New PDF release: Logic, Methodology and Philosophy of Science III

By B. Van Rootselaar and J.F. Staal (Eds.)

ISBN-10: 0444534164

ISBN-13: 9780444534163

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Example text

This concludes the sketch of our proof. . a . + * Membership of 2(e) in E; is not required for this, it is only needed for Lemmas 5 and 6. NUMBERED FIELDS YU. L. ERSHOV Mathematical Institute, Novosibirsk, USSR General definitions of the theory of numbered algebras may be found in MALCEV11961). The special theory of numbered fields was considered in works of FROLICH and SHEPHERDSON [1958], MOSCHOVAKIS [1965] and RABIN[1960]. NERODE [1963] used numbered fields for constructing an algorithm for p-adic integral zeros of diophantine equations.

Jprede. ~ K I = ~ , cessors, so it has exactly p a predecessors. Finally, one shows that an arbitrary be B is either caught between some pair b,,,,, , and b(a,, o, whence it has exactly p a predecessors, or else it is less than all the elements b,,,,,, whence it has at most IK( predecessors. Since K i s exactly the set of all the cardinals p a , c1

Z. , < i . , in. > ( ~ i ~ ~ i rci,l,> The theory C resembles sets of formulas used by SILVER[1966] for other purposes, but here we have doubly indexed constants. The sentences (c) will insure that the constants cij are cofinal in the set of all terms applied to constants. The sentences (d) state that when the value of a term is less than cij, its value is independent of the second indices of the constants ckl where k > i. These sentences will insure that no element has too many predecessors.

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Logic, Methodology and Philosophy of Science III by B. Van Rootselaar and J.F. Staal (Eds.)

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