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Noah Schweber
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This is really a comment, but it's too long:

I'm not entirely sure what you're asking, but I think there are a couple things worth pointing out:

  • In terms of what a model $M$ "knows" about a structure Ackermann-coded into one of its subsets $A$, note that this question best makes sense if $M$ knows about $A$ to begin with. But most of the time, the $A$s we'll be interested in won't be definable in $M$: indeed, no model $W$ of ZF can be Ackermann-coded into any nonstandard model $M$ of PA in a definable way (since such an encoding would give evidence in $M$ of the non-standardness of whatever element of $M$ is coding the $\omega$ of $W$). So you have to be very careful when you ask "What does $M$ know?" about such a situation - what exactly do you mean, given that the most basic information about such a configuration will generally be undefinable in $M$?

  • In terms of Parikh's model, I think you're overestimating its relevance to Ackermann coding. Ackermann coding relies only on being able to prove that binary expansions of natural numbers exist and are unique; this is ultimately only a statement about the map $x\mapsto 2^x$. This is legal, since - in any model $M$ of PA - we have a definable exponentiation map, which PA proves satisfies the recursive properties of exponentiation. The possible existence of other maps which satisfy the recursive properties of exponentiation isn't relevant, since those maps won't be definable: any two definable maps satisfying the recursive properties of exponentiation must be the same, by induction.

Noah Schweber
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