Logic Public Service Announcement!

[danman3459]

Reminder: Both the Generalized Continuum Hypothesis and Trichotomy imply the Axiom of Choice.

Comments

[noetickerf]

They're all equivalent in ZF, right? Though that could be a reason for wondering about ZF...

[danman3459]

No, these are not all equivalent. GCH implies AC, and T is equivalent to AC, but AC does not imply GCH. I believe this is provable in all standard formulations of set theory (i.e. just using naive notions).

[noetickerf]

Sorry, braindead. Of course AC does not imply GC. I meant the well-ordering theorem, which is what I'm actually concerned about...

[danman3459]

ah, yes, WO is equivalent to AC also. I think that is provable in a naive theory too.

[paresink]

I am doing research for my university thesis, thanks for your great points, now I am acting on a sudden impulse.