# Tag Info

Accepted

### Is there a form of choice that can elude Kunen's inconsistency theorem?

Work of Usuba combined with work of Woodin shows that if there is a Reinhardt cardinal $\kappa$ that is a limit of Lowenheim-Skolem cardinals, then there is a forcing extension in which $\kappa$ ...
• 5,796

### Statements in differential geometry independent from ZFC

[Using the comments for context on undecidability/independence of ZFC] A computably undecidable problem is whether or not a homology sphere has a metric of positive scalar curvature [Page 79 of ...
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• 15.7k
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### Can the axiom of choice be proved with ZF+Tarski axiom?

Following the link found in the Wikipedia article about the Tarski–Grothendieck set theory, the required proof (by Tarski himself!) can be found beginning on p.181 of his article "On the well-...
• 5,005

### Does Urysohn's Lemma imply Dependent Choice?

In Versions of normality and some weak forms of the axiom of choice Paul Howard et al exhibit a model of MC (Multiple Choice) and not-DC, see page 381. In that model Urysohn's Lemma (NU) holds, so it ...
• 8,365

### Minimum transitive models and V=L

This is not a full answer, but I found it interesting to notice that if we relax the c.e. requirement somewhat, then there is a sweeping positive answer. Theorem. Every complete theory extending ZFC + ...
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### Bernstein's proof of the continuum hypothesis

This is really a long comment. This paper has been reviewed twice by zbMATH: one by H. B. Curry, which is not informative; another by W. Ackermann, which is in German. The following is the (...
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### Consequences of foundation/regularity in ordinary mathematics (over ZF–AF)?

I think Frucht's theorem, the statement "every group is isomorphic to the automorphism group of a graph", is a great example of what you're looking for; from the statement, it'd be hard to ...
• 106
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### Minimum transitive models and V=L

Yes, I claim you can in fact get one whose minimum model is a set forcing extension of a segment of $L$. Let $L_\alpha$ be least modelling ZFC. Let $\mathbb{P}=\mathbb{P}^{L_\alpha}$ be Jensen's ...
• 6,154
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Accepted

### Can we have this sequence where choice fails and returns?

Sure. Start with countably many inaccessible cardinals, $\kappa_n$, and now take the full support product adding $\kappa_n^+$ subsets to each $\kappa_n$. Then the $n$th model is the symmetric ...
• 36.6k
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### Is ordinal definability in terms of stages of cumulative size hierarchy equivalent to the usual one?

The answer is yes, in a very general way. What I claim, first, is that the Lévy-Montague reflection theorem holds in ZF for any definable continuous cumulative hierarchical representation of the set-...
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### Elementary countable submodels in Gödel's universe

Very clearly not. Take some countable elementary submodel $M_0$ of $L_{\omega_2}$, and take $M_1$ to be another one, but with $M_1$ a end extension of $M_0$. We can find such models by first finding ...
• 36.6k
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### Are there premice that are $\omega_1$-iterable but not $(\omega_1+1)$-iterable?

It is consistent (relative to large cardinals). There is an example given in Example 3.6 here. For a brief summary: the model is the minimal proper class mouse $S$ such that $\mathbb{R}^S$ is closed ...
• 6,154

### Complexity of infinitary satisfiability, part 2

Here is a partial answer; it deals with those admissibles $\kappa$ large enough to see that $\mathrm{cof}(|\kappa|)=\omega$ and such that $L_\kappa$ has largest cardinal $\theta$. That is, let $\kappa$...
• 6,154

### Can we iteratively reflect on self elementary embeddable stages of the cumulative hierarchy?

Any model of $ZF$+stationary proper class of $I3$ ordinals (whose consistency strength is at most $ZF$+$I2$) where $W_α$ is listing of $V_{κ}$ where $κ$ is $I3$ and $j_α$ the witness of $I3(V_κ)$ will ...
• 708
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• 15.1k