Re: THIS STATEMENT HAS NO PROOF IN ANY SYSTEM = true or false?

tchow_at_lsa.umich.edu
Date: 01/17/05 

Date: 17 Jan 2005 03:29:32 GMT

In article <cs8ug1$mu7$1@ra.nrl.navy.mil>,
Ralph Hartley <hartley@aic.nrl.navy.mil> wrote:
>tchow@lsa.umich.edu wrote:
[Re: claim that mathematicians usually mean "X is provable" when they say "X"]
>> As a matter of sociological fact, this is definitely false.
[...]
>I did say "usually".

Again, as a matter of sociological fact, this is just totally false. *You*
might usually mean "X is provable" when you say "X," but your claim was
about mathematicians, and the vast majority of mathematicians do not share
your point of view. You might be right and they might be wrong, but I am
only pointing out the sociological fact about mathematicians.

>I think there are proofs (not in ZFC) that ZFC is consistent. I'm not sure
>exactly what those proofs depend on, but if ZFC is inconsistent, then those
>proofs must not be sound either.

You can prove, for example, in "ZFC + there exists a strongly inaccessible
cardinal" that ZFC is consistent. So of course, if you give up ZFC, then
you would have to give up "ZFC + some other axioms." This would not make
much of a difference to most of mathematics, because very little mathematics
makes use of those other axioms. As I said, most of mathematics doesn't use
anywhere near the full power of ZFC. First-order Peano arithmetic, for
example, is good enough for a huge majority of mathematics.

[Re: If ZFC is abandoned]
>Someone would have to go back through every proof and make sure it
>is still valid (or a valid proof exists).

Nope. You make it sound like the way a mathematician checks a proof is to
verify that it is provable in ZFC. This never happens outside of logic and
automated theorem proving, and even in logic papers it rarely happens.
Mathematicians just check proofs by reading them and convincing themselves
of their logical correctness. They do not verify their provability within
any specific formal system, unless they are actually verifying "X is
provable in ZFC" as opposed to verifying X itself.

Since they never did what you seem to think they did in the first place,
they certainly aren't going to go back and do it over.

>It would make a *big* difference where the problem was. Just giving up
>choice would be easy.

Yes, it would. If the contradiction were in a very weak system, then that
would make a difference. But anyone other than someone with an extremely
skeptical philosophical stance knows with certainty that this can't happen.
There is hardly anything in the world that we know with greater certainty
than things we have proved mathematically.

>When someone says "sqrt(2) is irrational" it may be a bit ambiguous. Is it
>a statement about their intuitive (but imprecise) idea of the reals, or
>about the much more exactly defined objects of a formal system?

It need not be a statement about the reals; indeed, its most natural
reading is equivalent to, "Given any strictly positive integers a and b,
the integers a^2 and 2b^2 are distinct." This statement is about as precise
a statement as there is in this world. If you regard this statement as
imprecise, then the definition of a "formal system" is also imprecise,
since formal systems are defined in terms of things like "symbols," "rules,"
"strings," and so forth, which are no more precise than "integers." 

-- 
Tim Chow       tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth.  ---Galileo, Dialogues Concerning Two New Sciences

Relevant Pages

  • Re: THIS STATEMENT HAS NO PROOF IN ANY SYSTEM = true or false?
    ... Again, as a matter of sociological fact, this is just totally false. ... about mathematicians, and the vast majority of mathematicians do not share ... You can prove, for example, in "ZFC + there exists a strongly inaccessible ... any specific formal system, unless they are actually verifying "X is ...
    (sci.math)
  • Re: THIS STATEMENT HAS NO PROOF IN ANY SYSTEM = true or false?
    ... Again, as a matter of sociological fact, this is just totally false. ... about mathematicians, and the vast majority of mathematicians do not share ... You can prove, for example, in "ZFC + there exists a strongly inaccessible ... any specific formal system, unless they are actually verifying "X is ...
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    (comp.theory)
  • Re: Platonism
    ... Most mathematicians just say "theorem" without referencing a system, ... Retreating to ZFC is unlikely. ... happens in SOME physics classes. ... The set of reals is not hard to construct in ZFC, ...
    (sci.math)
  • Re: Platonism
    ... Retreating to ZFC is unlikely. ... My assessment of the beliefs of mathematicians comes from conversations ... reason to be suspicious of formalism. ... and not a problem with the mathematicians concept of reals or of ...
    (comp.theory)