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Definition: The cardinal number of a class K is the class of all classes equinumerous with K.

I understand that cardinal numbers represent the number of elements in a class, and I understand that equinumerous classes hold the same number of elements.

Am I therefore to interpret this definition as stating: The number of elements in a class k is the same number of elements in the class of all classes with the same number of elements as k?

If this is not correct, and I'm assuming it is not, what is the correct laymen translation of that definition? My problem is understanding how the concept of a class translates to a cardinal number.

  • Note that the modern definition in ZFC set theory is that the cardinal k is the least ordinal a such that |k| = |a|. This definition avoids the use of proper classes (which don't have formal meaning in ZFC) and makes a cardinal into a well-defined set. en.wikipedia.org/wiki/Cardinal_number. – user4894 May 11 '16 at 19:00
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    I'm voting to close this question as off-topic because it is purely mathematical. – user2953 May 11 '16 at 21:58
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    @Keelan It is mathematical but it also has a philosophical part. See the answer below. – Eliran May 11 '16 at 23:22
  • @Keelan it is a question about the logical construction of a mathematical concept, and I believe that logic ultimately belongs to philosophy. – IgnorantCuriosity May 12 '16 at 1:00
  • The line is blurry, but this particular issue is closer to the technical side of set theory, and therefore to Math SE. I assume what was intended is something like "the number of elements in a set k is the equivalence class of all sets equinumerous with k" (the OP sentence is circular and quaint), which is "morally" right. But "equivalence class of all sets" uses the unrestricted comprehension schema which produces Russell's paradox, so it is disallowed in ZFC. – Conifold May 12 '16 at 4:27
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First note (as @user4894 commented) that this isn't how cardinal numbers are defined in set theory (ZFC). The definition that you have given is closer to Frege's definition of number.

What does the definition mean?

That a cardinal number is itself a class that contains, as elements, all of the classes that are equinumerous with each other. (You can think of this as the equivalence class of the relation of equinumerousity.)

Example

The cardinal number of the class {Frege, Russell} is 2. Now 2 is itself a class that looks like this:

{{Frege, Russell}, {Plato, Aristotle}, {me, you}, ... [and so on for every class with 2 members] ...}

(Note that this isn't circular as it might seem and can be given a precise definition, in Frege's system for example.)


Update: some context

The main idea behind this conception is that numbers somehow involve second order concepts. That is, having a certain number is a property of 'plain' (first order) concepts. For example, the earth falls under the concept 'planet of the solar system', which itself falls under the concept 'has 8 members'.

Frege used this conception of number in his logicist project, attempting to show that arithmetic is reducible to logic. He developed a system in which he derived some theorems of arithmetic but this system was found inconsistent since it was subject to Russell's Paradox.

You can read more and see some technical details here.

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    The definition "closer to Frege's definition of number" is due to Bertrand Russell. – Mauro ALLEGRANZA May 11 '16 at 19:32
  • @MauroALLEGRANZA I'm pretty sure this appears (informally) in Frege's Grundlagen (1884), which is prior to Russell. – Eliran May 11 '16 at 19:33
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    Thanks for the explanation! I have a followup question; what use does this definition of number have? And what additional insight into the concept of number does this definition provide, if any? – IgnorantCuriosity May 11 '16 at 19:54
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    Great answer, especially for introducing the idea of second order concept! – user20153 May 11 '16 at 20:35
  • @IgnorantCuriosity: my 2 cents is that it ties the concept of natural number, inductively defined, to the more general notion of cardinal number. Equinumerosity alone does not give numbers, only 1-1 correspondence. SSZ is 2 but is not a set. But if you define S and Z in terms of sets you get Nat and cardinality. Also under this view numbers are essentially structural relations rather than quantities or magnitudes. – user20153 May 11 '16 at 20:51
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Your interpretation is off-the-mark, but your question is on-the-mark.

The number of a class k is another class, α, whose members are classes similar to the class k. This definition does not mean that k and α have the same number of elements.

Two classes are said to be similar to each other if they are the domain and converse domain of a one-one relation - i.e., in plain English, have the same number - but in order to avoid circular definition let's pretend we do not know what a number is at this point. One-one relation's definition is based on the concept of relation and the definition of 1; 1 is defined as the class of all unit classes; unit class is defined in terms of identity: the unit class of a term is the class of terms which are identical with the term in question.

Suppose there are only three individuals in the world: Kramer, Newman and Elaine. Then the unit class of Kramer is {Kramer}; 1 is the class of all unit classes: {{Kramer}, {Newman}, {Elaine}}. Let k = {Kramer, Newman}, then the number of k is 2, where 2 is defined as the class of all couples: {{Kramer, Newman}, {Newman, Elaine}, {Elaine, Kramer}} - let's call it α. We can see that α has three elements, k has only 2, but each of α's element is similar to k.

In Principia Mathematica, Whitehead & Russell successfully deduced ordinary arithmetic from this definition of number.

The philosophical insight behind this definition is that numbers are universals, i.e. they are properties. The mind can single out a property - called universal in philosophy - by looking at multiple things that have this property in common. Take the colour red for example. No one can see red alone detached from the things that have this colour. When you say to a child, "this is red," while pointing at a red apple, he is not sure what you mean: the whole object or the colour of it or the shape of it? But if you continue to point at other red things, such as red books and red towels, and say "this is red," eventually he will know what red means: it is some sort of visual sensation. In other words, red is what all these red things have in common.

Similarly, one is what all singles - viewed as unit classes - have in common; two is what all couples have in common; three is what all triples have in common.

Most people are somewhat surprised when they crack open a double-yolk egg: due to their previous experiences of seeing the content of an egg, there is a property so unconsciously anticipated that, when it is different, we will be surprised. That property is 1.

If you've ever been a receptionist of some event, you are very likely to hear exclamation like this: "they all come in pairs today!" Such exclamations show that humans have the ability to single out a property by looking at what several groups have in common, and the impression of this property is so striking that an exclamation is triggered.

A class is uniquely defined by a property that is possessed by all its members and no one else. W&R used classes to represent properties: red is represented by the class of all red things; 1 is represented by the class of all unit classes. In Principia Mathematica, W&R showed that representing a property by a class is sufficient to deduce ordinary mathematics.

This way of defining number is consistent with how the mind works: humans and many other animals can sense numbers just like they sense colours and tunes; and numbers stimulate the senses just like colours and tunes do. The platonic heaven is superfluous.

For a brief sketch, see definition of cardinal, ratio and reals.

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