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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).

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?

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    These are called Zermelo ordinals:"With this definition each natural number is a singleton set. So, the property of the natural numbers to represent cardinalities is not directly accessible; only the ordinal property (being the nth element of a sequence) is immediate. Unlike von Neumann's construction, the Zermelo ordinals do not extend to infinite ordinals." That's why they are insufficient.
    – Conifold
    Apr 9 at 19:21
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    "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." I didn't read your post too closely, but it seems related to impredicativity. en.wikipedia.org/wiki/Impredicativity
    – J D
    Apr 9 at 19:32
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    Re your "The following sequence is called the “pure” or “irreducible” power sets of the empty set", the sequence is neither pure not irreducible power sets of empty sets, unlike ZF set theory, in Zermelo set theory these are just canonical representation perhaps realized as a well-ordered set constructed using the empty set, any ordinal itself is defined in terms of an order type (not a set) of all well-ordered sets in its equivalence class following Hume's principle... Apr 9 at 22:23
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    Every well-ordered set is isomorphic to a unique ordinal, that is why we say ordinals "measure" the length of well-orderings. Apr 9 at 22:31
  • Ok. Yeah I got confused about how to describe the Zermelo ordinals I guess. But yeah, seems like it’s a common question: math.stackexchange.com/questions/2975907/… Apr 13 at 0:05

2 Answers 2

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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

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  • they don't work at all when one wants to look at "well-ordered sets and their structure-preserving maps", that is, monotone maps: there are two monotone maps from (1, <) to (2, <), but only one from {{}} to {{{}}}, so there's a lot of information loss” - that’s interesting - but why is it better that we have 2 monotone maps instead of one? You’re saying the zermelo ordinals are perfectly adequate as a “rank function”, but the VN-ordinals express more - then I wonder why we can’t consider Z-ordinals as canonical representation of natural numbers, and VN-ordinals of ordered sets - keep both Apr 13 at 0:19
  • there are two monotone maps from any ordered 1-element to any ordered 2-element set, it's not a question of being "better" or anything. regarding the ranking function: zermelo's don't really work for transfinite rank (that's the content of the edit). in any case, we want to speak of/work with functions, and zermelo's ordinals are all singletons, so completely inadequate
    – ac15
    Apr 13 at 0:46
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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.

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  • Would not a set of zero elements, Set = {} still be an ordinal set?
    – Gerry
    Apr 13 at 23:18
  • @Gerry I'm not sure what you mean exactly, but the empty set is both the first von Neumann ordinal and first Zermelo ordinal.
    – Invariance
    Apr 18 at 13:07

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