Re: ZFC
- From: tchow@xxxxxxxxxxxxx
- Date: 06 Jun 2005 13:06:35 GMT
In article <haberg-0606051004590001@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
Hans Aberg <haberg@xxxxxxxxxx> wrote:
>I got into the subject by writing on a theorem prover, based on extending
>and adapting a Prolog program using Hilbert's axioms, even though the
>underlying computational engine does not have any such limitations. Then I
>cross into those questions of metamathematical type theory, higher order
>theories and such, and it is not very clear what to use as basis.
Well, this much is certainly true, but it's misleading to express
this concept by saying "the foundations of mathematics are unclear."
To say that the foundations of mathematics are unclear suggests that the
subject is obscure and is plagued by logical difficulties that people
don't understand very well and are still trying to clarify. That's very
different from saying that for the purposes of building a theorem prover,
there are many choices and no clearcut winner. When building a theorem
prover, one has to consider many issues such as efficiency, ease of use,
peculiarities of the particular problem domain of interest, elegance,
and so forth.
The same goes for investigations into alternative foundations for
mathematics besides ZFC. The existence of these investigations doesn't
mean that there's anything logically suspect about ZFC. But depending
on what your goals are, you may find some other approach more natural,
elegant, efficient, etc.
--
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
.
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