New answers tagged large-cardinals
8
votes
Accepted
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. ...
- 218k
14
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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 ...
- 1,688
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 $j:V\to M$ ...
- 7,359
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 ...
- 218k
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 $\...
- 18.9k
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 ...
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