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Re: How big is infinity?

Subject: Re: How big is infinity?
From: Lester Zick
Date: Wed, 30 Aug 2006 12:22:07 -0700
Newsgroups: sci.logic, sci.math
On Wed, 30 Aug 2006 18:45:35 +0300, Aatu Koskensilta
<aatu.koskensilta@xxxxxxxxx> wrote:

>Virgil wrote:
>> Axioms are not required to be true, except for the purpose of seeing 
>> what would be derivable from them if they were. No assertion of the 
>> absolute truth of any axiom system is any part of any axiom system.
>
>Setting aside "absolute", whatever it's force is supposed to be, it's a 
>trivial observation that great many axioms in mathematics are presumed 
>to be true - in the ordinary mathematical sense, which might or might 
>not qualify as "absolute truth" -, ranging from induction for naturals 
>to the existence of infinitely many Woodin cardinals. It's a tedious and 
>pointless task to derive conclusions from some random bunch of axioms 
>unless the axioms are taken to be true - or to at least have some 
>possibly weaker soundness property -, in which case these conclusions - 
>e.g. Fermat's last theorem, Goldbach's conjecture, ... - are also true; 
>or the axioms have some metamathematical interest.

Please stop reiterating the obvious. It's not polite to tease the
animals. They're animals for a reason.

>As an example: why would a number theorist be interested in a proof of, 
>say, Goldbach's conjecture in set theory unless he thought the basic 
>principles of set theory are correct? After all, the statement ZFC |- GC 
>has no intrinsic number theoretical interest in itself.

~v~~

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