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Philosophy

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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
    Commented 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
    Commented 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...
    – Double Knot
    Commented 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.
    – Michael Carey
    Commented 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/…
    – Julius Hamilton
    Commented Apr 13 at 0:05