# Questions tagged [large-cardinals]

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### Absoluteness and the scale property for $Π^2_2$ or $Σ^2_2$

Under the diamond principle $◊$ and large cardinal axioms, which of the two pointclasses $Π^2_2$ or $Σ^2_2$ is expected to have the scale property? Because conditional $Σ^2_2$ absoluteness under $◊$ ...
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### How to understand the interface of the consistency strength hierarchy, reverse mathematics, and proof-theoretic ordinal analysis?

I am aware of three major "hierarchies" of mathematical theories, but I don't know how to relate these hierarchies to one another. Here are the hierarchies I have in mind: Consistency strength. My ...
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### Proving independence with large cardinals?

Suppose I want to prove some statement S is independent of ZFC. Now instead of the usual approach of making models, I do the following: - Take two large cardinal axioms L1 and L2 - Prove that ZFC + L1 ...
914 views

### “Bootstrapping” an unbounded class of inaccessible cardinals

The "richness principle" of set theory asserts roughly that "everything that happens once should happen an unbounded number of times". An example would be the existence of an unbounded class of ...
156 views

### Stationary sets and $\kappa$-complete normal ultrafilters

Let $\kappa$ be a measurable cardinal, and let $u$ be a normal $\kappa$-complete ultrafilter over $\kappa$. It is a standard easy fact that every closed unbounded set must belong to $u$ (notice that ...
196 views

### Uniqueness of countable version of $L[U]$?

Suppose $\alpha$ is a countable ordinal and $U_0,U_1,\kappa$ are such that $L_\alpha[U_i] \models \mathrm{ZFC} + U_i$ is a normal ultrafilter on $\kappa$. Does $U_0 = U_1$? The argument for ...
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### What is the strength of claiming that the class of all $V_\kappa$ stages that are $H_\kappa$ when $\kappa$ is regular, is inaccessible?

[EDIT] This posting had been edited to assert that we are speaking about regular mutual stages. Let $H_{\kappa}$ be the set of all sets that are hereditarily strictly smaller in cardinality than ...