As is well known, the following theory is equiconsistent with $PA$:

$ZFC$ with the axiom of infinity replaced by its negation.

Since this theory is equiconsistent with $PA$, it would seem reasonable to infer (wouldn't it?) that the consistency of '$ZFC$ with the axiom of infinity replaced by its negation' could be provable in "$PRA$ + $TI({\epsilon_0})$.

So what 'theory of sets'(?) is mutually interpretable with "$PRA$ + $TI({\epsilon_0})$? Also, can one define a notion of forcing in the aforementioned theory?

(If this seems too silly a question, please let me know and I will delete....)

  • 2
    $\begingroup$ What do the underlines and single quote marks mean? Or can we delete them? $\endgroup$ – Matt F. Dec 20 '16 at 15:20
  • $\begingroup$ @MattF.: Feel free to edit according to whatever the standard form for writing $T$ + $Axiom$ is, where $T$ is some formal theory. Thanks for the help. $\endgroup$ – Thomas Benjamin Dec 20 '16 at 15:38
  • $\begingroup$ ZFC with the axiom of infinity replaced by its negation is not a fragment of ZFC (it is even inconsistent with ZFC). A fragment of ZFC equiconsistent with PA is ZFC with the axiom of infinity dropped. $\endgroup$ – Emil Jeřábek Dec 20 '16 at 15:44
  • $\begingroup$ @EmilJeřábek: You are, of course, correct. Please let me know if the corrections are suitable, and if not, what would be, and I will re-edit. Thanks. $\endgroup$ – Thomas Benjamin Dec 20 '16 at 15:54
  • $\begingroup$ Yes, this is fine. $\endgroup$ – Emil Jeřábek Dec 20 '16 at 16:12

Yes, the consistency of "ZFC with the axiom of infinity replaced by its negation" is provable in "PRA + TI($\epsilon_0$)". Technically one has to also show that "PRA + TI($\epsilon_0$)" can prove the equiconsistency result (since it already proves the consistency of PA), but these are fairly natural theories so that shouldn't be a problem.

I'm not aware of a natural fragment of ZFC equiconsistent with "PRA + TI($\epsilon_0$)", and can't think of any reason there would be one. It's sort of remarkable that ZFC has a fragment which matches up perfectly with PA (which is mostly a reflection of how PA is a very natural theory).

  • $\begingroup$ Thanks. Is it also possible to define notions of forcing in $PRA$ + $TI({\epsilon_0})$ over the ground model "ZFC with the axiom of infinity replaced by its negation"? Also, since it's possible to interpret $Q$ in terms of sets and $PRA$ is an extension of $Q$, has anyone used the set interpretation of $Q$ to derive a set interpretation of $PRA$ + $TI({\epsilon_0})$? $\endgroup$ – Thomas Benjamin Dec 21 '16 at 11:22
  • $\begingroup$ @ThomasBenjamin: It's surely possible to define some kind of notion of forcing in PRA+TI($\epsilon_0$) (or even just PRA), but I'm not aware of any work doing this, nor am I aware of work on set interpretations of TI($\epsilon_0$). $\endgroup$ – Henry Towsner Dec 21 '16 at 16:30

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