Chang & Keisler Exercise 4.3.9
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I'm having a lot of trouble sorting out an exercise from Chang & Keisler's Model Theory:
4.3.9: Let $I$ be an infinite set of power $alpha$. If $E subset P(I)$, $|E| leq alpha$, and the filter generated by $E$ is uniform, then $E$ can be extended to an $alpha$-regular ultrafilter $D$ over $I$.
In chapter 4.3, CK proves that for any infinite $alpha$, there exists an $alpha$-regular ultrafilter on $alpha$. Their construction involves working with the set of finite subsets of $alpha$ and I tried mimicking that argument, but I'm not sure how to employ the given hypotheses. Any and all hints / suggestions are appreciated.
logic set-theory model-theory
asked Dec 19 '18 at 2:41
SpencerSpencer
2199
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add a comment |
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I'm having a lot of trouble sorting out an exercise from Chang & Keisler's Model Theory:
4.3.9: Let $I$ be an infinite set of power $alpha$. If $E subset P(I)$, $|E| leq alpha$, and the filter generated by $E$ is uniform, then $E$ can be extended to an $alpha$-regular ultrafilter $D$ over $I$.
In chapter 4.3, CK proves that for any infinite $alpha$, there exists an $alpha$-regular ultrafilter on $alpha$. Their construction involves working with the set of finite subsets of $alpha$ and I tried mimicking that argument, but I'm not sure how to employ the given hypotheses. Any and all hints / suggestions are appreciated.
logic set-theory model-theory
asked Dec 19 '18 at 2:41
SpencerSpencer
2199
$endgroup$
add a comment |
$begingroup$
I'm having a lot of trouble sorting out an exercise from Chang & Keisler's Model Theory:
4.3.9: Let $I$ be an infinite set of power $alpha$. If $E subset P(I)$, $|E| leq alpha$, and the filter generated by $E$ is uniform, then $E$ can be extended to an $alpha$-regular ultrafilter $D$ over $I$.
In chapter 4.3, CK proves that for any infinite $alpha$, there exists an $alpha$-regular ultrafilter on $alpha$. Their construction involves working with the set of finite subsets of $alpha$ and I tried mimicking that argument, but I'm not sure how to employ the given hypotheses. Any and all hints / suggestions are appreciated.
logic set-theory model-theory
asked Dec 19 '18 at 2:41
SpencerSpencer
2199
$endgroup$
I'm having a lot of trouble sorting out an exercise from Chang & Keisler's Model Theory:
4.3.9: Let $I$ be an infinite set of power $alpha$. If $E subset P(I)$, $|E| leq alpha$, and the filter generated by $E$ is uniform, then $E$ can be extended to an $alpha$-regular ultrafilter $D$ over $I$.
In chapter 4.3, CK proves that for any infinite $alpha$, there exists an $alpha$-regular ultrafilter on $alpha$. Their construction involves working with the set of finite subsets of $alpha$ and I tried mimicking that argument, but I'm not sure how to employ the given hypotheses. Any and all hints / suggestions are appreciated.
logic set-theory model-theory
logic set-theory model-theory
asked Dec 19 '18 at 2:41
SpencerSpencer
2199
asked Dec 19 '18 at 2:41
SpencerSpencer
2199
asked Dec 19 '18 at 2:41
SpencerSpencer
2199
asked Dec 19 '18 at 2:41
SpencerSpencer
2199
asked Dec 19 '18 at 2:41
SpencerSpencer
2199
2199
add a comment |
add a comment |
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