The relation between the limit cardinal $alpha$ and a sequence of cardinal numbers strictly less than $alpha$
$begingroup$
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
$endgroup$
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?)
$endgroup$
– 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
1
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 |
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3060337%2fthe-relation-between-the-limit-cardinal-alpha-and-a-sequence-of-cardinal-numb%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
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 |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3060337%2fthe-relation-between-the-limit-cardinal-alpha-and-a-sequence-of-cardinal-numb%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
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