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Results tagged with cardinal-arithmetic
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user 109573
11
votes
1
answer
491
views
Does $\text{AC}_{\text{WO}}$ prove $\Theta \neq \aleph_{\omega+1}?$
The choice principle $\text{AC}_{\text{WO}}$ proves a large amount of cardinal arithmetic. It's well-known to imply DC, that successor cardinals are regular, and that for all $X$, there is $\lambda$ s …
- 4,316
5
votes
Accepted
Does $\text{AC}_{\text{WO}}$ prove $\Theta \neq \aleph_{\omega+1}?$
($\text{ZF + AC}_{\text{WO}}$) For any cardinals $\kappa_1, \kappa_2,$ there is $\lambda$ such that $\aleph(^{\kappa_2}\kappa_1)=\lambda^+$ and $\text{cf}(\lambda)>\kappa_2.$
Pf: Let $\lambda$ be such …
- 4,316
5
votes
Accepted
A surjection from square onto power: Is limit Hartogs/Lindenbaum number necessary?
In the Feferman-Levy model $M$ for $\mathbb{R}$ being a countable union of countable sets, there is $X$ with $|\mathcal{P}(X)| =^* |X^2|$
and $\aleph(X)=\aleph^*(X)=\omega_1.$
In this model, we can ex …
- 4,316