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I think the mystery is not that some infinities can be larger than others, but that there are infinite sets of equal ‘size’ that conflict with intuition. Some

I think the mystery is not that some infinities can be larger than others, but that there are infinite sets of equal ‘size’ that conflict with intuition. Some examples that ‘prove’ some infinities are larger than others to laymen: - there are twice as many integers as odd integers - there are more points on a plane then on a line - there are more points on a line than on a circle - there are more points on a plane then on a semiplane - there are more rationals than integers - there are more reals than rationals It’s only in the intermediate state of a mathematician’s education, where they have just accepted that, for infinite sets, ‘more’ isn’t the best way to determine size equivalence that it becomes a surprise that for the last one, “the size of the set of reals is larger than that of the rationals” is true, and can be proven to be.

anonGone73 4y ago
Infinite is un-enumerable, but you can carve it up in more or less ways.
SkyPuncher 4y ago
I've found it easier to reason about when I think of it as "how quickly does the set of number grow as you add more to it". It becomes particularly apparent when you think about it on a finite scale. 0 to 100 * 100 integers * 50 odd numbers, 50 even numbers * infinite floats (unless you only represent part of the float)
dwater 4y ago
It's been a long time since my number theory class, but aren't there the same number of integers and odd integers? Take the set of odd integers {... -3, -1, 1, 3, 5, ...} For each item, subtract one and divide the result by 2. Now you have the set of all integers without any insertions or deletions: {... -2, -1, 0, 1, 2, ...} Therefore the set of odd integers can be mapped 1:1 onto the set of all integers so they are of equal length.
- ShamelessC 4y ago
  They have the same cardinality.
- renewiltord 4y ago
  Your proof is correct.
- deleted 4y ago
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- RheingoldRiver 4y ago
  That's why 'Prove' is in quotation marks. Not all of GP's statements are true, though it would've been helpful if they'd added (true) and (false) tags next to them.
rtkwe 4y ago
> there are twice as many integers as odd integers > there are more rationals than integers Both of these are incorrect you can create a 1-1 mapping between all of these sets so they are the same "size". Things get unintuitive when you're dealing with infinites things that feel like they should be larger aren't when you examine them rigorously. For integers to odd integers the mapping is easy for each natural number n map n -> 2n+1. Mapping integers to rational numbers is more difficult to write into an equation but if you lay them out in a a grid defined by numerators and denominators x/y you can snake along this grid to eventually map every rational number to a corresponding natural number (ie positive integers which has the same cardinality as integers). http://www.cwladis.com/math100/Lecture5Sets.htm#:~:text=the%20natural%20numbers%3F-,The%20Cardinality%20of%20the%20Set%20of%20Rational%20Numbers,-We%20recall%20that http://www.cwladis.com/math100/Lecture5Sets.htm#:~:text=the%... > there are more points on a plane then on a line Going further R (points on a line) to R^2 (points on a plane) is also the same cardinality. The proofs are over my head as a 10 years past math minor but they're out there. Including this one that goes from R^3 to R. https://math.stackexchange.com/posts/183383/revisions https://math.stackexchange.com/posts/183383/revisions
- hallucy 4y ago
  This is literally the point of the comment. To a layperson not familiar with this approach to measuring infinities, all of the examples are “correct”. To a mathematician that hasn’t heard of uncountability, all are “incorrect”. (And learning the last is actually “correct” is surprising).
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  - jimsimmons 4y ago
    That’s not what GP says though. They say their last statement is correct when it’s not. There are only two types of infinities: countable and uncountable. Both reals and rationals are not countable
    - 4y ago
vacuumcl 4y ago
Half your examples are wrong, but maybe it is your point. Although it wasn't clear to me the first time I read your comment. - The cardinality of the odd and even integers is the same. - It is true there are more points on a plane then on a line (Cantor's theorem.) - The circle is the compactified real line, i.e. it can be represented as the real numbers with one additional point (the point at infinity). In terms of cardinality they are the same since they just differ by one point which does not change the cardinality. - There are not more points on a plane than a half-plane, you can find a bijective mapping between them easily. - There are more rationals than integers: not true, they are both countable sets of the same cardinality. - There are more reals than rationals, this is true (again Cantor.)
- function_seven 4y ago
  Parent is making the same point you are. It’s a surprise to the intermediate mathematician when the last bullet point turns out to be true.
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- jdkee 4y ago
  "- It is true there are more points on a plane then on a line (Cantor's theorem.)" There is a bijection between the points on a line and the points on a plane or in any n-dimensional space.
  - vacuumcl 4y ago
    Yes, that was a bad mistake on my part. Thanks for pointing it out!..
jltsiren 4y ago
Isn't that more like first-year material in mathematics and in more theoretically oriented CS programs? Once you start talking about injections, surjections, and bijections, you may as well prove some basic results about the sizes of sets of numbers.
- yodsanklai 4y ago
  There are basically no prerequisites to get a grasp of these ideas. I remember being introduced to this in one of Raymond Smullyan's book when I was a kid.
sleepyams 4y ago
Good points, a large chunk of math is different formalizations of the notion of size, for example (in order of increasing absurdity): https://en.wikipedia.org/wiki/Cardinality https://en.wikipedia.org/wiki/Cardinality https://en.wikipedia.org/wiki/Measure_(mathematics) https://en.wikipedia.org/wiki/Measure_(mathematics) https://en.wikipedia.org/wiki/Ultrafilter_(set_theory) https://en.wikipedia.org/wiki/Ultrafilter_(set_theory) https://en.wikipedia.org/wiki/Euler_characteristic https://en.wikipedia.org/wiki/Euler_characteristic https://golem.ph.utexas.edu/category/2008/02/metric_spaces.html https://golem.ph.utexas.edu/category/2008/02/metric_spaces.h... https://golem.ph.utexas.edu/category/2006/10/euler_characteristic_of_a_cate.html https://golem.ph.utexas.edu/category/2006/10/euler_character... (and of course there are certainly many that I'm missing) For more fun, see these slides from John Baez: https://math.ucr.edu/home/baez/counting/counting.pdf https://math.ucr.edu/home/baez/counting/counting.pdf
m3kw9 4y ago
So infinity-1 is smaller than infinity?
- Twisol 4y ago
  Not in terms of cardinality, at least. Consider the set of natural numbers {0, 1, 2, ...}. This set is clearly of infinite size. If we remove 0, then we have a set {1, 2, ...} with, in principle, one less element than we started with -- yet it is also clearly infinite. However, these sets have exactly the same number of elements (i.e. "infinity-1 is the same size as infinity") -- I can pair 0 with 1, 1 with 2, 2 with 3, and so on, so that every number in the first set is uniquely paired with a number in the second set, and vice versa. Arithmetically, I can write `f(x) = x + 1` to go from the first set to the second set, and `g(x) = x - 1` to go from the second set back to the first set, and "clearly" we don't lose any information performing `f` followed by `g` -- no two elements land in the same place in either direction.
opportune 4y ago
You need to consider the term “bijection” and what it means in practice. If I create a bijection of integers to integers f(x)=2x you can see that each member of the right side had exactly one member on the right, and vice versa, despite the left side containing some elements not on the right. There is no such bijection between ints and reals (by cantors theorem), but there is clearly a surjection of reals to ints (round down), so reals are of a larger infinite size. I think a layman could easily understand this personally, without fancy math terms like bijection or surjection, the main exception being Cantors Theorem which fortunately has a plethora of nice visual proofs. Anyway, the article is not really even about that but about potentially modifying ZFC to accommodate a particular outcome of the CH. I am quite interested in mathematics but to me this is a pointless philosophical argument because there don’t seem to be any useful or empirically verifiable results that depend on CH. Discussing CH to me seems to be more of a way for set theorists to troll each other and trigger cranks.
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hinkley 4y ago
Standup Maths recently tackled, "Is an infinite stack of ￡1 notes worth more than an infinite stack of ￡100 notes?" The answer is they are both worth ￡∞.
- lostmsu 4y ago
  Is it right for any systems that include infinity? What about surreals?
  - hinkley 4y ago
    He’s looking at C X n and saying B x n = C x n when n = infinity (the same infinity on both sides of the equation).
yarg 4y ago
Some of those infinities are the same, others are not. There's a one to one mapping between the integers and the cartesian product of the integers (I^I == I), but (I believe) that's not true for the reals (someone correct me if I'm wrong about that). However, you can take some subset of R^n and stretch it to infinity across all dimensions, giving you back R^n. So: There are the same number of integers as odd integers, There are more points on a continuous plane than on a continuous line. The number of points on a line is the same as the number of points on the edge of a circle. The number of points in a plane is the same as the number of points in a semiplane. There are the same number of rationals as integers. There are more reals than integers.
- ahahahahah 4y ago
  I think it was clear from there their comment that they not only understood that, but picked those examples specifically because of that understanding. Your comment would be fine if it were framed as clarifying that for those who didn't understand, but as it is it just seems to out you as them.
  - yarg 4y ago
    He may have stated the things that are not true to illustrate that they don't seem that different from those that are, but could have been clearer in delineating the two sets.
- paulddraper 4y ago
  Correct...except that there are the same number of points on a line as in a plane. Aleph_0^2 = Aleph_0 Aleph_1^2 = Aleph_1
  - yarg 4y ago
    How though (genuine question)? I can see space filling curves through I^I, but I can't see them covering R^I. How does a space-filling curve cover a plane without being filled with infinitesimal discontinuities?

anonGone73 4y ago

Infinite is un-enumerable, but you can carve it up in more or less ways.

SkyPuncher 4y ago

I've found it easier to reason about when I think of it as "how quickly does the set of number grow as you add more to it". It becomes particularly apparent when you think about it on a finite scale. 0 to 100 * 100 integers * 50 odd numbers, 50 even numbers * infinite floats (unless you only represent part of the float)

dwater 4y ago

It's been a long time since my number theory class, but aren't there the same number of integers and odd integers? Take the set of odd integers {... -3, -1, 1, 3, 5, ...} For each item, subtract one and divide the result by 2. Now you have the set of all integers without any insertions or deletions: {... -2, -1, 0, 1, 2, ...} Therefore the set of odd integers can be mapped 1:1 onto the set of all integers so they are of equal length.

ShamelessC 4y ago
They have the same cardinality.
renewiltord 4y ago
Your proof is correct.
deleted 4y ago
[deleted]
RheingoldRiver 4y ago
That's why 'Prove' is in quotation marks. Not all of GP's statements are true, though it would've been helpful if they'd added (true) and (false) tags next to them.

rtkwe 4y ago

> there are twice as many integers as odd integers > there are more rationals than integers Both of these are incorrect you can create a 1-1 mapping between all of these sets so they are the same "size". Things get unintuitive when you're dealing with infinites things that feel like they should be larger aren't when you examine them rigorously. For integers to odd integers the mapping is easy for each natural number n map n -> 2n+1. Mapping integers to rational numbers is more difficult to write into an equation but if you lay them out in a a grid defined by numerators and denominators x/y you can snake along this grid to eventually map every rational number to a corresponding natural number (ie positive integers which has the same cardinality as integers). http://www.cwladis.com/math100/Lecture5Sets.htm#:~:text=the%20natural%20numbers%3F-,The%20Cardinality%20of%20the%20Set%20of%20Rational%20Numbers,-We%20recall%20that http://www.cwladis.com/math100/Lecture5Sets.htm#:~:text=the%... > there are more points on a plane then on a line Going further R (points on a line) to R^2 (points on a plane) is also the same cardinality. The proofs are over my head as a 10 years past math minor but they're out there. Including this one that goes from R^3 to R. https://math.stackexchange.com/posts/183383/revisions https://math.stackexchange.com/posts/183383/revisions

hallucy 4y ago
This is literally the point of the comment. To a layperson not familiar with this approach to measuring infinities, all of the examples are “correct”. To a mathematician that hasn’t heard of uncountability, all are “incorrect”. (And learning the last is actually “correct” is surprising).
- deleted 4y ago
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- jimsimmons 4y ago
  That’s not what GP says though. They say their last statement is correct when it’s not. There are only two types of infinities: countable and uncountable. Both reals and rationals are not countable
  - 4y ago

vacuumcl 4y ago

Half your examples are wrong, but maybe it is your point. Although it wasn't clear to me the first time I read your comment. - The cardinality of the odd and even integers is the same. - It is true there are more points on a plane then on a line (Cantor's theorem.) - The circle is the compactified real line, i.e. it can be represented as the real numbers with one additional point (the point at infinity). In terms of cardinality they are the same since they just differ by one point which does not change the cardinality. - There are not more points on a plane than a half-plane, you can find a bijective mapping between them easily. - There are more rationals than integers: not true, they are both countable sets of the same cardinality. - There are more reals than rationals, this is true (again Cantor.)

function_seven 4y ago
Parent is making the same point you are. It’s a surprise to the intermediate mathematician when the last bullet point turns out to be true.
deleted 4y ago
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jdkee 4y ago
"- It is true there are more points on a plane then on a line (Cantor's theorem.)" There is a bijection between the points on a line and the points on a plane or in any n-dimensional space.
- vacuumcl 4y ago
  Yes, that was a bad mistake on my part. Thanks for pointing it out!..

jltsiren 4y ago

Isn't that more like first-year material in mathematics and in more theoretically oriented CS programs? Once you start talking about injections, surjections, and bijections, you may as well prove some basic results about the sizes of sets of numbers.

yodsanklai 4y ago
There are basically no prerequisites to get a grasp of these ideas. I remember being introduced to this in one of Raymond Smullyan's book when I was a kid.

sleepyams 4y ago

Good points, a large chunk of math is different formalizations of the notion of size, for example (in order of increasing absurdity): https://en.wikipedia.org/wiki/Cardinality https://en.wikipedia.org/wiki/Cardinality https://en.wikipedia.org/wiki/Measure_(mathematics) https://en.wikipedia.org/wiki/Measure_(mathematics) https://en.wikipedia.org/wiki/Ultrafilter_(set_theory) https://en.wikipedia.org/wiki/Ultrafilter_(set_theory) https://en.wikipedia.org/wiki/Euler_characteristic https://en.wikipedia.org/wiki/Euler_characteristic https://golem.ph.utexas.edu/category/2008/02/metric_spaces.html https://golem.ph.utexas.edu/category/2008/02/metric_spaces.h... https://golem.ph.utexas.edu/category/2006/10/euler_characteristic_of_a_cate.html https://golem.ph.utexas.edu/category/2006/10/euler_character... (and of course there are certainly many that I'm missing) For more fun, see these slides from John Baez: https://math.ucr.edu/home/baez/counting/counting.pdf https://math.ucr.edu/home/baez/counting/counting.pdf

m3kw9 4y ago

So infinity-1 is smaller than infinity?

Twisol 4y ago
Not in terms of cardinality, at least. Consider the set of natural numbers {0, 1, 2, ...}. This set is clearly of infinite size. If we remove 0, then we have a set {1, 2, ...} with, in principle, one less element than we started with -- yet it is also clearly infinite. However, these sets have exactly the same number of elements (i.e. "infinity-1 is the same size as infinity") -- I can pair 0 with 1, 1 with 2, 2 with 3, and so on, so that every number in the first set is uniquely paired with a number in the second set, and vice versa. Arithmetically, I can write `f(x) = x + 1` to go from the first set to the second set, and `g(x) = x - 1` to go from the second set back to the first set, and "clearly" we don't lose any information performing `f` followed by `g` -- no two elements land in the same place in either direction.

opportune 4y ago

You need to consider the term “bijection” and what it means in practice. If I create a bijection of integers to integers f(x)=2x you can see that each member of the right side had exactly one member on the right, and vice versa, despite the left side containing some elements not on the right. There is no such bijection between ints and reals (by cantors theorem), but there is clearly a surjection of reals to ints (round down), so reals are of a larger infinite size. I think a layman could easily understand this personally, without fancy math terms like bijection or surjection, the main exception being Cantors Theorem which fortunately has a plethora of nice visual proofs. Anyway, the article is not really even about that but about potentially modifying ZFC to accommodate a particular outcome of the CH. I am quite interested in mathematics but to me this is a pointless philosophical argument because there don’t seem to be any useful or empirically verifiable results that depend on CH. Discussing CH to me seems to be more of a way for set theorists to troll each other and trigger cranks.

deleted 4y ago

[deleted]

hinkley 4y ago

Standup Maths recently tackled, "Is an infinite stack of ￡1 notes worth more than an infinite stack of ￡100 notes?" The answer is they are both worth ￡∞.

lostmsu 4y ago
Is it right for any systems that include infinity? What about surreals?
- hinkley 4y ago
  He’s looking at C X n and saying B x n = C x n when n = infinity (the same infinity on both sides of the equation).

yarg 4y ago

Some of those infinities are the same, others are not. There's a one to one mapping between the integers and the cartesian product of the integers (I^I == I), but (I believe) that's not true for the reals (someone correct me if I'm wrong about that). However, you can take some subset of R^n and stretch it to infinity across all dimensions, giving you back R^n. So: There are the same number of integers as odd integers, There are more points on a continuous plane than on a continuous line. The number of points on a line is the same as the number of points on the edge of a circle. The number of points in a plane is the same as the number of points in a semiplane. There are the same number of rationals as integers. There are more reals than integers.

ahahahahah 4y ago
I think it was clear from there their comment that they not only understood that, but picked those examples specifically because of that understanding. Your comment would be fine if it were framed as clarifying that for those who didn't understand, but as it is it just seems to out you as them.
- yarg 4y ago
  He may have stated the things that are not true to illustrate that they don't seem that different from those that are, but could have been clearer in delineating the two sets.
paulddraper 4y ago
Correct...except that there are the same number of points on a line as in a plane. Aleph_0^2 = Aleph_0 Aleph_1^2 = Aleph_1
- yarg 4y ago
  How though (genuine question)? I can see space filling curves through I^I, but I can't see them covering R^I. How does a space-filling curve cover a plane without being filled with infinitesimal discontinuities?