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8 votes
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Concept of bedrock and mantle in the multiverse view in the philosophy of mathematics

The confusion seems to arise from your statement "the mantle of $V$ which is the smallest ground for $V$." But this not quite right. In a bottomless model of ZFC, the mantle is not a ground. ...
Joel David Hamkins's user avatar
14 votes

A Löwenheim–Skolem–Tarski-like property

Here is a lower bound that improves Joel's by a bit. If the reflection property of the post holds at $\kappa$ then there is some measurable $\lambda<\kappa$ with $$V_\lambda\models``\text{there is ...
Andreas Lietz's user avatar
16 votes

A Löwenheim–Skolem–Tarski-like property

Here is an upper bound: Suppose $\kappa$ is $2$-fold supercompact. Then the property holds at $\kappa$. (Recall that $2$-fold supcompactness means that for each ordinal $\lambda$, there is $j:V\to M$ ...
Farmer S's user avatar
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16 votes
Accepted

A Löwenheim–Skolem–Tarski-like property

Let me improve somewhat on Farmer's lower bound. Theorem. If there is a cardinal $\kappa$ with the stated reflection property, then there are many measurable cardinals, measurable cardinals of very ...
Joel David Hamkins's user avatar
17 votes

A Löwenheim–Skolem–Tarski-like property

Here's a counterexample for $\kappa=\aleph_1$: let $B$ be the structure with underlying set $\mathbb{N}\sqcup\mathcal{P}(\mathbb{N})$, equipped with the usual ordering on $\mathbb{N}$ as well as the $\...
Noah Schweber's user avatar
21 votes
Accepted

Are Berkeley cardinals easier to refute in ZFC than Reinhardt cardinals?

Yes, it is easier to refute Berkeleys than Reinhardts. There is a very simple refutation of Berkeleys in ZFC that is due to Woodin. It is part of the motivation for his contention that Berkeley ...
Gabe Goldberg's user avatar

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