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Given the discussions on Refuting the Anti-Cantor Cranks in which I left a comment that I suspected would not get any attention, I decided to start a new thread instead.

Now, I can't read the mind of "anti-Cantorians" so I don't really know what their objections are. But I get the impression that most reasonable objections aren't based upon the validity of the proofs that $\mathbb{R}$ has higher cardinality than $\mathbb{N}$. That seems pretty undisputable.

But a viewpoint I personally haven't been able to shake is the notion of cardinality as capable of describing "sizes" of sets. Now, different people may have different views on which criteria this notion of size absolutely needs to adhere to and which we can discard, but I think it's certainly thinkable to require that any reasonable notion of size needs to satisfy $B \subset A \Rightarrow \text{size}(B) < \text{size}(A).$

Starting from the assumption that $B \subset A \Rightarrow \text{size}(B) < \text{size}(A)$, then the statement "if two sets can be put into one-to-one correspondence with each other, then they are of the same size" leads to a contradiction and can therefore not be true, leading us to a chicken-and-egg axiom argument with no right answer.

Which makes me wonder. Could the interpretation of cardinality as the "size of a set" simply be seen as a mneumonic? Am I wrong in assuming all mathematical results relating to cardinalities of sets will be as relevant without thinking of it as a "size"?

What does the interpretation really buy us and what would we lose by leaving the notion of the size of an infinite set undefined?

EDIT: The very nice discussion in Relative sizes of sets of integers and rationals revisited - how do I make sense of this? is probably sufficient for this question, but I'll reiterate my position since it apparently wasn't as clearly formulated as I thought given the down-votes.

We want a nice, intuitive notion of the "size" (or better yet, "numerosity") of a set. So we make a list of properties that we know hold for finite sets that ideally we would want to hold for infinite sets. Two of these:

1) If you can take all the elements of set $A$ and place each element next to a unique member of set $B$, then $A$ and $B$ are "of the same size".

2) If you take a set $A$ and proceed by removing some elements from it, then you will have a set smaller in size than you started out with.

Taking the first one as an axiom leads to the concept of "cardinality", causing property 2 to lead to a contradiction, while taking the second one as an axiom leads to the first one causing a contradiction. Clearly we can't have both of them.

From some answers and comments, you get the impression that expecting the second property to hold is a "fallacy", sprung through some naive expectation that properties of finite sets should automatically carry over to infinite sets. But it seems to me that the same could be said for the first property. The only reason to prefer using #1 as a definition seems to be "more interesting mathematics spring from it", which is clearly an excellent reason - but it doesn't make it a more natural candidate for capturing the notion of size than #2.

EDIT #2: There was an excellent point brought up in the question I linked above which is the formulation of various closure operations on sets - for example the convex hull of a $A$ is the smallest convex set containing $A$. So letting $A = \{0,1\}$, what's its convex hull? Well, $[0,1]$ certainly can't be the unique correct answer, since $\mathbb{R}$ is of the same size! Not that this invalidates the cardinality concept, but it shows that maybe we should be careful with equating cardinality with size.

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Benjamin Lindqvist
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  • $B\subset A$ $\Rightarrow$ size$(B)<$size$(A)$ is not correct. But $B\subset A$ $\Rightarrow$ size$(B)\leq$size$(A)$ would be better. Because $\Bbb N\subset\Bbb Z$ but size$(\Bbb N)=$size$(\Bbb Z)$. – Gregory Grant Dec 26 '15 at 20:34
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    That is certainly way the less intuitive definition in my mind. $B \subseteq A \Rightarrow$ size$(B) \leq $ size$(A)$ would be fine and correspond to both of our notions. – Benjamin Lindqvist Dec 26 '15 at 20:36
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    @Gregory: Sure, if you say that size means cardinality. But the question starts from the assumption that it would be interesting to have a notion of size for which that would be correct. –  Dec 26 '15 at 20:36
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    Could you put your question up front? It is a lot to read before getting to the question, and thus feels more like an invitation to discussion. I sometimes suggest to people having trouble that cardinality can also be seen as a measure of complexity. This explains why the real numbers are still non-denumerable when you consider them constructively. – Thomas Andrews Dec 26 '15 at 20:37
  • @MikeMiller Ok but isn't it trivial to show such a notion cannot exist, because otherwise you could create an infinite strictly decreasing sequence of natural numbers which is impossible. – Gregory Grant Dec 26 '15 at 20:44
  • @ThomasAndrews I actually think that sounds like a much more natural interpretation of cardinality rather than size. Not sure what the best edit is though, I feel like a mod maybe would do a better job. – Benjamin Lindqvist Dec 26 '15 at 20:44
  • The very relevant http://math.stackexchange.com/questions/1393/relative-sizes-of-sets-of-integers-and-rationals-revisited-how-do-i-make-sense?rq=1 just popped up on the sidebar... – Benjamin Lindqvist Dec 26 '15 at 20:50
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    In my opinion statements like "cardinality describes size of a set" are meant to provide intuition to the definition of cardinality. For finite sets, the definition captures our intuition of "size", so it's reasonable to keep using it for infinite sets. But if you disagree with this intuition for infinite sets, there are other ways to describe sizes of sets. – Wojowu Dec 26 '15 at 20:50
  • @Wojowu I understand and agree, but my requirement on the size operator holds for finite sets in the same way as the "one-to-one" definition does. So imo it is reasonable to expect this requirement to carry over to the infinite case in the same way. – Benjamin Lindqvist Dec 26 '15 at 20:55
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    @Benjamin: I disagree. Most of the problems that beginners have with infinite sets result from expecting them to behave like finite sets. The expectation is understandable, but I don’t think that it’s reasonable. – Brian M. Scott Dec 26 '15 at 20:59
  • @BrianM.Scott I'm saying it is no less reasonable to expect the proper subset property to carry over than it is reasonable to expect that "two sets have the same size if they can be placed in one-to-one correspondence to on another" will carry over. Both are intuitively appealing properties that trivially hold for finite sets. – Benjamin Lindqvist Dec 26 '15 at 21:03
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    @Benjamin: And it’s trivial to verify that for infinite sets they aren’t equivalent, so at that point it comes down to deciding which is a more useful notion of size, and on that score there’s no real question. I think that calling it size is more than a mnemonic, but I also think that calling it size has little to do with the problems that beginners have. (And the anti-Cantorian crackpots have other problems.) – Brian M. Scott Dec 26 '15 at 21:09
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    @GregoryGrant "Ok but isn't it trivial to show such a notion cannot exist, because otherwise you could create an infinite strictly decreasing sequence of natural numbers which is impossible." That would only be true if you assign infinite sets sizes which are natural numbers. But why would you do that? In the context of cardinality, we use the notion of transfinite cardinals to define the cardinality of infinite sets. I assume we'd have to develop some similar notion of infinitely large numbers here. – Keshav Srinivasan Dec 26 '15 at 21:26
  • Also relevant: http://math.stackexchange.com/questions/40309/cardinality-density/40318#40318 and http://math.stackexchange.com/a/182251/622 and http://math.stackexchange.com/questions/242057/why-the-principle-of-counting-does-not-match-with-our-common-sense/242064#242064 – Asaf Karagila Dec 26 '15 at 21:26
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    I'm not convinced that this question is actually about mathematics rather than philosophy. Of course it doesn't matter for any mathematics whether you choose to call cardinality "size" or not; it's just a word. – Eric Wofsey Dec 27 '15 at 01:47

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Why any reasonable notion of size should satisfy $A \subsetneq B \implies size(A) < size(B)$?

Mass is a good notion of "size", but we can have a bar of chocolate surrounded by vacuum and it does not respect your properties (ignoring any physics "mumbo-jumbo"). Mathematically, measure (in the context of measure theory) is also a good notion of "size", and it does not respect your property either.

Not only that, but cardinality is not a measurement of "size". It measures "correspondencicity". This is true even for finite sets. When we count them (for example, with fingers), we are corresponding each finger with the elements of the finite set. For instance, if you take $10$ balls of steel, and put them scattered in a closed room , the "size" that they are enclosing is very big. If you cluster them in the table in that room, not so much. However, I can correspond to each element of the previous arrangement an element of the new arrangement in a bijective manner. This is the intuition from finite sets.

As a sidenote, based on the comments, I feel this personal input may be useful: In my understanding, intuition is when you work, see or deal with something on a regular basis and has developed an acquaintance with the subject, sufficient enough for you to be able to infer something without a clear logical concatenation. This seems to aggree with the entry on the wiktionary for intuition:

Noun, intuition ‎(plural intuitions)

  • Immediate cognition without the use of conscious rational processes.

  • A perceptive insight gained by the use of this faculty.

and also to the "colloquial usage" section: "Intuition, as a gut feeling based on experience, (...)"

Therefore, a person who has dealed with finite sets has no intuition with infinite sets. Period. What he has is naivety:

Naivety (...) is the state of being naïve, that is to say, having or showing a lack of experience, understanding or sophistication, often in a context where one neglects pragmatism in favor of moral idealism.

or

Adjective, naive

Lacking worldly experience, wisdom, or judgement; unsophisticated.

(of art) Produced in a simple, childlike style, deliberately rejecting sophisticated techniques.

And one of the utmost goals of Mathematics is to get rid of naivety.

Aloizio Macedo
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  • I don't see how equating the size of a set to how much space its element would enclose if we scattered them makes much sense. We're trying to define numerosity, nothing else. Further, I think it's far from obvious that "measure" disagrees with "my" notion of size more than it disagrees with Cantor's. – Benjamin Lindqvist Dec 26 '15 at 21:16
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    You misinterpreted the point of my answer. You said that "any reasonable notion of size needs to satisfy..." implying any reasonable general, intuitive notion of size should satisfy the property you mentioned, for which I gave counter-examples.

    Now, what you say is precisely the point: "We're trying to define numerosity..." Note that this is a particular notion of "size". It shouldn't, and of course won't be equivalent to other notions of "size". However, as argued, there is no reason to think that it should satisfy your property. (...)

     – Aloizio Macedo Dec 26 '15 at 21:21
  • (...) The reasoning that it satisfies it for finite sets should imply it satisfies it for infinite sets is not an argument, is not intuitive, it is only a fallacy, much like saying that "If it holds for 1,2,3 then it holds for all natural numbers" is a fallacy. – Aloizio Macedo Dec 26 '15 at 21:21
  • Again, you're arguing as if cardinality is equivalent to numerosity. The whole point of my post is that cardinality starts with us expecting that the "one-to-one $\Leftrightarrow$ same size" property of finite sets to carry over to infinite sets. There's no "fallacy" involved, in either side of the argument, just two different relaxations on the notion of size. – Benjamin Lindqvist Dec 26 '15 at 21:28
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    @BenjaminLindqvist If that's the case, what is special about cardinality? This happens frequently in mathematics, so why not discuss all instances? There is something about "cardinality" that, for some specific reason unknown to me (maybe the fact that saying something loosely as "$\mathbb{N}$ has the same number of elements as $\mathbb{Q}$" is shocking and sounds intellectual) that people tend to discuss this issue so much, but this is not a particular event at all.

    For instance, let's proceed analogously as you did: (...)

     – Aloizio Macedo Dec 27 '15 at 02:00
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    (...) "We want a nice, intuitive notion of "\sin" of a complex number. So we make a list of properties that we know hold for $\sin$ in $\mathbb{R}$ that ideally we would want to hold for $\sin$ of complex numbers. Two of these:

    1. $\sin$ should be differentiable (and we now want this to hold in the "complex" sense of the word)

    2. $\sin$ should be bounded.

    Taking the first one as an axiom leads to the usual concept, causing property 2 to lead to a contradiction, while taking the second one as an axiom leads to the first one causing a contradiction. Clearly we can't have both of them." (...)

     – Aloizio Macedo Dec 27 '15 at 02:00
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    I therefore reiterate: Wanting to have something is not intuition. If you want to attain yourself to concepts which are not "as useful" as others, that is not intuition. And this is widely understood in most areas of mathematics, but for some reason this springs a lot of "philosophical debate" when we talk about sets and cardinality. – Aloizio Macedo Dec 27 '15 at 02:02
  • I'm sorry but this is just a pointless detour that I've no interest in following up further. Congratulations on finding one instance where we can't have all we wish for, but in general, observing a property for a special case and wishing it to hold for the general case is precisely how we construct useful definitions and in fact is precisely the method Cantor employed when he defined cardinality. Good day. – Benjamin Lindqvist Dec 27 '15 at 07:48
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    No. It is not observing a property. It is observing the useful property. And again, this is ubiquitous of Mathematics, not "one instance" that I've found. It was the first thing I thought of. Examples are endless. "We want a nice, intuitive notion of "compactness" of a space (...) Two of these: 1) Every closed, bounded set must be compact. 2) Every set such that every sequence must have a convergent subsequence is to be compact(...)". I'll refrain from commenting, since my arguments were attacked via ad hominem a while ago, and this is not healthy for any discussion. – Aloizio Macedo Dec 27 '15 at 14:55