I recently discovered a philosophical term that gives expression to a paradigm that had been circling in my head. G. E. Moore discussed the “paradox of analysis”, which is similar to what I think of as “abductive reasoning”. Especially in philosophical mathematics, we are like “the dragon eating its own tail”, because we want to define something, but we don’t know what we want to define. It relates to this recent Philosophy SE question as well.

I reflected this morning on that “continuity” may be one such aporia. We think there is such a thing, so we want to understand better what explicit criterion gives rise to something which behaves in the way we want “continuity” to behave. But because we don’t know what continuity is yet, we are not able to say with definiteness, “Find me the essential properties X which lead to the a fortiori property y”. We want to deduce logical preconditions on property y, without knowing anything about property y. Simultaneously, we want to deduce property y as following from key logical preconditions X, while at the same time having no point of reference for what premises X should look like, except for with recourse to the as-of-yet undetermined y.

This appears to relate to how the concept of “ordinal numbers” were first defined:

Definition of an ordinal as an equivalence class

The original definition of ordinal numbers, found for example in the Principia Mathematica, defines the order type of a well-ordering as the set of all well-orderings similar (order-isomorphic) to that well-ordering: in other words, an ordinal number is genuinely an equivalence class of well-ordered sets. This definition must be abandoned in ZF and related systems of axiomatic set theory because these equivalence classes are too large to form a set. However, this definition still can be used in type theory and in Quine's axiomatic set theory New Foundations and related systems (where it affords a rather surprising alternative solution to the Burali-Forti paradox of the largest ordinal).

• Wikipedia

Roughly, it seems like Russell had the motivation to define “ordinal numbers”, because he wanted to capture the general idea of “order”. His original formulation did not work, but Von Neumann provided one that is now common:

Von Neumann definition of ordinals

The first several von Neumann ordinals:

0   =   {}  =   ∅
1   =   {0} =   {∅}
2   =   {0,1}   =   {∅,{∅}}
3   =   {0,1,2} =   {∅,{∅},{∅,{∅}}}
4   =   {0,1,2,3}   =   {∅,{∅},{∅,{∅}},{∅,{∅},{∅,{∅}}}}

Rather than defining an ordinal as an equivalence class of well-ordered sets, it will be defined as a particular well-ordered set that (canonically) represents the class. Thus, an ordinal number will be a well-ordered set; and every well-ordered set will be order-isomorphic to exactly one ordinal number. For each well-ordered set T, a \mapsto T_{<a} defines an order isomorphism between T and the set of all subsets of T having the form T_{<a} := \{ x \in T \mid x < a \} ordered by inclusion. This motivates the standard definition, suggested by John von Neumann at the age of 19, now called the von Neumann ordinals: “each ordinal is the well-ordered set of all smaller ordinals". In symbols, \lambda =[0,\lambda ). Formally:

A set S is an ordinal if and only if S is strictly well-ordered with respect to set membership, and every element of S is also a subset of S.

The natural numbers are thus ordinals by this definition. For instance, 2 is an element of 4 = {0, 1, 2, 3}, and 2 is equal to {0, 1} and so it is a subset of {0, 1, 2, 3}.

What properties did Von Neumann’s definition accomplish?

The following sequence is called the “pure” or “irreducible” power sets of the empty set:

{}, {{}}, {{{}}}, {{{{}}}}, …

Conceptually, this sequence ‘represents’ order, to me. Why is it insufficient, as a definition of ordinal?