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By Abraham Kandel

ISBN-10: 0201117525

ISBN-13: 9780201117523

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In the case where cz is infinite, we are assuming M , E F ( O ~ ~ ~so +') proposition 2 does give Ma+5 F ( O ' " ' ~ + ~The ) . ~ ~Clearly + , F(w so by proposition 2 we have '"*x+1)EF(W'"2=+2) Def, ( F ( o w z K + 'E ) )P ( o W Z a + 2 ) . By setting X = Ma, Y = F(U'"'~+'), and 2 = F(oo2ui+2) the following proposition shows that for CI infinite, Def(M,) is itself firstorder definable over F ( W ' " ' ~ + Thus ~ ) . by a further application of proposition 2, we have: M,+ E F ( O ~ ~ ~ + ~ ) . , The following proposition thus will conclude the induction at successor ordinals for the main theorem.

Note that we are following Godel’s notation in which the elements of the domain of a function are the second elements of the ordered pair. **) Cnvz and Cnvs are permutations of ordered triples, see [2] p. 15. 7) This characterization was suggested by Profs. Martin Davis and Raymond M. Smullyan. 36 TH. A. First, assume A E Def,( U ) . , x,) that defines A over U. One can assume that q is written with -, A and 3 as the only logical constants. , ui,) can be obtained by a suitable sequence providing this can be done for all formulas $ with fewer logical constants.

By lemma 6 however there is a t e T such that: b N t = U, n F"s. All that remains t o be shown is the N-validity of the power set axiom. By lemma 6 it is sufficient to prove: LEMMA8. If we set: ~ ( x =) D f ~ . / 3U,x ~ for XE U, then z is M-definable by lemma 1. @,U, is a set of M , it follows that p=Df sup n"PNU, is an ordinal of M . The conclusion follows by: We close this section with an observation which will prove useful in connection with the forcing method. We chose to define the truth notion for RL ( k N ) simultaneously with the valuation (N(t)ltET>.

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Fuzzy Mathematical Techniques With Applications by Abraham Kandel


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