Does an element of a set, that can't be in a list, make that set uncountable?
We can map the infinite subsets of $mathbb N$ to the finite subsets of $mathbb N$
The finite subset will be a prefix, of the infinite subset (that it is paired with), that has not yet been used.
$1 mapsto { color{red}{1} ,2,3,4,5,6,7,8 dots } mapsto { color{red}{1} }$
$2 mapsto { color{red}{2} ,4,6,8,10,12,14 dots } mapsto { color{red}{2} }$
$3 mapsto { color{red}{1,3} ,5,7,9,11,13,15 dots } mapsto { color{red}{1,3} }$
$4 mapsto { color{red}{1,2} ,3,7,9,19,27,31 dots } mapsto { color{red}{1,2} }$
$5 mapsto { color{red}{1,2,3} ,4,21,22,25,32 dots } mapsto { color{red}{1,2,3} }$
$6 mapsto { color{red}{2,3} ,4,6,7,8,21,55,58 dots } mapsto { color{red}{2,3} }$
$7 mapsto { color{red}{2,3,4} ,6,7,8,9,21,55,58 dots } mapsto { color{red}{2,3,4} }$
$8 mapsto { color{red}{2,3,4,6} ,7,9,21,55,58 dots } mapsto { color{red}{2,3,4,6} }$
$9 mapsto { color{red}{2,3,4,6,7} ,6,7,8,21,55,58 dots } mapsto { color{red}{2,3,4,6,7} }$
$dots$
We can then find an infinite set that is not in this list (by diagonalization).
$N mapsto { color{red}{4,5,8,9,dots} } mapsto { color{red}{4} } lor { color{red}{4,5} } lor { color{red}{4,5,8} } lor{ color{red}{4,5,8,9} } dots$
We can then pair our infinite set, that can't be in our list, with ${ color{red}{4} }$ if it is not been used or ${ color{red}{4,5} }$ or ${ color{red}{4,5,8} }$ etc... until we find some finite set that has not been used.
Since we need sets with a finite number of elements, and we have sets with an infinite number of elements to choose from, we will always find a finite prefix from each infinite set.
So, if we can find a finite set for every infinite set IN or NOT IN our list, and our set of finite sets is countable then can we still say that our set of infinite sets is uncountable based on there being an element that can't be in a list?
elementary-set-theory
add a comment |
We can map the infinite subsets of $mathbb N$ to the finite subsets of $mathbb N$
The finite subset will be a prefix, of the infinite subset (that it is paired with), that has not yet been used.
$1 mapsto { color{red}{1} ,2,3,4,5,6,7,8 dots } mapsto { color{red}{1} }$
$2 mapsto { color{red}{2} ,4,6,8,10,12,14 dots } mapsto { color{red}{2} }$
$3 mapsto { color{red}{1,3} ,5,7,9,11,13,15 dots } mapsto { color{red}{1,3} }$
$4 mapsto { color{red}{1,2} ,3,7,9,19,27,31 dots } mapsto { color{red}{1,2} }$
$5 mapsto { color{red}{1,2,3} ,4,21,22,25,32 dots } mapsto { color{red}{1,2,3} }$
$6 mapsto { color{red}{2,3} ,4,6,7,8,21,55,58 dots } mapsto { color{red}{2,3} }$
$7 mapsto { color{red}{2,3,4} ,6,7,8,9,21,55,58 dots } mapsto { color{red}{2,3,4} }$
$8 mapsto { color{red}{2,3,4,6} ,7,9,21,55,58 dots } mapsto { color{red}{2,3,4,6} }$
$9 mapsto { color{red}{2,3,4,6,7} ,6,7,8,21,55,58 dots } mapsto { color{red}{2,3,4,6,7} }$
$dots$
We can then find an infinite set that is not in this list (by diagonalization).
$N mapsto { color{red}{4,5,8,9,dots} } mapsto { color{red}{4} } lor { color{red}{4,5} } lor { color{red}{4,5,8} } lor{ color{red}{4,5,8,9} } dots$
We can then pair our infinite set, that can't be in our list, with ${ color{red}{4} }$ if it is not been used or ${ color{red}{4,5} }$ or ${ color{red}{4,5,8} }$ etc... until we find some finite set that has not been used.
Since we need sets with a finite number of elements, and we have sets with an infinite number of elements to choose from, we will always find a finite prefix from each infinite set.
So, if we can find a finite set for every infinite set IN or NOT IN our list, and our set of finite sets is countable then can we still say that our set of infinite sets is uncountable based on there being an element that can't be in a list?
elementary-set-theory
I'm confused---what, exactly, is your question? If I understand it correctly, you are trying to determine if the set of finite subsets of $mathbb{N}$ is countable. That is, is the set $$bigcup_{n=1}^{infty} mathbb{N}^n$$ countable? If this is your question, then a countable union of countable sets is countable. Or are you asking about the set of infinite sequences with elements in $mathbb{N}$, which is uncountable by a diagonalization argument?
– Xander Henderson
2 days ago
1
math.stackexchange.com/questions/2873936/…
– Asaf Karagila♦
2 days ago
add a comment |
We can map the infinite subsets of $mathbb N$ to the finite subsets of $mathbb N$
The finite subset will be a prefix, of the infinite subset (that it is paired with), that has not yet been used.
$1 mapsto { color{red}{1} ,2,3,4,5,6,7,8 dots } mapsto { color{red}{1} }$
$2 mapsto { color{red}{2} ,4,6,8,10,12,14 dots } mapsto { color{red}{2} }$
$3 mapsto { color{red}{1,3} ,5,7,9,11,13,15 dots } mapsto { color{red}{1,3} }$
$4 mapsto { color{red}{1,2} ,3,7,9,19,27,31 dots } mapsto { color{red}{1,2} }$
$5 mapsto { color{red}{1,2,3} ,4,21,22,25,32 dots } mapsto { color{red}{1,2,3} }$
$6 mapsto { color{red}{2,3} ,4,6,7,8,21,55,58 dots } mapsto { color{red}{2,3} }$
$7 mapsto { color{red}{2,3,4} ,6,7,8,9,21,55,58 dots } mapsto { color{red}{2,3,4} }$
$8 mapsto { color{red}{2,3,4,6} ,7,9,21,55,58 dots } mapsto { color{red}{2,3,4,6} }$
$9 mapsto { color{red}{2,3,4,6,7} ,6,7,8,21,55,58 dots } mapsto { color{red}{2,3,4,6,7} }$
$dots$
We can then find an infinite set that is not in this list (by diagonalization).
$N mapsto { color{red}{4,5,8,9,dots} } mapsto { color{red}{4} } lor { color{red}{4,5} } lor { color{red}{4,5,8} } lor{ color{red}{4,5,8,9} } dots$
We can then pair our infinite set, that can't be in our list, with ${ color{red}{4} }$ if it is not been used or ${ color{red}{4,5} }$ or ${ color{red}{4,5,8} }$ etc... until we find some finite set that has not been used.
Since we need sets with a finite number of elements, and we have sets with an infinite number of elements to choose from, we will always find a finite prefix from each infinite set.
So, if we can find a finite set for every infinite set IN or NOT IN our list, and our set of finite sets is countable then can we still say that our set of infinite sets is uncountable based on there being an element that can't be in a list?
elementary-set-theory
We can map the infinite subsets of $mathbb N$ to the finite subsets of $mathbb N$
The finite subset will be a prefix, of the infinite subset (that it is paired with), that has not yet been used.
$1 mapsto { color{red}{1} ,2,3,4,5,6,7,8 dots } mapsto { color{red}{1} }$
$2 mapsto { color{red}{2} ,4,6,8,10,12,14 dots } mapsto { color{red}{2} }$
$3 mapsto { color{red}{1,3} ,5,7,9,11,13,15 dots } mapsto { color{red}{1,3} }$
$4 mapsto { color{red}{1,2} ,3,7,9,19,27,31 dots } mapsto { color{red}{1,2} }$
$5 mapsto { color{red}{1,2,3} ,4,21,22,25,32 dots } mapsto { color{red}{1,2,3} }$
$6 mapsto { color{red}{2,3} ,4,6,7,8,21,55,58 dots } mapsto { color{red}{2,3} }$
$7 mapsto { color{red}{2,3,4} ,6,7,8,9,21,55,58 dots } mapsto { color{red}{2,3,4} }$
$8 mapsto { color{red}{2,3,4,6} ,7,9,21,55,58 dots } mapsto { color{red}{2,3,4,6} }$
$9 mapsto { color{red}{2,3,4,6,7} ,6,7,8,21,55,58 dots } mapsto { color{red}{2,3,4,6,7} }$
$dots$
We can then find an infinite set that is not in this list (by diagonalization).
$N mapsto { color{red}{4,5,8,9,dots} } mapsto { color{red}{4} } lor { color{red}{4,5} } lor { color{red}{4,5,8} } lor{ color{red}{4,5,8,9} } dots$
We can then pair our infinite set, that can't be in our list, with ${ color{red}{4} }$ if it is not been used or ${ color{red}{4,5} }$ or ${ color{red}{4,5,8} }$ etc... until we find some finite set that has not been used.
Since we need sets with a finite number of elements, and we have sets with an infinite number of elements to choose from, we will always find a finite prefix from each infinite set.
So, if we can find a finite set for every infinite set IN or NOT IN our list, and our set of finite sets is countable then can we still say that our set of infinite sets is uncountable based on there being an element that can't be in a list?
elementary-set-theory
elementary-set-theory
edited 2 days ago
Ivan Hieno
asked 2 days ago
Ivan HienoIvan Hieno
1659
1659
I'm confused---what, exactly, is your question? If I understand it correctly, you are trying to determine if the set of finite subsets of $mathbb{N}$ is countable. That is, is the set $$bigcup_{n=1}^{infty} mathbb{N}^n$$ countable? If this is your question, then a countable union of countable sets is countable. Or are you asking about the set of infinite sequences with elements in $mathbb{N}$, which is uncountable by a diagonalization argument?
– Xander Henderson
2 days ago
1
math.stackexchange.com/questions/2873936/…
– Asaf Karagila♦
2 days ago
add a comment |
I'm confused---what, exactly, is your question? If I understand it correctly, you are trying to determine if the set of finite subsets of $mathbb{N}$ is countable. That is, is the set $$bigcup_{n=1}^{infty} mathbb{N}^n$$ countable? If this is your question, then a countable union of countable sets is countable. Or are you asking about the set of infinite sequences with elements in $mathbb{N}$, which is uncountable by a diagonalization argument?
– Xander Henderson
2 days ago
1
math.stackexchange.com/questions/2873936/…
– Asaf Karagila♦
2 days ago
I'm confused---what, exactly, is your question? If I understand it correctly, you are trying to determine if the set of finite subsets of $mathbb{N}$ is countable. That is, is the set $$bigcup_{n=1}^{infty} mathbb{N}^n$$ countable? If this is your question, then a countable union of countable sets is countable. Or are you asking about the set of infinite sequences with elements in $mathbb{N}$, which is uncountable by a diagonalization argument?
– Xander Henderson
2 days ago
I'm confused---what, exactly, is your question? If I understand it correctly, you are trying to determine if the set of finite subsets of $mathbb{N}$ is countable. That is, is the set $$bigcup_{n=1}^{infty} mathbb{N}^n$$ countable? If this is your question, then a countable union of countable sets is countable. Or are you asking about the set of infinite sequences with elements in $mathbb{N}$, which is uncountable by a diagonalization argument?
– Xander Henderson
2 days ago
1
1
math.stackexchange.com/questions/2873936/…
– Asaf Karagila♦
2 days ago
math.stackexchange.com/questions/2873936/…
– Asaf Karagila♦
2 days ago
add a comment |
2 Answers
2
active
oldest
votes
Your question is moot, since the motivating "fact" is false.
The culprit is your implicit claim in the sentence
until we find some finite set that has not been used.
How do we know such a thing exists at all?
For example, let $(F_i)_{iinmathbb{N}}$ enumerate the finite sets, and let $X_i=F_icup (max(F_i),infty)$ (I'm just taking the unions above in order to make each of my $X_i$s infinite). Note that $F_i$ is a prefix of $X_i$. I can map each $X_i$ to $F_i$, and this seems to work ... but now I've used up every finite set, and when I come to my next infinite set - say, the set of evens - I'm stuck.
Now you might argue that I built my map stupidly above, but the point is that the onus is on you to prove that there is a non-stupid way to "continue building forever." And indeed, one of the consequences of Cantor's theorem is that there isn't.
1
Incidentally, a similar instance of the "I don't know where exactly you'll run out of room, but you will eventually" is Fodor's lemma, but that's much more complicated.
– Noah Schweber
2 days ago
add a comment |
Even though that specific infinite subset isn't in our list, we can still have all its finite prefixes in our list of finite subsets.
For instance, say the 10th infinite subset in your list is ${4, ldots}$. Then the 20th is ${4, 5, ldots}$. Then the 30th is ${4, 5, 8, ldots}$. And so on. None of them have to be your infinite set, all of them can be different from your infinite set early enough to allow your set to be the result of the diagonalization. But all of your set's prefixes are in our list of finite subsets.
add a comment |
Your Answer
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%2f3066311%2fdoes-an-element-of-a-set-that-cant-be-in-a-list-make-that-set-uncountable%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
Your question is moot, since the motivating "fact" is false.
The culprit is your implicit claim in the sentence
until we find some finite set that has not been used.
How do we know such a thing exists at all?
For example, let $(F_i)_{iinmathbb{N}}$ enumerate the finite sets, and let $X_i=F_icup (max(F_i),infty)$ (I'm just taking the unions above in order to make each of my $X_i$s infinite). Note that $F_i$ is a prefix of $X_i$. I can map each $X_i$ to $F_i$, and this seems to work ... but now I've used up every finite set, and when I come to my next infinite set - say, the set of evens - I'm stuck.
Now you might argue that I built my map stupidly above, but the point is that the onus is on you to prove that there is a non-stupid way to "continue building forever." And indeed, one of the consequences of Cantor's theorem is that there isn't.
1
Incidentally, a similar instance of the "I don't know where exactly you'll run out of room, but you will eventually" is Fodor's lemma, but that's much more complicated.
– Noah Schweber
2 days ago
add a comment |
Your question is moot, since the motivating "fact" is false.
The culprit is your implicit claim in the sentence
until we find some finite set that has not been used.
How do we know such a thing exists at all?
For example, let $(F_i)_{iinmathbb{N}}$ enumerate the finite sets, and let $X_i=F_icup (max(F_i),infty)$ (I'm just taking the unions above in order to make each of my $X_i$s infinite). Note that $F_i$ is a prefix of $X_i$. I can map each $X_i$ to $F_i$, and this seems to work ... but now I've used up every finite set, and when I come to my next infinite set - say, the set of evens - I'm stuck.
Now you might argue that I built my map stupidly above, but the point is that the onus is on you to prove that there is a non-stupid way to "continue building forever." And indeed, one of the consequences of Cantor's theorem is that there isn't.
1
Incidentally, a similar instance of the "I don't know where exactly you'll run out of room, but you will eventually" is Fodor's lemma, but that's much more complicated.
– Noah Schweber
2 days ago
add a comment |
Your question is moot, since the motivating "fact" is false.
The culprit is your implicit claim in the sentence
until we find some finite set that has not been used.
How do we know such a thing exists at all?
For example, let $(F_i)_{iinmathbb{N}}$ enumerate the finite sets, and let $X_i=F_icup (max(F_i),infty)$ (I'm just taking the unions above in order to make each of my $X_i$s infinite). Note that $F_i$ is a prefix of $X_i$. I can map each $X_i$ to $F_i$, and this seems to work ... but now I've used up every finite set, and when I come to my next infinite set - say, the set of evens - I'm stuck.
Now you might argue that I built my map stupidly above, but the point is that the onus is on you to prove that there is a non-stupid way to "continue building forever." And indeed, one of the consequences of Cantor's theorem is that there isn't.
Your question is moot, since the motivating "fact" is false.
The culprit is your implicit claim in the sentence
until we find some finite set that has not been used.
How do we know such a thing exists at all?
For example, let $(F_i)_{iinmathbb{N}}$ enumerate the finite sets, and let $X_i=F_icup (max(F_i),infty)$ (I'm just taking the unions above in order to make each of my $X_i$s infinite). Note that $F_i$ is a prefix of $X_i$. I can map each $X_i$ to $F_i$, and this seems to work ... but now I've used up every finite set, and when I come to my next infinite set - say, the set of evens - I'm stuck.
Now you might argue that I built my map stupidly above, but the point is that the onus is on you to prove that there is a non-stupid way to "continue building forever." And indeed, one of the consequences of Cantor's theorem is that there isn't.
edited 2 days ago
answered 2 days ago
Noah SchweberNoah Schweber
122k10149284
122k10149284
1
Incidentally, a similar instance of the "I don't know where exactly you'll run out of room, but you will eventually" is Fodor's lemma, but that's much more complicated.
– Noah Schweber
2 days ago
add a comment |
1
Incidentally, a similar instance of the "I don't know where exactly you'll run out of room, but you will eventually" is Fodor's lemma, but that's much more complicated.
– Noah Schweber
2 days ago
1
1
Incidentally, a similar instance of the "I don't know where exactly you'll run out of room, but you will eventually" is Fodor's lemma, but that's much more complicated.
– Noah Schweber
2 days ago
Incidentally, a similar instance of the "I don't know where exactly you'll run out of room, but you will eventually" is Fodor's lemma, but that's much more complicated.
– Noah Schweber
2 days ago
add a comment |
Even though that specific infinite subset isn't in our list, we can still have all its finite prefixes in our list of finite subsets.
For instance, say the 10th infinite subset in your list is ${4, ldots}$. Then the 20th is ${4, 5, ldots}$. Then the 30th is ${4, 5, 8, ldots}$. And so on. None of them have to be your infinite set, all of them can be different from your infinite set early enough to allow your set to be the result of the diagonalization. But all of your set's prefixes are in our list of finite subsets.
add a comment |
Even though that specific infinite subset isn't in our list, we can still have all its finite prefixes in our list of finite subsets.
For instance, say the 10th infinite subset in your list is ${4, ldots}$. Then the 20th is ${4, 5, ldots}$. Then the 30th is ${4, 5, 8, ldots}$. And so on. None of them have to be your infinite set, all of them can be different from your infinite set early enough to allow your set to be the result of the diagonalization. But all of your set's prefixes are in our list of finite subsets.
add a comment |
Even though that specific infinite subset isn't in our list, we can still have all its finite prefixes in our list of finite subsets.
For instance, say the 10th infinite subset in your list is ${4, ldots}$. Then the 20th is ${4, 5, ldots}$. Then the 30th is ${4, 5, 8, ldots}$. And so on. None of them have to be your infinite set, all of them can be different from your infinite set early enough to allow your set to be the result of the diagonalization. But all of your set's prefixes are in our list of finite subsets.
Even though that specific infinite subset isn't in our list, we can still have all its finite prefixes in our list of finite subsets.
For instance, say the 10th infinite subset in your list is ${4, ldots}$. Then the 20th is ${4, 5, ldots}$. Then the 30th is ${4, 5, 8, ldots}$. And so on. None of them have to be your infinite set, all of them can be different from your infinite set early enough to allow your set to be the result of the diagonalization. But all of your set's prefixes are in our list of finite subsets.
edited 2 days ago
answered 2 days ago
ArthurArthur
111k7107189
111k7107189
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%2f3066311%2fdoes-an-element-of-a-set-that-cant-be-in-a-list-make-that-set-uncountable%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
I'm confused---what, exactly, is your question? If I understand it correctly, you are trying to determine if the set of finite subsets of $mathbb{N}$ is countable. That is, is the set $$bigcup_{n=1}^{infty} mathbb{N}^n$$ countable? If this is your question, then a countable union of countable sets is countable. Or are you asking about the set of infinite sequences with elements in $mathbb{N}$, which is uncountable by a diagonalization argument?
– Xander Henderson
2 days ago
1
math.stackexchange.com/questions/2873936/…
– Asaf Karagila♦
2 days ago