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Thus the "Borel Continuuum Hypothesis" arises, and is a rather basic and striking classical result in descriptive set theory. See [Ke95] and the discussion below in H1. Sometimes a highly abstract statement not only causes no difficulties, but it even is obviously equivalent to a much more concrete statement. See the discussion below in H14. To elaborate on these points in some detail, we discuss the levels of Concreteness associated with Hilbert's famous list of 23 problems, 1900. org/wiki/Hilbert's_problems#Table_of_pr oblems Our treatment is rough at certain spots, and it would be very interesting to improve and extend it.

3) (∀x,y)(x ≥ y → l2(x) ≥ l2(y)). Obvious from π at l2. (4) l2(1) = 0. Obvious from π at l2. (5) (∀x)(x ≥ 1 → l2(2x) = l2(x)+1). Obvious from π at l2 and iii). (6) (∀x)(x ≥ 1 → 2l_2(x) = λ2(x)). Obvious from π at l2, λ2. (7) (∀x)(l2(2x) = x). Obvious from π at l2. (8) (∀x)(2x+1 = 2x + 2x). By iii). (9) (∀x)(x ≥ 1 → 2x-1 ≥ x). Let E = {x: 2x+1 ≥ x} ∪ {0}. Obviously 0 ∈ E. Suppose x ∈ E. Then x+1 ∈ E. Hence by iv), E is everything. (10) (∀x)(if x is a multiple of φ(m) then 2x-1 is a multiple of m), where m is an odd positive integer.

X-y as the additive inverse. d|x ↔ (∃y)(x = dy). where d ≥ 2. ) leads to the complete axiomatization a. (Z,0,+,-) is an Abelian group, with inverse - and identity 0. b. < is a strict linear ordering. c. x+y < x+z → y < z. d. d|x ↔ (∃y)(x = dy). e. 1 is the immediate successor of 0. f. x > 0 → (∃y)(0 ≤ y < d(1) ∧ x ≡d y). where d ≥ 2. It is easy to see that the result of applying π to a-f is provable in i-iv. Hence i-iv is an axiomatiation of (Z,<,+). To see that (Z,<,+) is not finitely axiomatizable, we argue that a-f is not logically equivalent to any finite subset of a-f.

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