Why are pure powers of the empty set insufficient as a definition for ordinals?

Question

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 posteriori 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?

Answer 1

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?

this sequence is not of powersets, as successive applications of powerset increse cardinality; rather, these are Zermelo's ordinals, and while they seem to match with von Neumann's ordinals as far as (the) ordinal rank (function) is concerned, they don't work at all when one wants to look at "well-ordered sets and their structure-preserving maps", that is, (possibly strict) monotone maps: there are two monotone maps from (1, <) to (2, <), but only one map from {{}} to {{{}}} (at all), so there's a lot of information loss, while the von Neumann's ordinals do work, so an answer for

What properties did Von Neumann’s definition accomplish?

is that they are a 'reasonable' skeleton for "well-ordered sets and monotone maps"

edit: just noticed I have no idea how Zemelo's ordinals look like at ω or higher, so the bit about rank is suspicious, but really, since they are no good at the first finite levels already, it's better to just forget about them

Answer 2

As Conifold and Michael Carey mentioned in the comments, the Zermelo ordinals do not extend to infinite sets. I want to emphasize this because it's very important to the way ordinal numbers are used. Very often (perhaps nearly always) when one is using the von Neumann ordinals, the infinite versions of the ordinals come into play.

For example, we typically use the ordinals to define cardinality in that |X| is the least ordinal number bijective to X, so we'd need to find some other way to formalize size. With the von Neumann ordinals, we get a series of infinite sets of increasing cardinality (starting with ω₀, ω₁, ω₂, ω₃, and continuing on for every ordinal subscript) and we know that every infinite set is bijective to ωₙ for some ordinal n (assuming the axiom of choice).

There are some other convenient things about the von Neumann ordinals, like how A ≤ B just means A ⊆ B, or how |X| = X for finite X, but infinity is a big one.

Answer 3

Though the question as stated does not indicate a concern with the matter, I should like to say that if we founded the world of sets on a Quine atom, then it would not be possible to use the singleton operation on this atom to pass to a higher/greater set, because a Quine atom is a self-singleton. That is why, to get a cumulative hierarchy built "on top of" a Quine atom, they (tend to) use the powerset operation instead. (Or some generic-sounding Wf(X) class operation, see here.) (And for all I know, there are a zillion things like this that I don't know about.)

Now, if Q = {Q}, then even so, Q has an empty subset, which is still tantamount to zero, because generally "there is a set that doesn't have any of Q's elements, or any other elements, here, for that matter," and that plays the role of the ur-empty set, etc. If we let Q = {Q} := 1 (i.e. if that was our alternative to the Zermelo or von Neumann finite ordinals as such), then to get to a set with two distinct elements, we can't just take {{Q}}, because that's still just {Q} and then Q, ultimately. But so if we take ℘(Q), we get {0, Q}. We can let this be 2, can't we? Unfortunately, if we want to "land on" each and every ordinal "in a row, and not in implicit whole chunks at a time," this will not do, for our next stop is {0, {0}, {Q}, {0, Q}} (I think), etc.

Still, we wouldn't get a facsimile of the finite ordinals at all from singleton or tupleton operations on Quine atoms as such, it seems to me.