Is there an alternative to Cantor's cardinalities that makes proper subsets smaller than their sets?

Question

Cantor defined an infinite set as a set whose subset can be placed in a one-to-one correspondence with its subset. That is, take the set of all natural numbers: {0, 1, 2, 3, 4,...}. From that set, you can form a subset of all even numbers: {0, 2, 4, 6,...}. There is a one-to-one correspondence between both these sets. After all, you can multiply each member in the set of all natural numbers by 2, and you’ll get the set of all even numbers. Because of this one-to-one correspondence, the two sets have the same cardinality, despite the set of even numbers being a subset of the set of natural numbers. This is counter-intuitive to me. How can a subset of a set be as big as the set itself? To make the intuition more concrete, consider the set of multiples of 1,000,000: {0, 1,000,000, 2,000,000,...}. There is a one-to-one correspondence between this set and the set of all natural numbers. So they have the same cardinality. This is really counter-intuitive.

From as far as I can tell, there are basically two axioms at play here:

A1: If all members of set A can be placed in a one-to-one correspondence with all the members of set B, then both sets have the same cardinality.

A2: If set A is a proper subset of set B, then A has a smaller cardinality than set B.

How Cantor defined infinite sets is that he accepted A1 and rejected A2. That is counter-intuitive to me. But mathematicians might say that my intuition is just wrong. So my question is whether we can define infinite sets in another way, i.e. accept A2 but reject A1. That is, the set of even numbers is smaller than the set of all natural numbers, even though both sets have a one-to-one correspondence. If you think this is counter-intuitive because sets with one-to-one correspondence with each other should have the same cardinality, and thus think this is wrong, why can’t I say the same when Cantor accepted A1 and rejected A2? Either way, there is something counter-intuitive and paradoxical.

So, my questions are:

1. Can we accept A2 and reject A1 for the definition of an infinite set?
2. What arguments, if any, are given for accepting A1 and rejecting A2?

Answer 1

The answer is affirmative. The only hard fact is that the Hume's principle (bijective sets have equal sizes) and the part-whole axiom of Euclid (the whole is greater than its part) are incompatible for infinite sets. It is not that your intuition is "wrong", but rather that any extension of "size" to infinite sets will be paradoxical, it has to discard one or the other. One can select one group of intuitions to keep, or one can select another. Cantor made one choice, and it was adopted in modern mathematics (for its technical benefits, among other things), but an alternative group of intuitions can be coherently taken instead. We just can not keep it all. If there is anything wrong here, it is expecting that our intuitive concept of size, developed from experience with finite sets, would carry over in full beyond the context in which it was developed.

This psychological obstacle is reflected in the long historical struggles with the concept of size for infinite sets, described e.g. in Mancosu's Measuring the Size of Infinite Collections of Natural Numbers numbers: Was Cantor's Theory of Infinite Number Inevitable? It is interesting that Bolzano, Cantor's precursor on set theory, rejected the Hume's principle, exactly because it conflicted with the part-whole axiom. Gödel , on the other hand, gave an influential argument that Cantor's choice in favor of the Hume's principle was "inevitable". Nonetheless, a coherent alternative, called numerosity, that adopts the part-whole axiom instead, was introduced by Benci in 1995, and developed by him, Di Nasso and Forti in 2000-s, see their Aristotelian Notion of Size. However, since the Hume's principle is now rejected we can not apply it to sets proper, they need to have additional counting structure, in this case they have to be labeled sets.

The idea is to split a set into boxes, each one containing only finitely many elements (with the same label), and count them by counting the content of the boxes, in sequence. The numerosity is built from the sequence whose n-th term counts the number of elements with labels up to n, i.e. in the first n boxes. This only works for countable sets, but there is an extension to arbitrary sets using ordinals. All integers (labeled by themselves) produce 1,2,3,4,5,6..., even integers (also labeled by themselves) produce 1,1,2,2,3,3..., which is strictly smaller starting from the second term, so their numerosity is strictly smaller.

The general construction is technical and is similar to the ultrapower construction of the hyperreal numbers - one takes numerosities to be equivalence classes of nondecreasing integer sequences that are equivalent modulo a Ramsey ultrafilter (whatever this means). But the numerosities do satisfy the part-whole axiom, and have nice algebraic properties. One can define arithmetic on them, and they add when disjoint labeled sets are put together. The trade-off is that there are plenty of bijective sets with different numerosities.

Here is Mancosu's commentary on Gödel's argument and how it fails for numerosities:

"Gödel’s reflection aims at showing that in generalizing the notion of number from the finite to the infinite one inevitable ends up with the Cantorian notion of cardinal number. The key step in the argument is the premise and the theory of numerosities can help us see that the premise already contains in itself the Cantorian solution. In fact, the premise takes as evident the request that “the number of objects belonging to some class does not change if, leaving the objects the same, one changes in any way whatsoever their properties or mutual relations (e.g., their colors or their distribution in space).” While the premise constitutes no problem when dealing with finite sets, one might question its acceptability in the realm of the infinite. Indeed, in the theory of numerosities we cannot grant the premise when it comes to infinite sets.

For, while it is possible to abstract from the nature of the objects themselves there is one type of relation that affects the counting, namely the way in which the elements are grouped. Such grouping makes no difference in the realm of finite sets of integers. But when we move to infinite sets a rearrangement of the grouping will in general affect the approximating functions and thus the numerosity of the set. Someone committed to the counting embodied in the theory of numerosities might thus reasonably resist accepting the premise on which Godel bases his argument and thus also resist the claim that the generalization of number from the finite to the infinite must perforce end up with the notion of cardinal number. In short, having a different way of counting infinite sets shows that while Gödel gives voice to one plausible intuition about how to generalize “number” to infinite sets there are coherent alternatives."

Answer 2

Axiom

A2: If set A is a proper subset of set B, then A has a smaller cardinality than set B

is intuitive.

Correct; since ancient time [see Euclid's Elements] it was assumed :

Common notion 5.

The whole is greater than the part.

But is also quite ancient the discovery that this principle may lead to problems in some cases; see the so-called Galileo's paradox.

Thus, Cantor's intuition to define a new way of "counting" the elements of a collection : the notion of one-to-one correspondence is prior to (and independent of) the concept of counting number.

The benefit of this approach was its applicability to every type of collections: finite and infinite.

In the case of finite collection*, the new approach was consistent with the usual one, based on counting numbers.

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Having said that, there are some alternative approaches to mathematical infinity; see e.g.

• Jan Mycielski, Analysis Without Actual Infinity, JSL (1981)

and

• Shaughan Lavine, Understanding the Infinite, Harvard UP (1994).

See also Finitism.

Answer 3

This is just a question of disambiguating “bigger” (or “smaller” or “size”).

There are two kinds of objects involved here:

• Sets of Xs (where X is some kind of objects). Such a set is a selection of Xs. You can picture it as a bag of points where each point has a different label (telling us which X it represents).

• Cardinal numbers. They are “shapes” of sets. You can picture a cardinal number as a bag of points without labels.

There is a mapping from sets of Xs to cardinal numbers, associating to each such set its cardinality. In the bag metaphor, it amounts to removing the labels, i.e. forgetting which Xs are members of the set. You’re left with the set’s “shape” (or cardinal number). This removing the labels is the abstraction of the nature of the elements in Cantor’s famous quote:

We will call by the name “power” or “cardinal number” of M the general concept which, by means of our active faculty of thought, arises from the aggregate M when we make abstraction of the nature of its various elements m and of the order in which they are given. Cantor starts with sets enumerated in some order, unlike us, which is why we don’t need to forget also the order in which they are given.

Now, there is a standard order relation on sets of Xs, namely inclusion: a set of Xs is smaller than or equal to another one if all Xs in the first are also in the second. They are the same if they have exactly the same elements.

There is also a standard order relation on cardinal numbers. Here, since the points don’t have any labels, you can’t ask for two cardinal numbers to have the same elements, they don’t have any. You can just see if you can map points of the first injectively into points of the second, in which case the first is smaller than or equal to the second. They are the same if you can map them bijectively. In practice, cardinal numbers are usually presented to us via some concrete presentation as some set of Xs, so that we define these injections or bijections in terms of Xs. (N.B. any spatial arrangement that may come to your mind is just extra baggage due to the bag metaphor, the points of a cardinal number have no spatial arrangement, they are not spatial at all.)

This gives rise to two different notions of “bigger” for sets of Xs:

1. A is bigger₁ than B if it’s bigger for the inclusion order of sets. For instance, the set of all natural numbers is bigger₁ than the set of even numbers. The only set of natural numbers which is the same size₁ as that of even numbers is itself. N.B. If you want to reify size₁, you can just take the size₁ of A to be A itself (that would be the alternative to cardinality of the question in your title).

2. A is bigger₂ than B if A’s cardinality is bigger than B’s, for the standard order on cardinal numbers. For instance, the set of all natural numbers is not bigger₂ than the set of even numbers, they are the same size₂, as witnessed by the “divide by 2” bijection between them, which makes the corresponding cardinal numbers equal.

There is nothing paradoxical if we don’t forget the indices (or use different words). Both notions of size coexist peacefully in mathematics. We just need to specify what we mean by “bigger” (mathematicians would usually be explicit, using, if the context isn’t enough, “is included in” or “has a greater cardinality than”).

The answers to your questions 1 and 2 would be, as written, “no” and “none”, but that’s just nitpicking: if you reformulate your A1 by substituting “size” for “cardinality” and your question 1 by substituting “for the definition of size of a set” for “for the definition of an infinite set”, then the answers are “yes” (just use size₁ instead of size₂) and “it depends on what notion of size you use”.

Answer 4

This is counter-intuitive to me. How can a subset be as large as the set itself?

This is because we are using two different notions of measure here and they are not complete commensurate.

Here's a physical example: take a rubber band and then pull it. It's now larger than it was before. Do you see this as being counter-intuitive? Of course not, it's something that's both readily visible and understandable from your own experience.

The language of sets is useful. And whilst Cantorian cardinality is a marvel, the main problem I have with it is that it has little use outside of its own domain. Compare this with the calculus which is still breaking new ground and proving its usefulness.