[FOM] Embedding Z (or other ZFC fragments) in HOL

[Rupert McCallum]

Z proves Ex(x=omega)&Ax(Ey y=~Px). For each positive integer n, it is not true in general that an n-large type would satisfy this theorem.
 

     On Thursday, 8 January 2015, 7:03, Mario Carneiro <di.gama at gmail.com> wrote:
   

 Hello all,

I'm trying to locate any research done on what kinds of subsystems of ZFC can be embedded into higher order logic. My initial approach on the subject leads me to believe that HOL can embed any proof in Z, using the following method:

HOL contains types of cardinality om (omega), ~P om, ~P ~P om, etc for any finite number of ~P's (the powerset operation). Since any particular proof in Z will use the theorems "om e. V" (omega exists) and "x e. V -> ~P x e. V" (a powerset exists) finitely many times, you could define a function on types (does such a thing exist?) saying that a type A is "n-large" meaning there is an injection from ~P ~P ... ~P om to A (where there are n ~P symbols), and then a proof in Z would get V mapped to an arbitrary type A, and if there are n ~P's used in the proof you would preface the entire proof with "A is n-large". When you are done and ready to map the model back to the regular HOL notions, you replace A with ind->bool->...->bool (n times), and then prove that this type is n-large to discharge the extra assumption.

Does anyone know of any papers or work done in the direction of embedding fragments of ZFC in HOL like this?

Mario Carneiro

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