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Consider NFU set theory as presented in this: http://math.boisestate.edu/~holmes/holmes/head.pdf

On page 15 of that pdf it states that the following is an axiom: The set $\{X\colon X=X\}$ exists.

Let $V = \{X\colon X=X\}$.

$V$ is an element of $V$ as $V=V$.

How is this not circular?

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    Have you read [the relevant Wikipedia article](http://en.wikipedia.org/wiki/New_Foundations)?2011-11-30
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    Thanks for the link. The wiki article says V exists as an entailment of the Comprehension axiom whereas the Holmes book is an axiom. {X:X=X} is an element of {X:X=X}. How is that different from something like defining the set y by the equation y={y}?2011-12-01
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    It is possible, without the axiom of foundation (also known as regularity) to have $x=\{x\}$.2011-12-01
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    Do you also view the fact that $0\cdot 0=0$ in arithmetic to be circular?2011-12-01
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    @JDH No mainly because that isn't the definition of zero. In NFU, V is defined to be {X:X=X}. As Asaf pointed out, this would only be "circular" if there were an axiom of foundation which would prohibit such things as V∈V.2011-12-06
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    @Tanner: Do we really need a tag for [nfu]?2012-10-07
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    @Asaf Well, I thought we could use one, but that was before I realized we had only two questions about it. I do plan to ask a third, for what it's worth.2012-10-07
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    @Tanner: I don't know if it merits a tag...2012-10-07
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    @Asaf I've posted a question on meta about it: http://meta.math.stackexchange.com/questions/6309/should-we-have-a-tag-for-the-set-theory-nfu2012-10-07

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