The relation between the limit cardinal $alpha$ and a sequence of cardinal numbers strictly less than $alpha$
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For a limit cardinal $alpha$ can we find a sequence of sets $(X_n)$ with $card X_1< card X_2<...< card X$and $card X= card X_1+ card X_2+...?$
set-theory
asked Jan 3 at 7:36
aliali
39218
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add a comment |
$begingroup$
For a limit cardinal $alpha$ can we find a sequence of sets $(X_n)$ with $card X_1< card X_2<...< card X$and $card X= card X_1+ card X_2+...?$
set-theory
asked Jan 3 at 7:36
aliali
39218
$endgroup$
2
$begingroup$
What do the $dots$ mean here? (Yes, I get it, an infinite sequence, but how long exactly?)
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– Asaf Karagila♦
Jan 3 at 8:17
add a comment |
$begingroup$
For a limit cardinal $alpha$ can we find a sequence of sets $(X_n)$ with $card X_1< card X_2<...< card X$and $card X= card X_1+ card X_2+...?$
set-theory
asked Jan 3 at 7:36
aliali
39218
$endgroup$
For a limit cardinal $alpha$ can we find a sequence of sets $(X_n)$ with $card X_1< card X_2<...< card X$and $card X= card X_1+ card X_2+...?$
set-theory
set-theory
asked Jan 3 at 7:36
aliali
39218
asked Jan 3 at 7:36
aliali
39218
asked Jan 3 at 7:36
aliali
39218
asked Jan 3 at 7:36
aliali
39218
asked Jan 3 at 7:36
aliali
39218
39218
2
$begingroup$
What do the $dots$ mean here? (Yes, I get it, an infinite sequence, but how long exactly?)
$endgroup$
– Asaf Karagila♦
Jan 3 at 8:17
add a comment |
2
$begingroup$
What do the $dots$ mean here? (Yes, I get it, an infinite sequence, but how long exactly?)
$endgroup$
– Asaf Karagila♦
Jan 3 at 8:17
2
2
$begingroup$
What do the $dots$ mean here? (Yes, I get it, an infinite sequence, but how long exactly?)
$endgroup$
– Asaf Karagila♦
Jan 3 at 8:17
$begingroup$
What do the $dots$ mean here? (Yes, I get it, an infinite sequence, but how long exactly?)
$endgroup$
– Asaf Karagila♦
Jan 3 at 8:17
add a comment |
1 Answer
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Your question suggests you are looking for a countable sequence; then the answer is: NO.
$aleph_{omega_1}$ is a limit cardinal, but not the sum of countably many smaller cardinals.
You may want to study the notion `cofinality of a cardinal number'.
answered Jan 3 at 9:25
hartkphartkp
1,30965
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add a comment |
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1 Answer
1
active
oldest
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1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Your question suggests you are looking for a countable sequence; then the answer is: NO.
$aleph_{omega_1}$ is a limit cardinal, but not the sum of countably many smaller cardinals.
You may want to study the notion `cofinality of a cardinal number'.
answered Jan 3 at 9:25
hartkphartkp
1,30965
$endgroup$
add a comment |
$begingroup$
Your question suggests you are looking for a countable sequence; then the answer is: NO.
$aleph_{omega_1}$ is a limit cardinal, but not the sum of countably many smaller cardinals.
You may want to study the notion `cofinality of a cardinal number'.
answered Jan 3 at 9:25
hartkphartkp
1,30965
$endgroup$
add a comment |
$begingroup$
Your question suggests you are looking for a countable sequence; then the answer is: NO.
$aleph_{omega_1}$ is a limit cardinal, but not the sum of countably many smaller cardinals.
You may want to study the notion `cofinality of a cardinal number'.
answered Jan 3 at 9:25
hartkphartkp
1,30965
$endgroup$
Your question suggests you are looking for a countable sequence; then the answer is: NO.
$aleph_{omega_1}$ is a limit cardinal, but not the sum of countably many smaller cardinals.
You may want to study the notion `cofinality of a cardinal number'.
answered Jan 3 at 9:25
hartkphartkp
1,30965
answered Jan 3 at 9:25
hartkphartkp
1,30965
answered Jan 3 at 9:25
hartkphartkp
1,30965
answered Jan 3 at 9:25
hartkphartkp
1,30965
1,30965
add a comment |
add a comment |
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2
$begingroup$
What do the $dots$ mean here? (Yes, I get it, an infinite sequence, but how long exactly?)
$endgroup$
– Asaf Karagila♦
Jan 3 at 8:17