7 votes

Does every ordinal appearing in a model of ZF appear in a model of ZFC?

As Noah mentioned in the comments, the answer to the first question is "yes". The answer to the second question is consistently "no", assuming a little consistency: if there is a ...
  • 6,549
3 votes

When can interpretations be "optimized"?

The answer to Question 2 is negative and the answer to Question 1 is positive. In fact the usual interpretation $\Phi$ of $\mathsf{PA}$ in $\mathsf{ZFC}$ is not even weakly optimal. Indeed, consider ...
2 votes

Equational theories determined by "identities without variables"

This question was edited on March 12, 2023 and I was asked to comment on the edited form. Let me copy the essential part of the new question: I believe $\ldots$ that a variety $\mathscr V$ is of the ...
1 vote

Need proof on a model being elementarily equivalent but non-isomorphic

To show that the models are not isomorphic assume an isomorphism $f:\frak{A}\to\frak{B}$, take $(a_n)_{n\in\omega}$ be an increasing sequence such $a_i,a_j$ are from a different copy of $\Bbb Z$ for ...
  • 1,097
1 vote

Is there a metamathematical $V$?

This may be more in line with Vincent R.B. Blazy's answer, but one thing I would like to add is the principle of inexhaustibility. This principle appears in Maddy's "Believing the Axioms. I",...
  • 482
1 vote

dp-minimality and stability

This comes a bit late. I just wanted to mention that an interesting family of examples are superflat graphs, proved to be stable by Podewski and Ziegler. See "Computations of Vapnik-Chervonenkis ...

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