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5 votes
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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 …
Elliot Glazer's user avatar
5 votes
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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 …
Elliot Glazer's user avatar
11 votes
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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 …
Elliot Glazer's user avatar