6 research outputs found
Intermediate arithmetic operations on ordinal numbers
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 will 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.Comment: 18 pages, 3 table
Ordinal Measures of the Set of Finite Multisets
Well-partial orders, and the ordinal invariants used to measure them, are relevant in set theory, program verification, proof theory and many other areas of computer science and mathematics. In this article we focus on a common data structure in programming, finite multisets of some well partial order. There are two natural orders one can define on the set of finite multisets of a partial order: the multiset embedding and the multiset ordering. Though the maximal order type of these orders is already known, other ordinal invariants remain mostly unknown. Our main contributions are expressions to compute compositionally the width of the multiset embedding and the height of the multiset ordering. Furthermore, we provide a new ordinal invariant useful for characterizing the width of the multiset ordering
Long reals
The familiar continuum R of real numbers is obtained by a well-known procedure which, starting with the set of natural numbers N=\omega, produces in a canonical fashion the field of rationals Q and, then, the field R as the completion of Q under Cauchy sequences (or, equivalently, using Dedekind cuts). In this article, we replace \omega by any infinite suitably closed ordinal \kappa in the above construction and, using the natural (Hessenberg) ordinal operations, we obtain the corresponding field \kappa-R, which we call the field of the \kappa-reals. Subsequently, we study the properties of the various fields \kappa-R and develop their general theory, mainly from the set-theoretic perspective. For example, we investigate their connection with standard themes such as forcing and descriptive set theory