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From: Brent Meeker <meekerdb.domain.name.hidden>

Date: Sat, 08 Nov 2008 15:56:02 -0800

A. Wolf wrote:

*>> 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
*

*>> 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
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*> are set, and the binary relation of membership). You write the axioms
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*> using the language of predicate calculus, but that's just a
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*> convenience to be able to refer to them.
*

*>
*

*>> Well not entirely by itself - one still needs the rules of inference to get to
*

*>> Russell's paradox.
*

*>
*

*> Not true! The paradox arises from the axioms alone (and it isn't a
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*> true paradox, either, in that it doesn't cause a contradiction among
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*> the axioms...it simply reveals that certain sets do not exist). The
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*> set of all sets cannot exist because it would contradict the Axiom of
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*> Extensionality, which says that each set is determined by its elements
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*> (something can't both be in a set and not in the same set, in other
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*> words).
*

I thought you were citing it as an example of a contradiction - but we digress.

What is your objection to the existence of list-universes? Are they not

internally consistent "mathematical" structures? Are you claiming that whatever

the list is, rules of inference can be derived (using what process?) and thence

they will be found to be inconsistent?

Brent

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Received on Sat Nov 08 2008 - 18:56:14 PST

Date: Sat, 08 Nov 2008 15:56:02 -0800

A. Wolf wrote:

I thought you were citing it as an example of a contradiction - but we digress.

What is your objection to the existence of list-universes? Are they not

internally consistent "mathematical" structures? Are you claiming that whatever

the list is, rules of inference can be derived (using what process?) and thence

they will be found to be inconsistent?

Brent

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You received this message because you are subscribed to the Google Groups "Everything List" group.

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

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Received on Sat Nov 08 2008 - 18:56:14 PST

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