[FOM] Infinity and the "Noble Lie"

Andrew Boucher Helene.Boucher at wanadoo.fr
Tue Dec 13 01:25:33 EST 2005

On  12 Dec 2005, at 9:12 PM, joeshipman at aol.com wrote:
> Anyone who claims to doubt the existence of "actually
> infinite sets" has no right to assert RH, or cite it as a needed lemma
> in his own work.  But many more people seem to be willing to say they
> doubt the existence of infinite sets than are willing to take this
> doubt to its consistent conclusion

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?

> and say that theorems like the Graph
> Miinor theorem have not really been proven, and that credit for the
> proof of the Prime Number Theorem properly belongs to Erdos/Selberg  
> and
> not to Hadamard/de la Vallee Poussin.

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.

> There are only 4 alternatives I can see which avoid hypocrisy:  accept
> the full Axiom of Infinity, or accept an axiom such as Con(ZFC), or
> protest that certain widely accepted theorems have not really been
> proven, or claim that such theorems don't have definite truth values.

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.

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