Download PDF by Jech, T.J; Scott, D.: Axiomatic Set Theory, Volume 1 (Symposium in Pure

By Jech, T.J; Scott, D.

ISBN-10: 0821802453

ISBN-13: 9780821802458

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Extra info for Axiomatic Set Theory, Volume 1 (Symposium in Pure Mathematics Los Angeles July, 1967)

Example text

Las by t h e i r q u a n t i f i e r - r a n k ~ i~ , of for- 7 V{7q) : q)((~} ; VWqo , last two e q u i v a l e n c e s many variables pairs equivalent: A~, These that the f o l l o w i n g m is d e f i n e d inductively as follows: sub(~) It is e a s i l y = {~} if K = {l~} u sub(m); sub(A#) = {A~} u U { s u b (~) : ~E~}; sub(V~) = {V~} u U { s u b (m) : m(~}; sub(VW~) = {VW~} u sub(~o); sub(3Wm) : {3Win} u sub(~). proved that is s i n g u l a r to a f o r m u l a junctions of is atomic; sub(1~) if m is r e g u l a r then Isub(m)] < <} L i = {~(L h : If ~ L

A) we could assume that but it is n e c e s s a r y to c o n s i d e r finite K for part (b). 54 (a) First assume that I: ~ We show by i n d u c t i o n sequence of ~ < from and for any A on f o r m u l a s K' variables, f ( I I: ~[~] If ~ is a t o m i c isomorphisms The al, and h e n c e inductive so we just do the is a s e q u e n c e of Let ~-termed a be a at e v e r y a~, where is an forth c and c~ with K atomic sequence that ~ < K, for ~ I are f ( I y property. be d e f i n e d ~ I= ~[~ ,c] § , By the g ( I Then, trivi- where desired Then is some ~ < e.

E The next t h e o r e m shows, on there is some w e l l - o r d e r e d 40 which is 2-partially not w e l l - o r d e r e d . 1). 1 T H E O R E M <~'<> ~2 ~B [12]). 6 a | ~ mB ~ | ~ lin- product iff is the lin- times. exponentiation. ~,a> we d e f i n e h ls for of a | IB some of f i n i t e l y ~ and B + i ~ ~, g E IB with interval s many with ~6 h(0) s f ~ g some Then and <6,a> of of all functions order-preserving = <0,0>. of s u b m o d e l s f E I6+i, and and of implies form collection intervals the b a c k - a n d - f o r t h let of the < <~B-(6+l),a>}.

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Axiomatic Set Theory, Volume 1 (Symposium in Pure Mathematics Los Angeles July, 1967) by Jech, T.J; Scott, D.


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