[FOM] Replacement after removal of Extensionality.

[Zuhair Abdul Ghafoor Al-Johar]

Dear Sirs,

The usual presentation of ZF with the customary form of Replacement weakens
down to Z once Extensionality is removed, this is a known result due to Dana Scott. 

If instead of Replacement one uses Separation and Collection schemata, then one gets
a theory that does interpret full ZF, even in absence of Extensionality. This subject 

had been investigated and a reference to those is present in this FOM posting: 

https://cs.nyu.edu/pipermail/fom/2015-April/018645.html 

I recently attempted to establish an interpretation of ZF in a theory that 
has axioms of Foundation, Pairing, Union, Power, Infinity, and Replacement* 
where the latter is the following: 

Replacement*: If phi is a formula in which only the symbols "x","z" occur free, 
and those only occurring free, and in which the symbol "B" never occur, then: 

forall A exists B forall y [y in B <-> exists x in A forall z (z in y <-> phi)] 

is an axiom. 

A sketch of the proof is posted here: 

https://mathoverflow.net/questions/289943/equivalents-of-replacement-under-removal-of-extensionality 

However, I came to realize that if one tries to use the customary way of presenting Replacement 
i.e. in a pre-conditioned manner, then for the same specifics of formulation presented above 
one can use the following formula: 

forall A([forall x in A exists z forall y (phi(x,y) -> y in z)] -> exists B forall y (y in B <-> exists x in A phi(x,y))) 

now this will imply the above schema, and it would itself directly prove Pairing, Union, 
and Separation. 

What changed is that the limiting factor on replacement became "membership in a set" 
instead of being "equality to a set". But this seems to be in some sense similar to 
Separation being just a limited naive comprehension where the limiting factor is again
being "membership in a set" imposed on the defining formula of naive comprehension. 

This had lead me back also to Dana Scott's result of "Z" being actually interpretable 
in just a modified form of separation and infinity, where the former is: 

forall A exists x forall y (y in x <-> y C A /\ phi) is an axiom, 
for every formula phi not having the symbol "x" occurring in it. 

So the limit to comprehension did change here from "membership in a set" to 
"subset-hood of a set"! 

I was just thinking if one can do the same thing with Replacement since the 
formulation of it that can withstand removal of extensionality do use "membership in a set" 
as a limiting factor, so we can upgrade the above axiom to: 

forall A([forall x in A exists z forall y (phi(x,y) -> y C z)] -> exists B forall y (y in B <-> exists x in A phi(x,y))) 

Then this is still a very able scheme, it straightforwardly proves Pairing, Power, 
Separation, and Replacement*, and it does prove Union for sets in which each element
is a member of some transitive set, and this will build up the stages of the cumulative
hierarchy and thus interpret ZF. 


The common matter between the two is the using of 'subset-hood' as a limiting factor
to comprehension replacing the "membership" criterion. 


So it appears that full ZF can be interpreted in a first-order theory in the language
of set theory that has one axiom scheme and a single axiom, namely that of a modified form of
Replacement, and of Infinity. 

The point that I want to raise here is that the above versions of replacement
are indeed about Replacement! The concept behind Replacement* (and the subsequent
versions presented above) is the same one underlying Replacement, and so the sameidea behind replacement can result in versions that can interpret full ZF in absence
of Extensionality. This shows that there is nothing substantial in "Collection" schema

that enables it to interpret ZF in absence of extensionality!
 


Zuhair