You are right about the absence of the analogue of HOD in second order arithmetic and Kelley-Morse theory of classes. However, this is not because of the failure of reflection (since both second order arithmetic and Kelley-Morse theory of classes, similar to ZF, exhibit reflection phenomena), but rather it is because of the fact that the objects that “reflect” (i.e., classes), in contradistinction with the rank initial segments of the universe in ZF, are not provably of the order type of the class of ordinals.

On the other hand, there are extensions of second order arithmetic and Kelley-Morse theory of classes that incorporate an axiom that asserts that all classes are constructible (intuitively: V = L holds). These extensions support a global definable well-ordering [and in particular, they verify the statement “every nonempty definable class has a definable member”].

]]>Also, let me point out that the analogue of Theorem 1 (particularly: the equivalence between 2 and 3) fails when ZF is replaced by Z_2 (Second Order Arithmetic); as recently shown by Kanovei & Lyubetsky in the paper: https://arxiv.org/abs/1702.03566. Perhaps their machinery can be extended to KM (Kelley-Morse).

]]>I find this new result unusual, since often one finds that results generalize from arithmetic to models of $V=\text{HOD}$, rather than models of $V\neq\text{HOD}$. Perhaps it suggests a fully positive answer may be possible.

]]>The winning strategy in misère nim is to play exactly the same as in regular nim, by always moving to balance the position in the way I had described, except when the position would have all piles of height only one; in this special case, then make sure to leave an odd number of such piles, rather than an even number. So optimal play in misère nim is almost exactly the same as in regular nim, until the very end of the game, when all piles have height one.

]]>Thanks for your comment, which I have just now noticed.

I think your objection is totally right, and that proof is broken. Although the Henkin assertions make for a conservative extension, since the constants can be reinterpreted, nevertheless we cannot ensure that they are interpreted the same in $N$ as in $M$, since perhaps new witnesses are called for, and this will affect the $\Sigma_2$ assertions made in the theory.

In a sense, this might be good news, since I would like a positive answer to my question!

Back to the drawing board…

]]>Here is my reasoning: your proof (of the last theorem) will verbatim go through if we replace ZFC by PA, and Σ_2 by Σ_1. This would then result in a countable model M of PA such that for any end extension N of M, N is a Sigma_1 elementary end extension of M; but the existence of such a model M contradicts Woodin’s theorem discussed in your previous blogpost. Indeed, it was been long known [Wilkie 1977] that for every nonstandard model M of PA there is a model N that end extends M such that Th(M) = Th(N) and yet M is not a Σ_1 elementary submodel of N.

I think the gap in the proof (of your last theorem) is in the last paragraph, where it reads:

“At that stage, we had already added finitely many Σ_2 assertions ψ(d) to the theory, and so these assertions are true in M, and therefore also in N”.

But at that stage we had also committed (via Henkin axioms) to the *negations* of some Σ_2 assertions, such negations will of course hold in M, but not necessarily in N.

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