believe its interpretation as a statement about
the standard integers) even if we started out by believing that ZFC is
consistent. (In fact, Goedel's second incompleteness theorem assures
us that if ZFC is consistent then ~Con(ZFC) can at the very least be
added to the axioms of ZFC without fear of inconsistency, even if
~Con(ZFC) is not a theorem of ZFC.)
Or as another example, suppose someone proves on the basis of ZFC a
theorem that states that the Goldbach conjecture is false, i.e., a
theorem of the form "there exists an integer k that refutes the GC."
Again, a priori we still can't conclude automatically that the GC is
indeed false (for the standard integers) even if we assume that ZFC
is consistent.
In other words, consistency doesn't buy you everything that you might
want from a formal system---it tells you that models of the system
exist, but it doesn't tell you that there are any models where the
"integers" are isomorphic to the standard integers, so you can't
necessarily "trust" theorems of the system.
Does this mean that we can't trust *any* theorems of arithmetic that
ZFC generates? No. Notice that the above examples are all of the
form "for some k, P(k)" where P is some recursive property (roughly,
a property that can be checked "finitarily" for any given k). In
general there is no guarantee that a theorem of this form is true.
However, if T is any consistent system (containing some minimal
fragment of arithmetic---enough to "check" recursive properties for
any fixed integer), then any theorem of T of the form "for all k, P(k)"
*is* true. The reason for this is as follows: suppose it isn't true.
Then there exists a standard integer k for which P(k) is false, i.e.,
an explicit counterexample, and the details of this can easily be
encoded in T, i.e., if "for all k, P(k)" is false then its negation
is not only true---it is a *theorem* of T. But this contradicts the
assumed consistency of T.
So consistency guarantees the truth of *some* arithmetical theorems
but says nothing about a lot of other interesting ones.
Newsgroups: sci.logic,sci.math
From: oliver@sage.math.ucla.edu (Mike Oliver)
Subject: Re: Consistency is a piddling condition
Message-ID: <1994May10.202633.29500@math.ucla.edu>
Organization: UCLA Mathematics Department
References:
<2qnn6u$lpg@cantua.canterbury.ac.nz>
Date: Tue, 10 May 94 20:26:33 GMT
In article <2qnn6u$lpg@cantua.canterbury.ac.nz>
wft@math.canterbury.ac.nz (Bill Taylor) writes:
>1) Is it the case then that omega-consistency "buys what we want"? That is,
> if ZFC is omega-consistent, does this suffice to guarantee that
>any PA-statement provable in ZFC will be true in the standard integers?
No. For example ZFC "could" be omega-consistent but prove its own
omega-inconsistency, which is a first-order statement of arithmetic
(I assume that's what you mean by "PA-statement.")
(The word "could" in quotes means "If the theory 'ZFC+ZFC is omega-consistent'
is consistent, then it doesn't refute the statement in question.")
>[...] if PA |- thm , can we be sure that thm is true?
Well, as sure as we are that PA is true. But not in the sense
that such an inference can be proved in PA or even PA+Con(PA).
Specifically PA "could" be consistent but prove its own inconsistency.
More precisely, if PA+Con(PA) is consistent, then PA+Con(PA) doesn't
prove that PA doesn't prove not(Con(PA)).