This paper deals mainly with generalizations of results in finitary combinatorics to infinite ordinals. It is well-known that for finite ordinals ∑bT<αβ is the number of 2-element subsets of an α-element set. It is shown here that for any well-ordered set of arbitrary infinite order type α, ∑bT<αβ is the ordinal of the set M of 2-element subsets, where M is ordered in some natural way. The result is then extended to evaluating the ordinal of the set of all n-element (...) subsets for each natural number n ≥ 2. Moreover, series ∑β<αf are investigated and evaluated, where α is a limit ordinal and the function f belongs to a certain class of functions containing polynomials with natural number coefficients. The tools developed for this result can be extended to cover all infinite α, but the case of finite α appears to be quite problematic. (shrink)

In his celebrated paper of 1931 [7], Kurt Gödel proved the existence of sentences undecidable in the axiomatized theory of numbers. Gödel's proof is constructive and such a sentence may actually be written out. Of course, if we follow Gödel's original procedure the formula will be of enormous length.Forty-five years have passed since the appearance of Gödel's pioneering work. During this time enormous progress has been made in mathematical logic and recursive function theory. Many different mathematical problems have been proved (...) recursively unsolvable. Theoretically each such result is capable of producing an explicit undecidable number theoretic predicate. We have only to carry out a suitable arithmetization. Until recently, however, techniques were not available for carrying out these arithmetizations with sufficient efficiency.In this article we construct an explicit undecidable arithmetical formula,F(x, n), in prenex normal form. The formula is explicit in the sense that it is written out in its entirety with no abbreviations. The formula is undecidable in the recursive sense that there exists no algorithm to decide, for given values ofn, whether or notF(n, n)is true or false. MoreoverF(n, n)is undecidable in the formal (axiomatic) sense of Gödel [7]. Given any of the usual axiomatic theories to which Gödel's Incompleteness Theorem applies, there exists a value ofnsuch thatF(n, n)is unprovable and irrefutable. Thus Gödel's Incompleteness Theorem can be “focused” into the formulaF(n, n). Thus some substitution instance ofF(n, n)is undecidable in Peano arithmetic, ZF set theory, etc. (shrink)