By U. Felgner, T. B. Flannagan (auth.), Gert H. Müller, Dana S. Scott (eds.)
Read Online or Download Higher Set Theory: Proceedings, Oberwolfach, Germany, April 13–23, 1977 PDF
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Additional resources for Higher Set Theory: Proceedings, Oberwolfach, Germany, April 13–23, 1977
LEMMA 6. of The following is provable in +l ZF (in the 1-sets), (xl, E) ~ Y[y ] Proof: ZF : iff for all formulas F (x2,#) in the language p(y)[pf(~l)]. This is proved by induction on the formula We now wish to define an isomorphism within Y Y. (ZF + V = L)* between the l- sets and the 2-sets. By lemma 3, within index i(x I) (ZF + V = L)* of a l-set ordinal for which xI be we have (a,Y), x I E L-(a+l), and y i(x 2) be the index of a 2-set V1 whose index is into V 2. x 2. For V1 V 2. v Let the is the least l-set theoretic xI over (L-(a),E).
A theory consists of a set becomes a theory if a function into sets T(~). changes of variables. T ~(S) from The resulting is closed under is obtained from the schematic theory in T by formulas ~ E ~(S) "in the That is, we require that there be no clashes of variables. is univalent if ~(~) ~- G-(S) = ~(S)-. be given by consists of one element, for all S. be the classes of schemes given simultaneously by the following clauses. 1. Every 2. ,tk) is in S I N WI; • E S I N W I. are closed under conjunction, versal quantification.
1. = O) then If PA ~ (Vn)(~n)(F(n,m) HA ~ (Zn)(F(n,m) = 0). emma i, HA ~ --N (Zm)(F(n,m) : O). Sometimes the full Markov rule is considered: and . N (Zm)(~(n,m)) are provable, then = 0). By lemma 3, HA I- (Zm)(F(n,m) = O). if (Vn)(Vm)(q0(n~m) V -- ~(n~m)) (Zm)(q0(n,m)) is provable. We can ob- tain the full Markov rule using the following lemmas. I~99WA 4. HA is closed under Church's rule. then there is an Proof: LKMMA 5. , if HA ~ (Vn)(Zm)(4(n,m)) , HA ~ (Vn)(4(n,[e](n))). This is well known. If HA ~ %0(n,m) V N ~(n~m), then there is an F such that HA I- ~0(n,m) (Zr)(F(r,n,m) = 0).