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From: A. Wolf <a.lupine.domain.name.hidden>

Date: Sat, 8 Nov 2008 18:03:31 -0500

*> I can if there's no rule of inference. Perhaps that's crux. You are requiring
*

*> that a "mathematical structure" be a set of axioms *plus* the usual rules of
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*> inference for "and", "or", "every", "any",...and maybe the axiom of choice too.
*

Rules of inference can be derived from the axioms...it sounds circular

but in ZFC all you need are nine axioms and two undefinables (which

are set, and the binary relation of membership). You write the axioms

using the language of predicate calculus, but that's just a

convenience to be able to refer to them.

*> Well not entirely by itself - one still needs the rules of inference to get to
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*> Russell's paradox.
*

Not true! The paradox arises from the axioms alone (and it isn't a

true paradox, either, in that it doesn't cause a contradiction among

the axioms...it simply reveals that certain sets do not exist). The

set of all sets cannot exist because it would contradict the Axiom of

Extensionality, which says that each set is determined by its elements

(something can't both be in a set and not in the same set, in other

words).

Anna

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Received on Sat Nov 08 2008 - 18:03:43 PST

Date: Sat, 8 Nov 2008 18:03:31 -0500

Rules of inference can be derived from the axioms...it sounds circular

but in ZFC all you need are nine axioms and two undefinables (which

are set, and the binary relation of membership). You write the axioms

using the language of predicate calculus, but that's just a

convenience to be able to refer to them.

Not true! The paradox arises from the axioms alone (and it isn't a

true paradox, either, in that it doesn't cause a contradiction among

the axioms...it simply reveals that certain sets do not exist). The

set of all sets cannot exist because it would contradict the Axiom of

Extensionality, which says that each set is determined by its elements

(something can't both be in a set and not in the same set, in other

words).

Anna

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To post to this group, send email to everything-list.domain.name.hidden

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Received on Sat Nov 08 2008 - 18:03:43 PST

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