The class of all ordinal numbers can be partitioned into two subclasses in such a way that neither subclass contains an arithmetic progression of order type ω, where an arithmetic progression of order type τ means an increasing sequence of ordinal numbers γ r, δ ≠ 0.

There are two well‐known ways of doing arithmetic with ordinal numbers: the “ordinary” addition, multiplication, and exponentiation, which are defined by transfinite iteration; and the “natural” (or “Hessenberg”) addition and multiplication (denoted ⊕ and ⊗), each satisfying its own set of algebraic laws. In 1909, Jacobsthal considered a third, intermediate way of multiplying ordinals (denoted × ), defined by transfinite iteration of natural addition, as well as the notion of exponentiation defined by transfinite iteration of his multiplication, which (...) we denote. (Jacobsthal's multiplication was later rediscovered by Conway.) Jacobsthal showed these operations too obeyed algebraic laws. In this paper, we pick up where Jacobsthal left off by considering the notion of exponentiation obtained by transfinitely iterating natural multiplication instead; we shall denote this. We show that and that ; note the use of Jacobsthal's multiplication in the latter. We also demonstrate the impossibility of defining a “natural exponentiation” satisfying reasonable algebraic laws. (shrink)

The article investigates a system of polymorphically typed combinatory logic which is equivalent to Gödel’s T. A notion of (strong) reduction is defined over terms of this system and it is proved that the class of well-formed terms is closed under both bracket abstraction and reduction. The main new result is that the number of contractions needed to reduce a term to normal form is computed by an ε 0-recursive function. The ordinal assignments used to obtain this result are (...) also used to prove that the system under consideration is indeed equivalent to Gödel’s T. It is hoped that the methods used here can be extended so as to obtain similar results for stronger systems of polymorphically typed combinatory terms. An interesting corollary of such results is that they yield ordinally informative proofs of normalizability for sub-systems of second-order intuitionist logic, in natural deduction style. (shrink)

In this paper, we give an axiomatization of the ordinal number system, in the style of Dedekind’s axiomatization of the natural number system. The latter is based on a structure $(N,0,s)$ consisting of a set N, a distinguished element $0\in N$ and a function $s\colon N\to N$. The structure in our axiomatization is a triple $(O,L,s)$, where O is a class, L is a class function defined on all s-closed ‘subsets’ of O, and s is a class function $s\colon (...) O\to O$. In fact, we develop the theory relative to a Grothendieck-style universe (minus the power set axiom), as a way of bringing the natural and the ordinal cases under one framework. We also establish a universal property for the ordinal number system, analogous to the well-known universal property for the natural number system. (shrink)

The theorem that the arithmetic mean is greater than or equal to the geometric mean is investigated for cardinal and ordinal numbers. It is shown that whereas the theorem of the means can be proved for n pairwise comparable cardinal numbers without the axiom of choice, the inequality a2 + b2 ≥ 2ab is equivalent to the axiom of choice. For ordinal numbers, the inequality α2 + β2 ≥ 2αβ is established and the conditions for (...) equality are derived; stronger inequalities are obtained for finite and infinite sequences of ordinals under suitable monotonicity hypotheses. MSC: 03E10, 04A10, 03E25, 04A25. (shrink)