Chapter 1. Once 𝕍 is defined, we may call it an "ordinal number". Definition: ℕ : 1 → 𝔅 . Now ℕ : 1 → 𝔅 is defined. We've just constructed a set. Now ℕ is a construction. To be honest, it's a sort of an extension of the concept of a set, which, as we know, was defined (more than once, as we've seen) as a certain thing – a collection of elements. But, as we've already mentioned (or at least attempted to mention) many times, not only ℕ is not defined as a set but – and this is a really important point – it may not be something we recognize as a set! In our terminology, it would be unwise to define "Ordinal number" in the same manner in which we define "Set": We may write ℕ : 1 → 𝔅 or, in other words, for every 1 element we've got an ordinal number. That's right. We would be a little bit at a loss in case we'd try to write ℕ as {<1>, <2>, <3>, <4>, <5>, ...}. That's because, after all, we'd first need to define the set we're talking about and, then, the set itself would constitute an element in it – as we've learned, this would be a set and, therefore, after all this, we'd have at our disposal a set-theoretically defined set! 2.2. A (Set-Theoretically Defined) Generalization 𝕍 = 1 ℕ is "set-theoretically defined" too. But instead of using sets we're using ordinal numbers. But the following fact remains true in this case too: it is not defined as a set. So, once we've used ordinal numbers in place of "sets", the definition we wrote down earlier can be rewritten as follows: "For every ordinal number 𝕍 there is an ordinal number that comes after it." Here is an immediate consequence of the above observation: Definition: (In terms of ordinal numbers) A transfinite recursion is a sequence of "For every 𝕍 there is an 𝕌" that doesn't "die out": for every 𝕍 there is an 𝕌 so that the following holds: for all 𝕌 + 1, if there is an 𝕌𝕌 then there is an 𝕌𝕌𝕌. We want to emphasize that this fact says nothing but "For every 𝕍 there is an 𝕌". Such a generalized definition cannot help us avoid circularity. Consider the sequence: for every a there is an a' so that for every b, b' the following holds: a b if and only if a' b'. This definition would not be circular, but this fact wouldn't be interesting either. Definition: A (General) Transfinite Recursion. is a method for constructing a sequence {ϕ i : 𝓝} such that: ϕ1 is defined for some ordinal number 𝓝 ϕ1 , ϕ2 is defined for every 𝓝 + 1 ϕ2 , ϕ3 is defined for every 𝓝 + 2 ϕ3 , . . ., (...) ϕk is defined for every 𝓝 + k ϕk . If for every 𝓝, ϕi is defined as i 𝓝 then we'll call the sequence a (General) Transfinite Recursion. A set-theoretic example of a transfinite recursion is this: If 𝓝 is a set of ordinals, let 𝓜 be the following set: 𝓜 = { {<∆1>,...,<∆n> : ∆1,...∆n} : ∆1,...∆n ∈ 𝓝 } We will call this set a "The set of sequences of ordinals". Our task is to show that if 𝓜 is a set then 𝓜 can be identified as a certain transfinite recursion. To accomplish this we must identify the sequences of ordinals with finite sets of ordinals in such a manner as not to violate the axioms of the first-order set theory. What's the relationship between sets and sequences of ordinals? As we have seen before, it seems that a set can be seen as an initial part of a sequence of ordinals. In particular, we've seen that a set-theoretically defined set like the following one cannot be a set: { {<∆1>,...,<∆n> : ∆1,...∆n ∈ 𝓝} , {<∆n+1> : ∆1,...∆n ∈ 𝓝} } Here are a few additional remarks concerning the above fact. First, the definition above was a set-theoretic definition, so the notion that is being defined is the notion of the sequence of ordinals. Definition: A Set of Ordinals. is a pair of two ordinals 𝓝 and 𝓞 so that 𝓝 ∈ 𝓝, 𝓞 ∈ 𝓝, and 𝓞 has an immediate predecessor in 𝓝, i.e. there is 𝓞′ ∈ 𝓝 such that 𝓞 < 𝓞′ ∈ 𝓝. From this definition we get a notion of a "sequence of ordinals" – a sequence that has certain properties, and not just one sequence (for example, {, , }) – in this case we may just write this sequence as , , . Now consider the notion of "set of ordinals". If 𝓝 is a set-theoretically defined set then it's a set, but there is no sequence of ordinals that can be identified as the sequence of ordinals that appear in it. We have just two options here: either we identify a set with some sequence (which doesn't seem very interesting), or, in order to give the set-theoretic definition of a set of ordinals we'll have to do as follows: Let 𝓝 be a set. We'll define 𝓝 ∈ 𝓝 = {<𝓝>} because, after all, we can see each sequence as the sequence of ordinals that 𝓝 is a set of. Now, if 𝓝 is a set-theoretically defined set, then it is not a set in our generalized sense (as we've defined it above). So, if we want to identify a set with a sequence of ordinals, then we will have to define that sequence as a certain sequence of ordinals that is a member of the set. 2.2.2. Example: The Ordinals In this paragraph we want to clarify the distinction between the "set of ordinals" that we've introduced and the "usual ordinal numbers". First, the latter are, of course, sequences of natural numbers that satisfy the above definition of a sequence of ordinals – no two "usual" ordinals can have the same height and no two usual ordinals are in direct ascent from each other (as we know, they always differ by a power of the "usual" ordinal of ω). That's not the case of our "sets of ordinals". If 𝓝 is a "set of ordinals" than any two of its elements may differ by a power of the "usual" ordinal of ω and any two of its elements are in direct ascent from each other. For example, this is the case of our set of ordinals: 𝓝 = {<0, 0, 1, 1, 2, 2, 2, 2, 3, 4, 5, 6, ...>}. We also want to draw your attention to the fact that – contrary to the set of ordinals – our set-theoretic definition of a "set of ordinals" has nothing to do with any set-theoretic definition of the "usual" ordinal numbers. Moreover, our "set of ordinals" seems to have nothing to do with an usual notion of a "set". Let's try to make sense of this. Suppose 𝓞 is a set of ordinals. According to the definition of the set of ordinals we've just given, if 𝓞 ∈ 𝓝, then, as the height of every member of 𝓞 is 2ω, it must be that 𝓞 ∈ {<0, 1, 2, 3, ...>}. Moreover, we know that any two ordinals from 𝓝 have the same height, so we can write the set of ordinals that 𝓞 ∈ 𝓝 as ℕ\ℕ0 × {<0>} × <1> × {<2>} × {<3>} × {<4>} × {<5>} × ... × {<2