# [FOM] Plea for literature on recursion theory in V_\omega

Ali Enayat ali.enayat at gmail.com
Wed Apr 30 18:08:32 EDT 2008

This is a reply to Csaba Henk's query, who has asked for reference on:

> [snip] Or any book which discusses the well-known
> semantic equivalence (yes, this is a vague term, but you should know
> what I mean...) of <\omega, + , *> and <V_\omega, \epsilon> ?

1. The most recent paper on the above subject, with many references to
the literature, is the following:

Kaye, Richard and Wong, Tin Lok
On interpretations of arithmetic and set theory.
Notre Dame J. Formal Logic 48 (2007), no. 4, 497--510.

Note: you can find a more complete version of the paper below:
http://web.mat.bham.ac.uk/R.W.Kaye/papers/finitesettheory/

The above paper corrected the misconception that PA is
bi-interpretable with "finitistic ZF set theory" ZF_fin, defined via

ZF_fin: = ZF \ {Infinity} + {the negation of Infinity}.

More explicitly, Kaye and Wong show [by analysing the Ackermann
interpretation, and building an inverse for it] that PA is
bi-interpretable with ZF_fin + TC, where:

TC:= "every set has a transitive closure"

[indeed they show the stronger statement that the above two theories
are definitionally equivalent]. Note that it has long been known that
ZF_fin does not prove TC.

In a recent paper (almost ready for public dissemination) with Jim
Schmerl and Albert Visser, we have shown, among other things, that
even a modest form of bi-interpretability does not hold between PA and
ZF_fin.

2. You may also wish to consult the following text:

Mayberry, J. P.
The foundations of mathematics in the theory of sets.
Encyclopedia of Mathematics and its Applications, 82. Cambridge
University Press, Cambridge, 2000.

Best regards,

Ali Enayat