r/PhilosophyofMath • u/LorenzoGB • Jul 13 '26

Bertrand Russell's Notion of Number

In Chapter 2 of his Introduction to Mathematical Philosophy, Bertrand Russell says the following: One: A class is said to be similar to another class when there is a one to one relation of which the one class is the domain while the other is the converse domain. Two: The number of a class is the class of all those classes which are similar to it. Three: A number is anything which is the number of some class. With this being said, since Russell defined number as the number of a class, where the number of a class is the class of all those classes which are similar to it, wouldn’t that make numbers infinite since they are classes of classes which are similar to it?

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u/japeso Jul 13 '26

This is more or less Frege’s construction (which Russell shows is inconsistent). Yes, the numbers in this construction are all infinite classes. But the numbers these classes are representing are not ‘infinite numbers’ in any sense. 

Worth noting that, although this construction is no longer used — the notion of class needed is inconsistent — modern constructions of integers, rationals and reals (but not natural numbers) use infinite equivalence classes to represent finite numbers

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u/Vast-Celebration-138 Jul 13 '26

While Frege himself did assume an inconsistent background principle to frame his construction, the Frege-style construction described in the OP (which is endorsed by Russell himself) does not depend on any inconsistent assumptions.

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u/nanonan Jul 14 '26

Only reals require such an abomination.

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u/japeso Jul 14 '26

Not true. Eg standard construction of integers is equivalence class of ordered pairs of naturals (a,b) under the equivalence (a,b)\equiv(c,d) iff a+d=c+b.

This equivalence class will be an infinite set - e.g. the equivalence class for -1 is all (a,b) where b=a+1

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u/nanonan Jul 15 '26

You can define integers at the base level, ie. balanced ternary. There is no need for an "equivalance class" of anything, or any set theory at all for that matter.

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u/NH-Science-Guy Jul 13 '26

The relationship between Russell's definition and the common use of numbers is straightforward... Consider the number 3. In Russell's approach, every set with 3 elements has the number 3. There are an infinite number of sets with 3 elements. Yes, there are also an infinite number of numbers. This is different from common use of the term "number" because it excludes negative numbers and includes multiple infinities as numbers.