Re: ZFC

In article <1117987802.197022.257830@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
examachine@xxxxxxxxx wrote:

>tchow@xxxxxxxxxxxxx wrote:
>> In article <87oeapuwtx.fsf@xxxxxxxxxxxx>,
>> Jesse Alama <alama@xxxxxxxxxxxx> wrote:
>> >What strikes me as a more
>> >interesting claim is that the kinds of *reasoning* we perform to make
>> >judgments about these objects and their relations can also be treated
>> >within ZFC. Does that claim "ZFC is a foundation for mathematics"
>> >concerns merely the scope of the mathematical objects that can be
>> >treated within ZFC?
>>
>> No, the reasoning is also included in the scope.
>>
>> Roughly speaking, the completeness theorem of first-order logic is the
>> basis for thinking that our reasoning is adequately captured by the
>> usual formal rules. That is, the completeness theorem tells us that if
>> some statement S about the objects in question cannot be proved from
>> the given axioms using the known rules, then we can exhibit something
>> that satisfies the given axioms and not S; therefore, there is no valid
>> principle of reasoning whose addition to the known ones would let us
>> prove any new consequences of the axioms.
>
>I would think that the logic used to formalize a given set theory
>is the foundation, rather than the set theory itself. After all,
>when we look beneath set theory we find a logical language, but
>when we try to look beneath logic, there is nothing.

Not necessarily. The foundational proofs of the consistency and
completeness of FOL itself use models, those models being represented in
some set theory. So in this sense the logic itself stands on set theory
(which is of course expressed in a logic...) It's turtles all the way down.

>Regards,
>
>--
>Eray
>

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Relevant Pages

  • Re: ZFC
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    (comp.theory)
  • Re: Skolems Paradox and why is math the way it is?
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  • Re: Skolems Paradox and why is math the way it is?
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