MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

3 switched two words which were in the wrong order

Models of ZF-Infinity that arise from models of PA via binary bits - a method first introduced by Ackermann in 1940 to interpret arithmetic in 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:

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.

2 Improved formatting and added a postscript

Models of ZF-Infinity that arise from models of PA via binary bits - a method first introduced by Ackermann in 1940 to interpret arithmetic in set theory - 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} 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:

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.

1

Models of ZF-Infinity that arise from models of PA via binary bits - a method first introduced by Ackermann in 1940 to interpret arithmetic in set theory - 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: