Is it true that for every measurable cardinal

Emmy Knox

Emmy Knox

Answered question

2022-06-25

Is it true that for every measurable cardinal κ there is a normal, κ-complete, and non-principal ultrafilter on κ?

Answer & Explanation

Blaine Foster

Blaine Foster

Beginner2022-06-26Added 33 answers

Yes. Although this can be demonstrated in a more rigorous combinatorial manner, it's generally best to adhere to the proofs of
1. If κ is measurable, then κ is the critical point (i.e. least ordinal moved) of some nontrivial elementary embedding j : V  M where M is an inner model.
2. If j : V  M is a nontrivial elementary embedding with critical point κ ., then { X  κ : κ  j ( X ) } is a normal ultrafilter on κ .
All of this is in Jech (as is a proof not using the elementary embedding characterization).

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