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I was going through this link and a comment by user fkraiem mentioned that a characteristic property of an infinite set is that it has the same cardinality as at least one of its proper subsets. But how does that make any sense? For example, if $A=[1, \infty)$ has the same cardinality as $B=[1, 2)$ as per the bijection $f:A\rightarrow B, f(x)=\frac{-1}{x}+2$, then wouldn't that mean there are no real numbers in $[2, \infty)$?

Edit: I am in high school only and don't know much about infinite sets, so please clarify any issue with my question in the comments before downvoting.

Integreek
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    Not any of its subsets, it has the same cardinality as at least one of its proper subsets. – Thomas Andrews Sep 19 '24 at 03:46
  • @ThomasAndrews thanks for the suggestion, I edited my post. – Integreek Sep 19 '24 at 03:47
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    "then wouldn't that mean" No, why would it mean that? – Noah Schweber Sep 19 '24 at 03:49
  • @NoahSchweber Doesn't cardinality mean the relative size of a set, i.e., here, it can be thought of as the number of reals in the interval relative to another interval? – Integreek Sep 19 '24 at 03:50
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    The natural numbers ${0,1,2,\dots}$ are in bijection with the even natural numbers ${0,2,4,\dots}.$ That certainly doesn't mean that there are no odd numbers. Defining cardinality in terms of bijections means you have to throw away some of your intuitions about "size" from finite sets. – Thomas Andrews Sep 19 '24 at 03:55
  • So for infinite sets, what does cardinality signify? – Integreek Sep 19 '24 at 03:56
  • Size. But not all the intuition from finite subsets goes through. Starting with the definition of "infinite" itself! It tells you "there is something very different about 'size' when dealing with infinite sets from dealing with finite sets; user beware" – Arturo Magidin Sep 19 '24 at 04:00
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    It's still size. You just have to abandon the idea that adding additional elements always makes something bigger. (In constructive mathematics, cardinality becomes something more like "complexity." If you are having real trouble with these ideas, you can think of it that way.) – Thomas Andrews Sep 19 '24 at 04:00
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    "Doesn't cardinality mean the relative size of a set, i.e., here, it can be thought of as the number of reals in the interval relative to another interval?" I think you may be confusing "size:how many elements a set has" with "size:how large a physical space a set takes up". Those are entirely different concepts (and both badly and inaccurately worded). Finite means "fixed sum" you have a set amount and if you remove or add one you have another amount. Infinite-- not so. There is nothing contradictory about $[1,\infty), [1,2),[2,3),[3,4)$ all having the same cardinality of elements. – fleablood Sep 19 '24 at 05:13
  • @fleablood thank you for the clarification! – Integreek Sep 19 '24 at 06:22
  • @ThomasAndrews what is "complexity"? – Integreek Sep 19 '24 at 15:16
  • It is an intuition only. @RakshithPL – Thomas Andrews Sep 19 '24 at 15:22
  • @ThomasAndrews I am not able to understand, what is constructive mathematics and cardinality in it becomes complexity exactly mean? – Integreek Sep 19 '24 at 15:25
  • If that metaphor doesn't work for you, then I don't think explaining it further in comments is wise. I certainly don't mean for you to dig into what "constructive math" means, since the idea is just a metaphor. But I mean that $[1,2)$ is "the same complexity" as $[1,\infty)$ might make more sense than "the same size." – Thomas Andrews Sep 19 '24 at 16:03
  • @ThomasAndrews I have not learnt about it yet, so that's why I am not able to get it. Any help would be appreciated :) – Integreek Sep 19 '24 at 16:04
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    You don't need to know constructive math to treat cardinality as a measure of "complexity" rather than a measure of "size." Constructive math is just the source of the idea, not needed to understand the idea. Sorry to have brought it up. – Thomas Andrews Sep 19 '24 at 16:07
  • @ThomasAndrews Ok sir, but anyways thank you for your patience :) – Integreek Sep 19 '24 at 16:08
  • https://philosophy.stackexchange.com/questions/111475/is-infinity-a-concept-or-a-word-empty-of-meaning/111525#111525 – Anixx May 11 '25 at 02:24
  • https://mathoverflow.net/questions/445368/why-surreal-numbers-cannot-be-extended-further-in-this-way-using-measure-approac/445370#445370 – Anixx May 11 '25 at 02:25

2 Answers2

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When you are first introduced to the "size" of a set, you are told "sets have the same size if there's the same number of elements in them," which is fine as long as there's only finitely many things! When you say "the set $A = \{a,b,c,d,e,f,g\}$ has seven elements" what you really mean is that "there is a bijection from $A$ to the set $S_7 = \{1,2,3,4,5,6,7\}$". Here, you can read "bijection" as meaning there is a way of labelling each element of $A$ with an element of $S_7$ in such a way that every element of $A$ gets a label and every label gets used somewhere.

When we talk about a set $B$ being infinite (i.e., not finite) what we mean is that there is no way of labelling elements of $B$ with a finite set $S_n = \{1,2,\dots,n\}$. But we still want to know how to talk about sets having the same "size" (or perhaps the same "size-complexity" as mentioned in the comments) so we come up with a way to generalize the finite-context definition into a broader setting.

With that in mind, "bijections" become a useful way to describe whether sets have the "same size." It's a tool that works in the finite setting, but it doesn't inherently rely on any notion of "finiteness". We can talk about there being a bijection between $X$ and $Y$ without ever knowing whether $X$ and $Y$ are finite or not. All we need to know is whether we can use elements of $Y$ to label elements of $X$ in such a way that every element of $X$ gets a label, and every label from $Y$ is used.

Since you're in high school, I don't expect that you've seen what a "bijection" is, so this answer may not be very satisfying, but hopefully there's some insight you can glean here. (Edit: just realized you used "bijection" in your post, so ignore this last line! Well done diving into these deeper ideas already!)

Kyle
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  • What is size complexity? – Integreek Sep 19 '24 at 04:56
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    Heuristically, "complexity" is a way of measuring how much time or effort something might take. For instance, a set of 7 elements is "less complex in terms of size" than a set with 700000000 elements, since (heuristically) it takes less time to count or "process" them all in some way. – Kyle Sep 19 '24 at 17:42
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This is absolutely counter-intuitive, but it is mathematically rigorous. The concept of cardinality is based on the idea of Cantor-Hume principle while what you are referring to in your question is called Euclid's principle, which can be formulated "the whole is greater than a part".

In short: cardinality rejects the Euclid's principle because it would make cardinality non-translation-invariant. By simple movement you can make a set into its subset (for instance, a set $\{0,1,2,3,\dots\}$ into $\{2,3,4,5,\dots\}$). Cantor wanted cardinality to be kept unchanged regardless of movements.

Of course, one can beuld a measure of labelled or ordered sets that respects Euclid's principle instead by sacrificing translation invariance.

Anixx
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