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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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