In this note we partly answer a question left open in [1, p. 52] by proving the following
theorem.Theorem. Supposep0≧2, ωn+1·p0≦σ<ωn+1(p0+1). Then W(σ) is uniformly many-one …

It is noted that this number x can always be found. For, obviously there exists a number u*
such that the pr function< 1 (u*, a) has the property (p1 (u*, ac)= a_-for i= 1,***, r; then the …

Two questions were left open in [1]. 1) Do W(σ) and W(r) have the same many-one degree for
any σ, τ such that ωη+1.2 ≤ σ < τ < ωη+2? 2) Is W(σ) many-one reducible to W(r) for any σ, …

In a theorem and its corollary [1] Friedberg gave an enumeration of all the recursively enumerable
sets without repetition and an enumeration of all the partial recursive functions without …

Let 0 (n', x) stand for sg (A (n'))* x+ sg (B (n'))* K ((n') i) and 8 (n') stand for sg (A (n')). 2 (') o.
3 (n') 1+ 1. Then the definition of g can be writ-ten as g (0) 0, g (n) O= M ('(Wn') where 0 (n, x) …

In this paper a constant, namely an operation without variables, is defined in the Zermelo-Fraenkel
set theory ZF. This constant, which we denote by ∞, is such that (1) ∞ ∈ ω ⊂R and (2…

TNA PAIR of articles,'Professors White and Quine have e I eight definitions of the term"
analytic" and after giving considera-tion to each, they assert that all these definitions are …

This talk is based upon my previous paper [1], namely" A prooftheoretic approach to non-standard
analysis with emphasis on distinguishing between constructive and non-constructive …

76 SHIH-CHAO LIU those of the preceding type. Examples of gr well-ordering for each of the
four types are given in this note. Theorem 1 shows that gr well-orderings of type IV give only …

S. Feferman and C. Spector. Incompleteness along paths in progressions of theories. The
journal of symbolic logic, vol. 27 no. 4, pp. 383–390. RA DiPaola-1967-Journal of Symbolic …