Does an element of a set, that can't be in a list, make that set uncountable?












2














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







share|cite|improve this question
























  • 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
















2














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







share|cite|improve this question
























  • 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














2












2








2


0





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







share|cite|improve this question















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






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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


















  • 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










2 Answers
2






active

oldest

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5














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.






share|cite|improve this answer



















  • 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





















5














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.






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    2 Answers
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    2 Answers
    2






    active

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    active

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    active

    oldest

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    5














    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.






    share|cite|improve this answer



















    • 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


















    5














    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.






    share|cite|improve this answer



















    • 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
















    5












    5








    5






    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.






    share|cite|improve this answer














    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.







    share|cite|improve this answer














    share|cite|improve this answer



    share|cite|improve this answer








    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
















    • 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













    5














    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.






    share|cite|improve this answer




























      5














      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.






      share|cite|improve this answer


























        5












        5








        5






        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.






        share|cite|improve this answer














        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.







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited 2 days ago

























        answered 2 days ago









        ArthurArthur

        111k7107189




        111k7107189






























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            An IMO inspired problem

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            14 4 This problem from IMO 1988 is said to be one of the most elegant ones in functional equations. Problem : The function $f$ is defined on the set of all positive integers as follows: begin{align} f(1) = 1, f(3) &= 3, f(2n) = f(n), \ f(4n+1) &= 2f(2n+1) - f(n), \ f(4n+3) &= 3 f(2n+1) - 2f(n) end{align} Find the number of $n$ with $f(n) = n, 1 leq n leq 1988$. The main idea towards the solution is realizing that these conditions stand for the fact that $f(n)$ just reverses the digits in the binary representation of the number $n$. So, essentially the solution is finding the number of binary pallindromes $leq 1988_{10}$. My question is the following : How to reformulate the problem so that $f(n)$ reverses the digits of $n$ in its ternary representation. Or even better, can we reformulate it for a
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            Management

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            Management From Wikipedia, the free encyclopedia Jump to navigation Jump to search "Manager" redirects here. For other uses, see Management (disambiguation) and Manager (disambiguation). An organization chart for the United States Coast Guard shows the hierarchy of managerial roles in that organization. Business administration Management of a business Accounting Management accounting Financial accounting Financial audit Business entities Cooperative Corporation Limited liability company Partnership Sole proprietorship State-owned enterprise Corporate governance Annual general meeting Board of directors Supervisory board Advisory board Audit committee Corporate law Commercial law Constitutional documents Contract Corporate crime Corporate liability Insolvency law International trade law Mergers and acquisitions Economics Commodity Pu
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            Investment

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            Investment From Wikipedia, the free encyclopedia Jump to navigation Jump to search This article is about investment in finance. For investment in macroeconomics, see Investment (macroeconomics). For other uses, see Investment (disambiguation). "Invest" redirects here. For the term in meteorology, see Invest (meteorology). This article needs additional citations for verification . Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. ( February 2017 ) (Learn how and when to remove this template message) In general, to invest is to distribute money in the expectation of some benefit in the future – for example, investment in durable goods, in real estate by the service industry, in factories for manufacturing, in product development, and in research and development. However, this article focuses specifically on investment in financial
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