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Topic: Anti-foundation axiom
Replies: 11   Last Post: Mar 15, 2013 6:00 AM

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Posts: 1,635
Registered: 2/27/06
Re: Anti-foundation axiom
Posted: Mar 9, 2013 4:55 PM
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On Mar 6, 7:59 am, Zuhair <> wrote:
> The following theory Violates foundation in preference to somewhat
> plausible
> axioms.
> ZF - foundation
> +
> For all x. Exist H(x)
> +
> for all x. x subnumerous to H(x)
> +
> Anti-Foundation Axiom: An infinite Dedekindian finite set exists.
> Where H(x) is the set of all sets hereditarily subnumerous to x.
> Zuhair

You can't really have axioms for set theory. Axioms are a way to
formalize our understanding of a subject e.g. number theory or
geometry. They are so well developed and understood that we try to
codify our great understanding with a formal system.

But how well do we understand sets? There are a dozen versions of set
theory. You end up with conflicting theorems from one system to
another!! Can you imagine 2 different versions of Geometry or Number
Theory? At most one would be right.

You really have to treat sets as something we do NOT understand and so
the next step is to firm up our limited understanding. That means we
examine the issues and consider the options. The different set
theories would be possibilities to examine. With each author
declaring their version of set theory authentic - WITHOUT ANY
RESPONSIBILITY TO JUSTIFY IT - you simply have conflict and the
resulting chaos of misunderstandings that Wiki describes.

What are the fundamental questions?

Can a set contain itself? (Using Recursion Theory as a guide, the
answer is yes. I posted on FOM a constructible X such that X={X}
based on a proof in Recursion Theory.)

Do you consider relationships besides sets (relations)? Right now we

A. Everything is a set.
B. x ~e x is not a set.

Direct inconsistency!

This is exactly like when you (Zuhair) attempt to formalize the Liar
Paradox without first deciding the nature of the resolution. You
fumble around with " 'This is false.' is both true and false." while I
point out the analogies that show that it is neither.

Mathematics is supposed to be about the truth. With this free-for-all
at declaring what set theory is - rather than looking at the whole
problem and considering each possibility - it is only about authors
jockeying for power and influence, promoting themselves rather than
giving an honest appraisal of other approaches.



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