[FOM] Infinity and the "Noble Lie"

joeshipman@aol.com joeshipman at aol.com
Tue Dec 13 13:04:19 EST 2005

Reply to Boucher:

>Can you give examples of people who say they doubt the existence of
>infinite sets but would not doubt to the "consistent conclusion"?
>With quotes, if possible?

I will try to find some from the FOM archives.

>Others are more knowledgeable about this than me, but Erdos/Selberg
>use real analysis necessarily in their proof, so it seems that they
>also need the Axiom of Infinity.   There are "super-elementary"
>proofs, by Eda and Fogels, which have done away with all real
>analysis, but even with these (in fact, I just know the latter), it's
>not evident to me that they don't require some notion of infinity.

The analysis in their proof is straightforwardly eliminable, it was 
understood immediately that their proof was formalizable in Peano 
Arithmetic. Every real function they used can be replaced by formulas 
involving calculations with rational approximations. Obviously the 
Prime Number Theorem itself, as usually stated, involves the real 
function log(x), but an equivalent version can be stated that speaks 
only about finite objects, and the Erdos/Selberg proof transforms in 
the same way.

>Here you are using "proven" in some absolute sense.  When people say
>something has been "proven", they may just mean "proven in ZFC" or
>"proven using the usual axioms".  There is no hypocrisy, of course,
>in expressing doubt about a theorem and accepting that it has been
>"proven in X," where "X" is some fixed formal system.

But mathematicians commonly do use "proven" in an absolute sense. For 
every mathematician, there is a set of theorems she is willing to say 
are simply "proven", because they follow from axioms she accepts as 
"true". No one bothers to say, when referring to Euclid's theorem on 
the infinitude of the primes or Gauss's Quadratic Reciprocity Law, that 
those theorems have been proven from a particular set of axioms -- the 
set of axioms needed is so universally accepted that those are simply 
"proven theorems".  Whenever a mathematician says that theorem X "has 
been proven", with no qualification about the axiom system involved, 
that mathematician ought to assent to X, assert X, accept the truth of 
X, assent to "X is true", etc., etc.

What I am ultimately criticizing is what I (possibly incorrectly) 
perceive as an unwillingness among mathematicians who express doubt 
about the existence of actual infinities, to criticize (or even to 
themselves eschew) language which refers to theorems whose proofs use 
Infinity as "proven" or "true" without a caveat about the axioms used. 
If you disbelieve in actual infinities, you shouldn't give a statement 
proven from the Axiom of Infinity a higher epistemological status than 
the Axiom itself.

-- JS
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