Is my understanding of this proof about cardinality correct?
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In my textbook Introduction to Set Theory by Hrbacek and Jech, there is a theorem:
and its corresponding proof:
I would like to ask if my understanding of the proof in case $color{blue}{aleph_beta < operatorname{cf}(aleph_alpha)}$ is correct.
Let $S={X subseteq omega_alpha mid |X|=aleph_beta}$. We next prove that $Xin S implies X$ is bounded. Assume the contrary that there exists $X' in S$ such that $X'$ is unbounded and thus $sup X'=omega_alpha$. Let $(delta_xi mid xi<lambda)$ be an increasing enumeration of $X'$. It follows that $|lambda|=|X'|=aleph_beta$ and $lim_{xi to lambda}delta_xi=sup X'=omega_alpha$. By definition of cofinality, $operatorname{cf}(aleph_alpha) le lambda$.
We have $operatorname{cf}(aleph_alpha) le lambda implies |operatorname{cf}(aleph_alpha)| le |lambda| implies operatorname{cf}(aleph_alpha) le |lambda| = aleph_beta implies operatorname{cf}(aleph_alpha) le aleph_beta$. This contradicts the fact that $aleph_beta < operatorname{cf}(aleph_alpha)$.
Thank you for your help!
elementary-set-theory cardinals ordinals
asked Dec 29 '18 at 11:33
Le Anh DungLe Anh Dung
1,2131621
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add a comment |
$begingroup$
In my textbook Introduction to Set Theory by Hrbacek and Jech, there is a theorem:
and its corresponding proof:
I would like to ask if my understanding of the proof in case $color{blue}{aleph_beta < operatorname{cf}(aleph_alpha)}$ is correct.
Let $S={X subseteq omega_alpha mid |X|=aleph_beta}$. We next prove that $Xin S implies X$ is bounded. Assume the contrary that there exists $X' in S$ such that $X'$ is unbounded and thus $sup X'=omega_alpha$. Let $(delta_xi mid xi<lambda)$ be an increasing enumeration of $X'$. It follows that $|lambda|=|X'|=aleph_beta$ and $lim_{xi to lambda}delta_xi=sup X'=omega_alpha$. By definition of cofinality, $operatorname{cf}(aleph_alpha) le lambda$.
We have $operatorname{cf}(aleph_alpha) le lambda implies |operatorname{cf}(aleph_alpha)| le |lambda| implies operatorname{cf}(aleph_alpha) le |lambda| = aleph_beta implies operatorname{cf}(aleph_alpha) le aleph_beta$. This contradicts the fact that $aleph_beta < operatorname{cf}(aleph_alpha)$.
Thank you for your help!
elementary-set-theory cardinals ordinals
asked Dec 29 '18 at 11:33
Le Anh DungLe Anh Dung
1,2131621
$endgroup$
$begingroup$
Yes, your understanding is correct
$endgroup$
– Holo
Dec 29 '18 at 12:17
$begingroup$
Thank you so much for your verification @Holo!
$endgroup$
– Le Anh Dung
Dec 29 '18 at 12:40
add a comment |
$begingroup$
In my textbook Introduction to Set Theory by Hrbacek and Jech, there is a theorem:
and its corresponding proof:
I would like to ask if my understanding of the proof in case $color{blue}{aleph_beta < operatorname{cf}(aleph_alpha)}$ is correct.
Let $S={X subseteq omega_alpha mid |X|=aleph_beta}$. We next prove that $Xin S implies X$ is bounded. Assume the contrary that there exists $X' in S$ such that $X'$ is unbounded and thus $sup X'=omega_alpha$. Let $(delta_xi mid xi<lambda)$ be an increasing enumeration of $X'$. It follows that $|lambda|=|X'|=aleph_beta$ and $lim_{xi to lambda}delta_xi=sup X'=omega_alpha$. By definition of cofinality, $operatorname{cf}(aleph_alpha) le lambda$.
We have $operatorname{cf}(aleph_alpha) le lambda implies |operatorname{cf}(aleph_alpha)| le |lambda| implies operatorname{cf}(aleph_alpha) le |lambda| = aleph_beta implies operatorname{cf}(aleph_alpha) le aleph_beta$. This contradicts the fact that $aleph_beta < operatorname{cf}(aleph_alpha)$.
Thank you for your help!
elementary-set-theory cardinals ordinals
asked Dec 29 '18 at 11:33
Le Anh DungLe Anh Dung
1,2131621
$endgroup$
In my textbook Introduction to Set Theory by Hrbacek and Jech, there is a theorem:
and its corresponding proof:
I would like to ask if my understanding of the proof in case $color{blue}{aleph_beta < operatorname{cf}(aleph_alpha)}$ is correct.
Let $S={X subseteq omega_alpha mid |X|=aleph_beta}$. We next prove that $Xin S implies X$ is bounded. Assume the contrary that there exists $X' in S$ such that $X'$ is unbounded and thus $sup X'=omega_alpha$. Let $(delta_xi mid xi<lambda)$ be an increasing enumeration of $X'$. It follows that $|lambda|=|X'|=aleph_beta$ and $lim_{xi to lambda}delta_xi=sup X'=omega_alpha$. By definition of cofinality, $operatorname{cf}(aleph_alpha) le lambda$.
We have $operatorname{cf}(aleph_alpha) le lambda implies |operatorname{cf}(aleph_alpha)| le |lambda| implies operatorname{cf}(aleph_alpha) le |lambda| = aleph_beta implies operatorname{cf}(aleph_alpha) le aleph_beta$. This contradicts the fact that $aleph_beta < operatorname{cf}(aleph_alpha)$.
Thank you for your help!
elementary-set-theory cardinals ordinals
elementary-set-theory cardinals ordinals
asked Dec 29 '18 at 11:33
Le Anh DungLe Anh Dung
1,2131621
asked Dec 29 '18 at 11:33
Le Anh DungLe Anh Dung
1,2131621
asked Dec 29 '18 at 11:33
Le Anh DungLe Anh Dung
1,2131621
asked Dec 29 '18 at 11:33
Le Anh DungLe Anh Dung
1,2131621
asked Dec 29 '18 at 11:33
Le Anh DungLe Anh Dung
1,2131621
1,2131621
$begingroup$
Yes, your understanding is correct
$endgroup$
– Holo
Dec 29 '18 at 12:17
$begingroup$
Thank you so much for your verification @Holo!
$endgroup$
– Le Anh Dung
Dec 29 '18 at 12:40
add a comment |
$begingroup$
Yes, your understanding is correct
$endgroup$
– Holo
Dec 29 '18 at 12:17
$begingroup$
Thank you so much for your verification @Holo!
$endgroup$
– Le Anh Dung
Dec 29 '18 at 12:40
$begingroup$
Yes, your understanding is correct
$endgroup$
– Holo
Dec 29 '18 at 12:17
$begingroup$
Yes, your understanding is correct
$endgroup$
– Holo
Dec 29 '18 at 12:17
$begingroup$
Thank you so much for your verification @Holo!
$endgroup$
– Le Anh Dung
Dec 29 '18 at 12:40
$begingroup$
Thank you so much for your verification @Holo!
$endgroup$
– Le Anh Dung
Dec 29 '18 at 12:40
add a comment |
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$begingroup$
Yes, your understanding is correct
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– Holo
Dec 29 '18 at 12:17
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Thank you so much for your verification @Holo!
$endgroup$
– Le Anh Dung
Dec 29 '18 at 12:40