Countable Infinite and Uncountable Infinite sets
$begingroup$
Mark each statement as TRUE, FALSE, or UNKNOWN
(a) $|Bbb{R}| < aleph_1$
(b) $|Bbb{R}| = aleph_1$
(c) $|P(Bbb{R})| > aleph_1$
Could someone explain to me the reasoning based on whatever the answer is for each one because I do not fully understand Countable Infinite and Uncountable Infinite sets
infinite-groups
$endgroup$
add a comment |
$begingroup$
Mark each statement as TRUE, FALSE, or UNKNOWN
(a) $|Bbb{R}| < aleph_1$
(b) $|Bbb{R}| = aleph_1$
(c) $|P(Bbb{R})| > aleph_1$
Could someone explain to me the reasoning based on whatever the answer is for each one because I do not fully understand Countable Infinite and Uncountable Infinite sets
infinite-groups
$endgroup$
$begingroup$
Then read up on countable infinite vs. uncountable infinite. There is nothing we can say that can't be said better in a good text.
$endgroup$
– fleablood
Dec 19 '18 at 6:37
$begingroup$
@fleablood have read about it and even watched youtube clips, but still don't know how to answer the questions given.
$endgroup$
– Viserom
Dec 19 '18 at 6:41
$begingroup$
Then it's up to you to ask us a question we can answer for you. We can't read your mind and know why you don't understand it.
$endgroup$
– fleablood
Dec 19 '18 at 6:45
add a comment |
$begingroup$
Mark each statement as TRUE, FALSE, or UNKNOWN
(a) $|Bbb{R}| < aleph_1$
(b) $|Bbb{R}| = aleph_1$
(c) $|P(Bbb{R})| > aleph_1$
Could someone explain to me the reasoning based on whatever the answer is for each one because I do not fully understand Countable Infinite and Uncountable Infinite sets
infinite-groups
$endgroup$
Mark each statement as TRUE, FALSE, or UNKNOWN
(a) $|Bbb{R}| < aleph_1$
(b) $|Bbb{R}| = aleph_1$
(c) $|P(Bbb{R})| > aleph_1$
Could someone explain to me the reasoning based on whatever the answer is for each one because I do not fully understand Countable Infinite and Uncountable Infinite sets
infinite-groups
infinite-groups
asked Dec 19 '18 at 6:31
ViseromViserom
123
123
$begingroup$
Then read up on countable infinite vs. uncountable infinite. There is nothing we can say that can't be said better in a good text.
$endgroup$
– fleablood
Dec 19 '18 at 6:37
$begingroup$
@fleablood have read about it and even watched youtube clips, but still don't know how to answer the questions given.
$endgroup$
– Viserom
Dec 19 '18 at 6:41
$begingroup$
Then it's up to you to ask us a question we can answer for you. We can't read your mind and know why you don't understand it.
$endgroup$
– fleablood
Dec 19 '18 at 6:45
add a comment |
$begingroup$
Then read up on countable infinite vs. uncountable infinite. There is nothing we can say that can't be said better in a good text.
$endgroup$
– fleablood
Dec 19 '18 at 6:37
$begingroup$
@fleablood have read about it and even watched youtube clips, but still don't know how to answer the questions given.
$endgroup$
– Viserom
Dec 19 '18 at 6:41
$begingroup$
Then it's up to you to ask us a question we can answer for you. We can't read your mind and know why you don't understand it.
$endgroup$
– fleablood
Dec 19 '18 at 6:45
$begingroup$
Then read up on countable infinite vs. uncountable infinite. There is nothing we can say that can't be said better in a good text.
$endgroup$
– fleablood
Dec 19 '18 at 6:37
$begingroup$
Then read up on countable infinite vs. uncountable infinite. There is nothing we can say that can't be said better in a good text.
$endgroup$
– fleablood
Dec 19 '18 at 6:37
$begingroup$
@fleablood have read about it and even watched youtube clips, but still don't know how to answer the questions given.
$endgroup$
– Viserom
Dec 19 '18 at 6:41
$begingroup$
@fleablood have read about it and even watched youtube clips, but still don't know how to answer the questions given.
$endgroup$
– Viserom
Dec 19 '18 at 6:41
$begingroup$
Then it's up to you to ask us a question we can answer for you. We can't read your mind and know why you don't understand it.
$endgroup$
– fleablood
Dec 19 '18 at 6:45
$begingroup$
Then it's up to you to ask us a question we can answer for you. We can't read your mind and know why you don't understand it.
$endgroup$
– fleablood
Dec 19 '18 at 6:45
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
$aleph_1$ by definition is the first uncountable cardinal number.
So $|A| < aleph_1$ by definition means that $A$ is a countable set.
The reals are not countable.
We (should) know that $|mathbb{R}| = |P(mathbb{N})| = 2^{aleph_0}> aleph_0$. So in particular $2^{aleph_0} ge aleph_1$.
The so-called continuum hypothesis (which is independent of ZFC, so "unknown" in your list) states that $2^{aleph_0} = aleph_1$.
By Cantor's theorem $|P(mathbb{R})| = 2^{|mathbb{R}|}= 2^{2^{aleph_0}} > 2^{aleph_0} ge aleph_1$.
$endgroup$
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%2f3046074%2fcountable-infinite-and-uncountable-infinite-sets%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
$begingroup$
$aleph_1$ by definition is the first uncountable cardinal number.
So $|A| < aleph_1$ by definition means that $A$ is a countable set.
The reals are not countable.
We (should) know that $|mathbb{R}| = |P(mathbb{N})| = 2^{aleph_0}> aleph_0$. So in particular $2^{aleph_0} ge aleph_1$.
The so-called continuum hypothesis (which is independent of ZFC, so "unknown" in your list) states that $2^{aleph_0} = aleph_1$.
By Cantor's theorem $|P(mathbb{R})| = 2^{|mathbb{R}|}= 2^{2^{aleph_0}} > 2^{aleph_0} ge aleph_1$.
$endgroup$
add a comment |
$begingroup$
$aleph_1$ by definition is the first uncountable cardinal number.
So $|A| < aleph_1$ by definition means that $A$ is a countable set.
The reals are not countable.
We (should) know that $|mathbb{R}| = |P(mathbb{N})| = 2^{aleph_0}> aleph_0$. So in particular $2^{aleph_0} ge aleph_1$.
The so-called continuum hypothesis (which is independent of ZFC, so "unknown" in your list) states that $2^{aleph_0} = aleph_1$.
By Cantor's theorem $|P(mathbb{R})| = 2^{|mathbb{R}|}= 2^{2^{aleph_0}} > 2^{aleph_0} ge aleph_1$.
$endgroup$
add a comment |
$begingroup$
$aleph_1$ by definition is the first uncountable cardinal number.
So $|A| < aleph_1$ by definition means that $A$ is a countable set.
The reals are not countable.
We (should) know that $|mathbb{R}| = |P(mathbb{N})| = 2^{aleph_0}> aleph_0$. So in particular $2^{aleph_0} ge aleph_1$.
The so-called continuum hypothesis (which is independent of ZFC, so "unknown" in your list) states that $2^{aleph_0} = aleph_1$.
By Cantor's theorem $|P(mathbb{R})| = 2^{|mathbb{R}|}= 2^{2^{aleph_0}} > 2^{aleph_0} ge aleph_1$.
$endgroup$
$aleph_1$ by definition is the first uncountable cardinal number.
So $|A| < aleph_1$ by definition means that $A$ is a countable set.
The reals are not countable.
We (should) know that $|mathbb{R}| = |P(mathbb{N})| = 2^{aleph_0}> aleph_0$. So in particular $2^{aleph_0} ge aleph_1$.
The so-called continuum hypothesis (which is independent of ZFC, so "unknown" in your list) states that $2^{aleph_0} = aleph_1$.
By Cantor's theorem $|P(mathbb{R})| = 2^{|mathbb{R}|}= 2^{2^{aleph_0}} > 2^{aleph_0} ge aleph_1$.
answered Dec 19 '18 at 7:28
Henno BrandsmaHenno Brandsma
114k348124
114k348124
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%2f3046074%2fcountable-infinite-and-uncountable-infinite-sets%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
$begingroup$
Then read up on countable infinite vs. uncountable infinite. There is nothing we can say that can't be said better in a good text.
$endgroup$
– fleablood
Dec 19 '18 at 6:37
$begingroup$
@fleablood have read about it and even watched youtube clips, but still don't know how to answer the questions given.
$endgroup$
– Viserom
Dec 19 '18 at 6:41
$begingroup$
Then it's up to you to ask us a question we can answer for you. We can't read your mind and know why you don't understand it.
$endgroup$
– fleablood
Dec 19 '18 at 6:45