Why are infinite cardinals limit ordinals?
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My book states this as obvious, but it isn't so trivial to me. thanks
elementary-set-theory cardinals ordinals
edited Feb 16 '13 at 16:38
Asaf Karagila♦
309k33441775
asked Feb 16 '13 at 16:19
mrksmrks
412
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add a comment |
$begingroup$
My book states this as obvious, but it isn't so trivial to me. thanks
elementary-set-theory cardinals ordinals
edited Feb 16 '13 at 16:38
Asaf Karagila♦
309k33441775
asked Feb 16 '13 at 16:19
mrksmrks
412
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2
$begingroup$
Can you come up with a bijection between $alpha$ and $alpha + 1$, where $alpha$ is an infinite ordinal?
$endgroup$
– Alexander Thumm
Feb 16 '13 at 16:22
add a comment |
$begingroup$
My book states this as obvious, but it isn't so trivial to me. thanks
elementary-set-theory cardinals ordinals
edited Feb 16 '13 at 16:38
Asaf Karagila♦
309k33441775
asked Feb 16 '13 at 16:19
mrksmrks
412
$endgroup$
My book states this as obvious, but it isn't so trivial to me. thanks
elementary-set-theory cardinals ordinals
elementary-set-theory cardinals ordinals
edited Feb 16 '13 at 16:38
Asaf Karagila♦
309k33441775
asked Feb 16 '13 at 16:19
mrksmrks
412
edited Feb 16 '13 at 16:38
Asaf Karagila♦
309k33441775
asked Feb 16 '13 at 16:19
mrksmrks
412
edited Feb 16 '13 at 16:38
Asaf Karagila♦
309k33441775
edited Feb 16 '13 at 16:38
Asaf Karagila♦
309k33441775
edited Feb 16 '13 at 16:38
Asaf Karagila♦
309k33441775
309k33441775
asked Feb 16 '13 at 16:19
mrksmrks
412
asked Feb 16 '13 at 16:19
mrksmrks
412
asked Feb 16 '13 at 16:19
mrksmrks
412
412
2
$begingroup$
Can you come up with a bijection between $alpha$ and $alpha + 1$, where $alpha$ is an infinite ordinal?
$endgroup$
– Alexander Thumm
Feb 16 '13 at 16:22
add a comment |
2
$begingroup$
Can you come up with a bijection between $alpha$ and $alpha + 1$, where $alpha$ is an infinite ordinal?
$endgroup$
– Alexander Thumm
Feb 16 '13 at 16:22
2
2
$begingroup$
Can you come up with a bijection between $alpha$ and $alpha + 1$, where $alpha$ is an infinite ordinal?
$endgroup$
– Alexander Thumm
Feb 16 '13 at 16:22
$begingroup$
Can you come up with a bijection between $alpha$ and $alpha + 1$, where $alpha$ is an infinite ordinal?
$endgroup$
– Alexander Thumm
Feb 16 '13 at 16:22
add a comment |
2 Answers
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Recall that a cardinal is an initial ordinal, an ordinal which cannot be put in bijection into a smaller ordinal. Thanks to the Cantor-Bernstein theorem it is enough to show that there is no injection into a smaller ordinal instead.
If an infinite ordinal $alpha$ is not a limit ordinal then there is some $beta$ such that $alpha=betacup{beta}$.
The map $gcolonalphatobeta$ defined as: $$g(x)=begin{cases} 0 & x=beta\x+1 & x<omega\x & text{otherwise}end{cases}$$
is injective from $alpha$ into a smaller ordinal, and therefore $alpha$ cannot be an initial ordinal, that is a cardinal.
answered Feb 16 '13 at 16:38
Asaf Karagila♦Asaf Karagila
309k33441775
$endgroup$
add a comment |
$begingroup$
They are ordinals by definition, because we choose to use initial ordinals as the canonical representative of each class of sets-with-the-same-cardinality. This choice could in principle have been made differently, but it turns out to be a useful convention (though it only works in when we have the Axiom of Choice).
They are limit ordinals, because (except for 0) limit ordinals are precisely the ordinals that are not successor ordinals. And an infinite successor ordinal always has the same cardinality as the one it is a successor of (you can make a bijection that moves the last element away into the initial $omega$ segment with the standard Hilberts-Hotel trick), so a successor ordinal is never the first ordinal with a given cardinality.
(On the other hand, beware that not all limit ordinals are cardinals).
answered Feb 16 '13 at 16:33
Henning MakholmHenning Makholm
244k17312556
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add a comment |
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2 Answers
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2 Answers
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$begingroup$
Recall that a cardinal is an initial ordinal, an ordinal which cannot be put in bijection into a smaller ordinal. Thanks to the Cantor-Bernstein theorem it is enough to show that there is no injection into a smaller ordinal instead.
If an infinite ordinal $alpha$ is not a limit ordinal then there is some $beta$ such that $alpha=betacup{beta}$.
The map $gcolonalphatobeta$ defined as: $$g(x)=begin{cases} 0 & x=beta\x+1 & x<omega\x & text{otherwise}end{cases}$$
is injective from $alpha$ into a smaller ordinal, and therefore $alpha$ cannot be an initial ordinal, that is a cardinal.
answered Feb 16 '13 at 16:38
Asaf Karagila♦Asaf Karagila
309k33441775
$endgroup$
add a comment |
$begingroup$
Recall that a cardinal is an initial ordinal, an ordinal which cannot be put in bijection into a smaller ordinal. Thanks to the Cantor-Bernstein theorem it is enough to show that there is no injection into a smaller ordinal instead.
If an infinite ordinal $alpha$ is not a limit ordinal then there is some $beta$ such that $alpha=betacup{beta}$.
The map $gcolonalphatobeta$ defined as: $$g(x)=begin{cases} 0 & x=beta\x+1 & x<omega\x & text{otherwise}end{cases}$$
is injective from $alpha$ into a smaller ordinal, and therefore $alpha$ cannot be an initial ordinal, that is a cardinal.
answered Feb 16 '13 at 16:38
Asaf Karagila♦Asaf Karagila
309k33441775
$endgroup$
add a comment |
$begingroup$
Recall that a cardinal is an initial ordinal, an ordinal which cannot be put in bijection into a smaller ordinal. Thanks to the Cantor-Bernstein theorem it is enough to show that there is no injection into a smaller ordinal instead.
If an infinite ordinal $alpha$ is not a limit ordinal then there is some $beta$ such that $alpha=betacup{beta}$.
The map $gcolonalphatobeta$ defined as: $$g(x)=begin{cases} 0 & x=beta\x+1 & x<omega\x & text{otherwise}end{cases}$$
is injective from $alpha$ into a smaller ordinal, and therefore $alpha$ cannot be an initial ordinal, that is a cardinal.
answered Feb 16 '13 at 16:38
Asaf Karagila♦Asaf Karagila
309k33441775
$endgroup$
Recall that a cardinal is an initial ordinal, an ordinal which cannot be put in bijection into a smaller ordinal. Thanks to the Cantor-Bernstein theorem it is enough to show that there is no injection into a smaller ordinal instead.
If an infinite ordinal $alpha$ is not a limit ordinal then there is some $beta$ such that $alpha=betacup{beta}$.
The map $gcolonalphatobeta$ defined as: $$g(x)=begin{cases} 0 & x=beta\x+1 & x<omega\x & text{otherwise}end{cases}$$
is injective from $alpha$ into a smaller ordinal, and therefore $alpha$ cannot be an initial ordinal, that is a cardinal.
answered Feb 16 '13 at 16:38
Asaf Karagila♦Asaf Karagila
309k33441775
edited Feb 16 '13 at 16:46
answered Feb 16 '13 at 16:38
Asaf Karagila♦Asaf Karagila
309k33441775
answered Feb 16 '13 at 16:38
Asaf Karagila♦Asaf Karagila
309k33441775
answered Feb 16 '13 at 16:38
Asaf Karagila♦Asaf Karagila
309k33441775
309k33441775
add a comment |
add a comment |
$begingroup$
They are ordinals by definition, because we choose to use initial ordinals as the canonical representative of each class of sets-with-the-same-cardinality. This choice could in principle have been made differently, but it turns out to be a useful convention (though it only works in when we have the Axiom of Choice).
They are limit ordinals, because (except for 0) limit ordinals are precisely the ordinals that are not successor ordinals. And an infinite successor ordinal always has the same cardinality as the one it is a successor of (you can make a bijection that moves the last element away into the initial $omega$ segment with the standard Hilberts-Hotel trick), so a successor ordinal is never the first ordinal with a given cardinality.
(On the other hand, beware that not all limit ordinals are cardinals).
answered Feb 16 '13 at 16:33
Henning MakholmHenning Makholm
244k17312556
$endgroup$
add a comment |
$begingroup$
They are ordinals by definition, because we choose to use initial ordinals as the canonical representative of each class of sets-with-the-same-cardinality. This choice could in principle have been made differently, but it turns out to be a useful convention (though it only works in when we have the Axiom of Choice).
They are limit ordinals, because (except for 0) limit ordinals are precisely the ordinals that are not successor ordinals. And an infinite successor ordinal always has the same cardinality as the one it is a successor of (you can make a bijection that moves the last element away into the initial $omega$ segment with the standard Hilberts-Hotel trick), so a successor ordinal is never the first ordinal with a given cardinality.
(On the other hand, beware that not all limit ordinals are cardinals).
answered Feb 16 '13 at 16:33
Henning MakholmHenning Makholm
244k17312556
$endgroup$
add a comment |
$begingroup$
They are ordinals by definition, because we choose to use initial ordinals as the canonical representative of each class of sets-with-the-same-cardinality. This choice could in principle have been made differently, but it turns out to be a useful convention (though it only works in when we have the Axiom of Choice).
They are limit ordinals, because (except for 0) limit ordinals are precisely the ordinals that are not successor ordinals. And an infinite successor ordinal always has the same cardinality as the one it is a successor of (you can make a bijection that moves the last element away into the initial $omega$ segment with the standard Hilberts-Hotel trick), so a successor ordinal is never the first ordinal with a given cardinality.
(On the other hand, beware that not all limit ordinals are cardinals).
answered Feb 16 '13 at 16:33
Henning MakholmHenning Makholm
244k17312556
$endgroup$
They are ordinals by definition, because we choose to use initial ordinals as the canonical representative of each class of sets-with-the-same-cardinality. This choice could in principle have been made differently, but it turns out to be a useful convention (though it only works in when we have the Axiom of Choice).
They are limit ordinals, because (except for 0) limit ordinals are precisely the ordinals that are not successor ordinals. And an infinite successor ordinal always has the same cardinality as the one it is a successor of (you can make a bijection that moves the last element away into the initial $omega$ segment with the standard Hilberts-Hotel trick), so a successor ordinal is never the first ordinal with a given cardinality.
(On the other hand, beware that not all limit ordinals are cardinals).
answered Feb 16 '13 at 16:33
Henning MakholmHenning Makholm
244k17312556
answered Feb 16 '13 at 16:33
Henning MakholmHenning Makholm
244k17312556
answered Feb 16 '13 at 16:33
Henning MakholmHenning Makholm
244k17312556
answered Feb 16 '13 at 16:33
Henning MakholmHenning Makholm
244k17312556
244k17312556
add a comment |
add a comment |
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Can you come up with a bijection between $alpha$ and $alpha + 1$, where $alpha$ is an infinite ordinal?
$endgroup$
– Alexander Thumm
Feb 16 '13 at 16:22