🚀 go-pugleaf

In article <TORKEL.92Nov5103134@isis.sics.se> torkel@sics.se (Torkel Franzen) writes:
>In article <Bx81Ho.9HL@cantua.canterbury.ac.nz> wft@math.canterbury.ac.nz 
>(Bill Taylor) writes:
>
>   >A = { 1 , { 2 , { 3 , { 4 , {.....}}}}}
>   >B = { 1 , { 1 , { 1 , { 1 , {.....}}}}}
>   >Can B above, be obtained from A above, by ZF methods.
>
>  The question is vague in several respects, but the most likely
>answer is no. A, to put this into formal ZF terms, is, let us say, the
>range of a function f defined on the natural numbers, where
>f(n)={n,f(n+1)} for all n.  Assuming the existence of such a function
>in ZF (without foundation) does not in any obvious way (and hence not
>at all, by the argument from experience) imply the existence of a set
>x such that x={1,x}.  In particular, Replacement by itself is no help
>since to get x using replacement, we need to define a function h by
>h(n)=G(f(n)), where G({a,b})={1,G(b)}, and there is no way of defining
>G in ZF (without foundation) so that this is provable. In other words:
>Replacement allows us to make infinite "substitutions" in sets, but to
>make infinite substitutions in non-well-founded sets we need a
>corresponding non-well-founded recursion principle.
>
>  Of course, even if this is correct it doesn't follow that accepting
>A and rejecting B makes any sense. Or in other words: it doesn't help
>explain what mathematical objects, if any, you have in mind.

Neither A nor B can be shown to represent a unique object in ZFC-.
You are being hypnotized by your notation.  

There _can_ be sequences of sets xi and yi such that xi = {i,x[i+1}
and yi = {1,x[i+1]}  in ZFC-.  There can be many of each!  Thus A may
ambiguously denote many objects, and B may denote many objects -- nor
need an object denoted by B be an element of itself (it may contain a
_different_ object of the form B).

Aczel's AFA has a "strong extensionality" consequence which asserts
that there can be at most one object of the form A and at most one
object of the form B, in addition to its "non-well-foundedness"
consequence which asserts that there is at least one of each.

-- 
The opinions expressed		|     --Sincerely,
above are not the "official"	|     M. Randall Holmes
opinions of any person		|     Math. Dept., Boise State Univ.
or institution.			|     holmes@opal.idbsu.edu