ZFCfin (ZFC without the axiom of infinity, plus its negation) is biinterpretable with Peano Arithmetic.

Each countable model of PA has what is known as its "standard system": the collection of sets of standard natural numbers which can be coded in that system. Usually the coding is via prime exponentiation: $n$ is in the set coded by $c$ if the $n^{th}$ standard prime divides $c$. The standard system of the standard model has only finite sets of naturals in it; all models with infinite sets in their standard system are nonstandard models.

Nonstandard models of ZFCfin (those not isomorphic to HF) contain "externally infinite" sets; elements of the model for which the number of elements standing in the model's member-of relation to them is greater than any finite number.

Are these "externally infinite" sets of a nonstandard model of ZFCfin somehow connected to the standard system of a nonstandard model of HF?

I ask because I'm starting to work through the literature on nonstandard models of PA and I'm finding it much easier to think of the nonstandard numbers as externally-infinite sets of ZFCfin. I'm wondering how far astray that intuition is likely to lead me.

Edit: I should add that I'm aware the nonstandard sets of ZFCfin are frequently $\epsilon$-illfounded. I still find them easier to think of than infinite numbers (frankly I always thought it strange to rule out such sets in the first place).