Download A Set Theory Workbook by Iain T. Adamson PDF

Download A Set Theory Workbook by Iain T. Adamson PDF

By Iain T. Adamson

Presents a proper description of set conception in line with the Von Neumann-Bernays-Godel axiomatic technique utilizing the idea that of periods. Covers the basis of the idea, family, ordinals, cardinals, and the axiom of selection. Paper. DLC: Set thought.

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E. (V.. )((*, v) G y&(v,tιθ G y . ^ (tι,tι;) G y ) , (Vti,i;Gx)(MGyV(t;,u)ey), (Vw, v G x)((u, v) G y & (v, w) G y. -> u = v). 42 I. Arithmetic as Number Theory, Set Theory and Logic (3) y is a mapping of x into z if iZeZ(y), dom(y) = z, rng(y) C z and (Vu G x)(Vυ, ID € z)((u, ι>) G y & (u, ιu) G y. -+ v = u>) A mapping is an injection (or: is one-one) if 2,v) G y y is a bijection of a: onto 2: if, additionally, rng(y) = 2. (4) For each x, the natural ordering of a: is the unique linear order y on a: such that, for each iz, v G x, (tί, y ) 6 | / = « < ι ; .

5(1), u = v = 0. Thus x = y. 1. Basic Developments; Partial Truth Definitions 33 (9) Easy using commutativity and associativity of addition. (10) Use the same plus cancellation. ->x*zφy*z. By (7), we may assume x < y. x * z x x*z

For simplicity, let F(k) = G(H(k)) for each &, and let φ(xy y), ψ(x, y) define GyH respectively, φ,ψ G Σ\. Then JF is defined by the Σ formula (3z)(φ(x,z)&φ(z,y)). 39 Lemma. Σ\ relations are closed under bounded universal projections. Proof. Let R C JV2 be defined by a formula (3z)φ(x, y, z) where φ is Σo and let 5 C JV be defined by (Vx < y)(3z)φ(x, y, z). We show that 5 is also defined by (3tϋ)(Va; < y)(3z < w)(φ(x, y, z), which is Σ\. ) Clearly the latter formula implies the former. Thus assume k G 5; we find a q such that JV 1= (Vx < Έ)(3z < q)φ(xjz, z).

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