Infinite size is actually number density? [closed]
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Am I wrong in thinking that saying one infinity is larger than another is slightly disingenuous? My argument being the infinity is actually more dense than 'larger'. Obviously, we are already thinking very abstractly when ascribing a bigger size to the set of all positive irrationals vs the set of all naturals.
Because both are defined as having no end, it is really a matter of having more numbers in-between.
sequences-and-series infinity
asked Aug 21 at 13:37
hisairnessag3
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closed as unclear what you're asking by Lord Shark the Unknown, Hans Lundmark, Sil, Leucippus, Key Flex Aug 22 at 2:50
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
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Am I wrong in thinking that saying one infinity is larger than another is slightly disingenuous? My argument being the infinity is actually more dense than 'larger'. Obviously, we are already thinking very abstractly when ascribing a bigger size to the set of all positive irrationals vs the set of all naturals.
Because both are defined as having no end, it is really a matter of having more numbers in-between.
sequences-and-series infinity
asked Aug 21 at 13:37
hisairnessag3
943
closed as unclear what you're asking by Lord Shark the Unknown, Hans Lundmark, Sil, Leucippus, Key Flex Aug 22 at 2:50
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
1
What does "more" mean in " having more numbers in-between," if not cardinality? E.g. how do you distinguish between the rationals (countable) and the irrationals (uncountable) without using cardinality as usually understood? Also, what about cardinality of sets which cannot be "naturally" linearly ordered, like $mathcalP(mathbbR)$, or ones whose most natural order isn't dense, like $omega_1$?
– Noah Schweber
Aug 21 at 13:50
1
You're wrong. Sets are not just sets of real numbers.
– Asaf Karagila♦
Aug 21 at 14:18
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up vote
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up vote
0
down vote
favorite
Am I wrong in thinking that saying one infinity is larger than another is slightly disingenuous? My argument being the infinity is actually more dense than 'larger'. Obviously, we are already thinking very abstractly when ascribing a bigger size to the set of all positive irrationals vs the set of all naturals.
Because both are defined as having no end, it is really a matter of having more numbers in-between.
sequences-and-series infinity
asked Aug 21 at 13:37
hisairnessag3
943
Am I wrong in thinking that saying one infinity is larger than another is slightly disingenuous? My argument being the infinity is actually more dense than 'larger'. Obviously, we are already thinking very abstractly when ascribing a bigger size to the set of all positive irrationals vs the set of all naturals.
Because both are defined as having no end, it is really a matter of having more numbers in-between.
sequences-and-series infinity
asked Aug 21 at 13:37
hisairnessag3
943
asked Aug 21 at 13:37
hisairnessag3
943
asked Aug 21 at 13:37
hisairnessag3
943
asked Aug 21 at 13:37
hisairnessag3
943
943
closed as unclear what you're asking by Lord Shark the Unknown, Hans Lundmark, Sil, Leucippus, Key Flex Aug 22 at 2:50
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
closed as unclear what you're asking by Lord Shark the Unknown, Hans Lundmark, Sil, Leucippus, Key Flex Aug 22 at 2:50
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
1
What does "more" mean in " having more numbers in-between," if not cardinality? E.g. how do you distinguish between the rationals (countable) and the irrationals (uncountable) without using cardinality as usually understood? Also, what about cardinality of sets which cannot be "naturally" linearly ordered, like $mathcalP(mathbbR)$, or ones whose most natural order isn't dense, like $omega_1$?
– Noah Schweber
Aug 21 at 13:50
1
You're wrong. Sets are not just sets of real numbers.
– Asaf Karagila♦
Aug 21 at 14:18
add a comment |Â
1
What does "more" mean in " having more numbers in-between," if not cardinality? E.g. how do you distinguish between the rationals (countable) and the irrationals (uncountable) without using cardinality as usually understood? Also, what about cardinality of sets which cannot be "naturally" linearly ordered, like $mathcalP(mathbbR)$, or ones whose most natural order isn't dense, like $omega_1$?
– Noah Schweber
Aug 21 at 13:50
1
You're wrong. Sets are not just sets of real numbers.
– Asaf Karagila♦
Aug 21 at 14:18
1
1
What does "more" mean in " having more numbers in-between," if not cardinality? E.g. how do you distinguish between the rationals (countable) and the irrationals (uncountable) without using cardinality as usually understood? Also, what about cardinality of sets which cannot be "naturally" linearly ordered, like $mathcalP(mathbbR)$, or ones whose most natural order isn't dense, like $omega_1$?
– Noah Schweber
Aug 21 at 13:50
What does "more" mean in " having more numbers in-between," if not cardinality? E.g. how do you distinguish between the rationals (countable) and the irrationals (uncountable) without using cardinality as usually understood? Also, what about cardinality of sets which cannot be "naturally" linearly ordered, like $mathcalP(mathbbR)$, or ones whose most natural order isn't dense, like $omega_1$?
– Noah Schweber
Aug 21 at 13:50
1
1
You're wrong. Sets are not just sets of real numbers.
– Asaf Karagila♦
Aug 21 at 14:18
You're wrong. Sets are not just sets of real numbers.
– Asaf Karagila♦
Aug 21 at 14:18
add a comment |Â
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Yes, I am afraid you are wrong in thinking so. Being 'more dense' is not a valid guide to size where infinity is concerned - for example, the rationals are obviously 'more dense' on the line than the integers, but one can specify a 1-to-1 mapping between them, so in that sense they are actually the same size. On the other hand, one CANNOT specify any such mapping between the IRrationals and the integers, so they are of different sizes [the irrationals being larger since one can map the integers INto them].
More broadly: Historically people have appealed to two different principles when judging relative sizes of discrete sets of objects. The first principle is 'If I can pair them off exactly, they are the same size'. The second principle is 'The whole is greater than the part - i.e the size of a set is bigger than the size of any proper subset'. The problem is that, when considering infinite sets, these two principles usually give different answers about relative sizes. It used to be that people cited this fact to conclude that the whole notion of 'size' for infinite sets was just nonsense. It was Cantor's brilliant insight that sense COULD be made of 'size of infinite set' if one simply accepted that the second principle doesn't actually apply.
answered Aug 21 at 15:23
PMar
291
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Infinite sets are quite different from finite sets with respect to "sizes". Everyone knows that if $A$ is a proper subset of a finite set $B$ then $A$ has fewer members than $B$, in that you cannot match the members of $A$, one-to-one, with the members of $B$ without having at least one member of $B$ left out. But we can match the even positive integers with $all$ the positive integers by matching $ 2n$ with $n,$ for each $nin Bbb Z^+.$
We can even map $Bbb Z^+$ onto the set of all $rationals.$ But we cannot map $Bbb Z^+$ onto all the points of the open real interval $(0,1)$, for reasons that are not obvious.
I suggest an introductory book on set theory. The preface in a math text will usually say what the book's intended "audience " is, which can help you to choose..... I recall a short book by Suppes that had no pre-requisites. And for some ideas and some fun, try "Stories About Sets" by Vilenkin.
answered Aug 22 at 2:40
DanielWainfleet
32k31644
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2 Answers
2
active
oldest
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2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
Yes, I am afraid you are wrong in thinking so. Being 'more dense' is not a valid guide to size where infinity is concerned - for example, the rationals are obviously 'more dense' on the line than the integers, but one can specify a 1-to-1 mapping between them, so in that sense they are actually the same size. On the other hand, one CANNOT specify any such mapping between the IRrationals and the integers, so they are of different sizes [the irrationals being larger since one can map the integers INto them].
More broadly: Historically people have appealed to two different principles when judging relative sizes of discrete sets of objects. The first principle is 'If I can pair them off exactly, they are the same size'. The second principle is 'The whole is greater than the part - i.e the size of a set is bigger than the size of any proper subset'. The problem is that, when considering infinite sets, these two principles usually give different answers about relative sizes. It used to be that people cited this fact to conclude that the whole notion of 'size' for infinite sets was just nonsense. It was Cantor's brilliant insight that sense COULD be made of 'size of infinite set' if one simply accepted that the second principle doesn't actually apply.
answered Aug 21 at 15:23
PMar
291
add a comment |Â
up vote
2
down vote
Yes, I am afraid you are wrong in thinking so. Being 'more dense' is not a valid guide to size where infinity is concerned - for example, the rationals are obviously 'more dense' on the line than the integers, but one can specify a 1-to-1 mapping between them, so in that sense they are actually the same size. On the other hand, one CANNOT specify any such mapping between the IRrationals and the integers, so they are of different sizes [the irrationals being larger since one can map the integers INto them].
More broadly: Historically people have appealed to two different principles when judging relative sizes of discrete sets of objects. The first principle is 'If I can pair them off exactly, they are the same size'. The second principle is 'The whole is greater than the part - i.e the size of a set is bigger than the size of any proper subset'. The problem is that, when considering infinite sets, these two principles usually give different answers about relative sizes. It used to be that people cited this fact to conclude that the whole notion of 'size' for infinite sets was just nonsense. It was Cantor's brilliant insight that sense COULD be made of 'size of infinite set' if one simply accepted that the second principle doesn't actually apply.
answered Aug 21 at 15:23
PMar
291
add a comment |Â
up vote
2
down vote
up vote
2
down vote
Yes, I am afraid you are wrong in thinking so. Being 'more dense' is not a valid guide to size where infinity is concerned - for example, the rationals are obviously 'more dense' on the line than the integers, but one can specify a 1-to-1 mapping between them, so in that sense they are actually the same size. On the other hand, one CANNOT specify any such mapping between the IRrationals and the integers, so they are of different sizes [the irrationals being larger since one can map the integers INto them].
More broadly: Historically people have appealed to two different principles when judging relative sizes of discrete sets of objects. The first principle is 'If I can pair them off exactly, they are the same size'. The second principle is 'The whole is greater than the part - i.e the size of a set is bigger than the size of any proper subset'. The problem is that, when considering infinite sets, these two principles usually give different answers about relative sizes. It used to be that people cited this fact to conclude that the whole notion of 'size' for infinite sets was just nonsense. It was Cantor's brilliant insight that sense COULD be made of 'size of infinite set' if one simply accepted that the second principle doesn't actually apply.
answered Aug 21 at 15:23
PMar
291
Yes, I am afraid you are wrong in thinking so. Being 'more dense' is not a valid guide to size where infinity is concerned - for example, the rationals are obviously 'more dense' on the line than the integers, but one can specify a 1-to-1 mapping between them, so in that sense they are actually the same size. On the other hand, one CANNOT specify any such mapping between the IRrationals and the integers, so they are of different sizes [the irrationals being larger since one can map the integers INto them].
More broadly: Historically people have appealed to two different principles when judging relative sizes of discrete sets of objects. The first principle is 'If I can pair them off exactly, they are the same size'. The second principle is 'The whole is greater than the part - i.e the size of a set is bigger than the size of any proper subset'. The problem is that, when considering infinite sets, these two principles usually give different answers about relative sizes. It used to be that people cited this fact to conclude that the whole notion of 'size' for infinite sets was just nonsense. It was Cantor's brilliant insight that sense COULD be made of 'size of infinite set' if one simply accepted that the second principle doesn't actually apply.
answered Aug 21 at 15:23
PMar
291
answered Aug 21 at 15:23
PMar
291
answered Aug 21 at 15:23
PMar
291
answered Aug 21 at 15:23
PMar
291
291
add a comment |Â
add a comment |Â
up vote
0
down vote
Infinite sets are quite different from finite sets with respect to "sizes". Everyone knows that if $A$ is a proper subset of a finite set $B$ then $A$ has fewer members than $B$, in that you cannot match the members of $A$, one-to-one, with the members of $B$ without having at least one member of $B$ left out. But we can match the even positive integers with $all$ the positive integers by matching $ 2n$ with $n,$ for each $nin Bbb Z^+.$
We can even map $Bbb Z^+$ onto the set of all $rationals.$ But we cannot map $Bbb Z^+$ onto all the points of the open real interval $(0,1)$, for reasons that are not obvious.
I suggest an introductory book on set theory. The preface in a math text will usually say what the book's intended "audience " is, which can help you to choose..... I recall a short book by Suppes that had no pre-requisites. And for some ideas and some fun, try "Stories About Sets" by Vilenkin.
answered Aug 22 at 2:40
DanielWainfleet
32k31644
add a comment |Â
up vote
0
down vote
Infinite sets are quite different from finite sets with respect to "sizes". Everyone knows that if $A$ is a proper subset of a finite set $B$ then $A$ has fewer members than $B$, in that you cannot match the members of $A$, one-to-one, with the members of $B$ without having at least one member of $B$ left out. But we can match the even positive integers with $all$ the positive integers by matching $ 2n$ with $n,$ for each $nin Bbb Z^+.$
We can even map $Bbb Z^+$ onto the set of all $rationals.$ But we cannot map $Bbb Z^+$ onto all the points of the open real interval $(0,1)$, for reasons that are not obvious.
I suggest an introductory book on set theory. The preface in a math text will usually say what the book's intended "audience " is, which can help you to choose..... I recall a short book by Suppes that had no pre-requisites. And for some ideas and some fun, try "Stories About Sets" by Vilenkin.
answered Aug 22 at 2:40
DanielWainfleet
32k31644
add a comment |Â
up vote
0
down vote
up vote
0
down vote
Infinite sets are quite different from finite sets with respect to "sizes". Everyone knows that if $A$ is a proper subset of a finite set $B$ then $A$ has fewer members than $B$, in that you cannot match the members of $A$, one-to-one, with the members of $B$ without having at least one member of $B$ left out. But we can match the even positive integers with $all$ the positive integers by matching $ 2n$ with $n,$ for each $nin Bbb Z^+.$
We can even map $Bbb Z^+$ onto the set of all $rationals.$ But we cannot map $Bbb Z^+$ onto all the points of the open real interval $(0,1)$, for reasons that are not obvious.
I suggest an introductory book on set theory. The preface in a math text will usually say what the book's intended "audience " is, which can help you to choose..... I recall a short book by Suppes that had no pre-requisites. And for some ideas and some fun, try "Stories About Sets" by Vilenkin.
answered Aug 22 at 2:40
DanielWainfleet
32k31644
Infinite sets are quite different from finite sets with respect to "sizes". Everyone knows that if $A$ is a proper subset of a finite set $B$ then $A$ has fewer members than $B$, in that you cannot match the members of $A$, one-to-one, with the members of $B$ without having at least one member of $B$ left out. But we can match the even positive integers with $all$ the positive integers by matching $ 2n$ with $n,$ for each $nin Bbb Z^+.$
We can even map $Bbb Z^+$ onto the set of all $rationals.$ But we cannot map $Bbb Z^+$ onto all the points of the open real interval $(0,1)$, for reasons that are not obvious.
I suggest an introductory book on set theory. The preface in a math text will usually say what the book's intended "audience " is, which can help you to choose..... I recall a short book by Suppes that had no pre-requisites. And for some ideas and some fun, try "Stories About Sets" by Vilenkin.
answered Aug 22 at 2:40
DanielWainfleet
32k31644
answered Aug 22 at 2:40
DanielWainfleet
32k31644
answered Aug 22 at 2:40
DanielWainfleet
32k31644
answered Aug 22 at 2:40
DanielWainfleet
32k31644
32k31644
add a comment |Â
add a comment |Â
1
What does "more" mean in " having more numbers in-between," if not cardinality? E.g. how do you distinguish between the rationals (countable) and the irrationals (uncountable) without using cardinality as usually understood? Also, what about cardinality of sets which cannot be "naturally" linearly ordered, like $mathcalP(mathbbR)$, or ones whose most natural order isn't dense, like $omega_1$?
– Noah Schweber
Aug 21 at 13:50
1
You're wrong. Sets are not just sets of real numbers.
– Asaf Karagila♦
Aug 21 at 14:18