It is well known how the intended model and [how the (countable) non-standard models of arithmetic look like](http://en.wikipedia.org/wiki/Non-standard_model_of_arithmetic#Structure_of_countable_non-standard_models).


It's also well known how the intended model of set theory with the axiom of infinity replaced by its negation (ZF-Inf) looks like: $\langle V\_\omega;\in\rangle$, the hereditarily finite sets with the $\in$-relation.



> 
> But how do (countable) non-standard
>  models of ZF-Inf look like?
> 
> 
> 



Non-standard models of finite set theory
Models of ZF-Infinity that arise from models of PA via binary bits - a method first introduced by Ackermann in 1940 to interpret set theory in arithmetic- end up satisfying the statement TC := "every set has a transitive closure". 


It is known that the strengthened theory ZF-Infinity+TC is bi-interpretable with PA, which in particular means that every model of ZF-Infinity+TC is an "Ackermann model" of a model of PA.


However, TC is essential: there are models of ZF-Infinity that do NOT satisfy TC; and therefore such models cannot arise via Ackermann coding on a model of PA. 


It is also known that there are "lots of" nonstandard model of ZF-Infinity [i.e., models not isomorphic to the intended model $V\_{\omega}$] that are ${\omega}$-models [i.e., they have no nonstandard integer]. 


It is possible for a nonstandard ${\omega}$-model of ZF-Infinity to have a computable epsilon relation. Indeed, there is an analogue of Tennenbaum's theorem here: all computable models of ZF-Infinity are ${\omega}$-models.


For more detail on the above, and references on the subject of finite set theory, you can consult the following paper:


<http://academic2.american.edu/~enayat/ESV%20%28May19,2009%29.pdf>


Ali Enayat


PS. In light of the comments about TC to my posting, it is worth pointing out that even though TC is not provable in ZF-Infinity, the theory ZF-Infinity is "smart enough" to interpret ZF-Infinity + TC via the inner model of sets whose transitive closure exists as a set [as opposed to a definable class; cf. the aforementioned paper for more detail]. 


Therefore the relation of TC to ZF-Infinity is analogous to the relation between Foundation (Regularity) to ZF without Foundation since ZF is interpretable in ZF without Foundation via an inner model.


