## 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. ...

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 ...

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 $j:V\to M$ ...

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 ...

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 $\...

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 ...

Top 50 recent answers are included

#### Related Tags

large-cardinals × 756set-theory × 729

lo.logic × 363

forcing × 142

reference-request × 68

inner-model-theory × 52

model-theory × 51

descriptive-set-theory × 38

axiom-of-choice × 35

ct.category-theory × 24

self-distributivity × 21

mathematical-philosophy × 20

determinacy × 18

soft-question × 15

infinite-combinatorics × 15

constructibility × 14

universal-algebra × 13

ultrafilters × 13

continuum-hypothesis × 12

inner-models × 12

co.combinatorics × 11

axioms × 11

ordinal-numbers × 10

ultrapowers × 10

measure-theory × 9