🚀 go-pugleaf

Just a few quick answers and questions.

>>Indeed, ZF intrinsically suggests it  [..nonselfmembership..]
>>*without* the bother of a foundation axiom, by limiting the set formation
>>possibilities so that self-membership is at least not immediately achieved
>
>What is the reasoning here?  That a theory T "intrinsically suggests"
>not-P for any P not in T?  If neither P nor not-P are in T, which of
>not-P or not-not-P (= P) does T "intrinsically suggest?"

Nice one ! Yes, I was a bit lax there. I should have said, "from the
definitionalist viewpoint ZF suggests...."  To expound the definitionalist
viewpoint, (a very lukewarm form of constructivism), would need a big post
of its own. Maybe one day.  I'll just remark here, that all the usual axioms
of ZF affirm set existence by actually giving a condition of membership.
i.e. a set is affirmed by saying what its members are. AC and AxFound and AFA
are all very much NOT like this. So the absence of any axioms of this type is
a kind of suggestion that all sets are those built up "from below" by the
member-definitional axioms.  Absence of evidence = evidence of absence ?

One could say that ZF is (seems to be) the natural theory with only
member-definitional axioms. (Though one could add some inaccessibles axioms).
A kind of minimal definitionalist set theory.

Enough waffling.


>>A = { 1 , { 2 , { 3 , { 4 , {.....}}}}} . Such a set seems very well defined

>If you are comfortable with A, you should not mind
>B = { 1 , { 1 , { 1 , { 1 , {.....}}}}} .

Yes, I *do* mind B, for the very reason you suggest; that it is a member of
itself. There is, I agree, not a lot of difference, on the face of it, between
B and A.  I'm not *devoted* to A. It *does* have a slightly fishy look. 
Even a hereditarily fishy look.  :)    That's why I put out my request.
Let me repeat it...

REQUEST: Can B above, be obtained from A above, by ZF methods.
~~~~~~~  I don't think Replacement can achieve it, but I'm open to correction.
Any ideas, anyone ?


Randall Holmes - a question.  You say that it is a *theorem* of ZF that

   "for any set x, for some ordinal a, x E V(a)" .

[where V(a) is defined by  V(0) = {}
                           V(a+1) = P{V{a}}
                           V(L) = U{a < L}[V(a)] (L a limit ordinal). ]

Is this really true ? Surely not. The collection of V's is a model for ZF, yes.
But the above isn't a theorem, is it? 
I expect it *is* a theorem of ZF+AxFndtn.

Can you confirm this point please; and give a reference, if you still say
it's a theorem of ZF only?
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            Bill Taylor              wft@math.canterbury.ac.nz
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                   I think, therefore I am; (I think).
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