way? The answer was found by Cantor: use ordinal arithmetic. 1 De nition of ordinals Countable ordinals can be represented as subsets of the real line so that one can only make a nite number of leftward jumps in the subset. We only care about the order of points in the ordinal, so two subsets ordered in the "same way" count as the same ordinal.

Every object we consider will be an ordinal unless otherwise stated. Definition 0.1. An ordinal is a transitive set of ordinals. Why is this not circular — don’t we need ordinals to exist in order to define any sets of ordinals? (No. The empty set is a set of ordinals.) Definition 0.2. AsetX is transitive if y 2 X ) y X. Remark 0.3.

ordinal arithmetic Ordinal arithmetic is the extension of normal arithmetic to the transfinite ordinal numbers. The successor operation S ⁢ x (sometimes written x + 1 , although this notation risks confusion with the general definition of addition ) is part of the definition of the ordinals , and addition is naturally defined by recursion over this:

30/09/2015 · However, with these details apart, the theory of ordinal arithmetic is a very beautiful and straightforward one, enjoyable and worth learning. Throughout, α, β, γ, δ range over ordinals and λ, μ range over limit ordinals. 2. The successor function and the order relation.

Ordinal Arithmetic ord-arithmetic.1 Introduction sth:ord-arithmetic:intro: sec In ??, we developed a theory of ordinal numbers. We saw in ?? that we can think of the ordinals as a spine around which the remainder of the hierarchy is constructed. But that is not the only role for the ordinals. There is also the task of performing ordinal arithmetic.

Ordinal arithmetic - Oxford Scholarship. This chapter defines operations of addition, multiplication, and exponentiation for ordinals. It takes as a model the recursive definitions of the corresponding operations for natural numbers (§ 5.4): the form the extended definitions should take at success or ordinals is clear; the form of the additional ...

Math 317 Week 06: Ordinal Arithmetics March 18, 2014 Table of contents References.....2 1. Ordering.....3 1.1. Relation.....3 1.2. Ordering: partial; total; well.....3 1.3.

Ordinal Arithmetic: Algorithms and Mechanization 3 were carried out with the Isabelle/ZF system [44, 46]. A version of the re ection theorem was also proved by Bancerek, using Mizar [3]. Another line of work is by Belinfante, who has used Otter to prove elementary theorems of ordinal number theory [4, 5, 6]. There is much

Chapter 4. Cardinal Arithmetic.∗ 4.1. Basic notions about cardinals. We are used to comparing the size of sets by seeing if there is an injection from one to the other, or a bijection between the two. Definition. Let x, y be sets. Le x y if there is an injection of x into y (i.e. a function f such that for all z 0, z 1 ∈ x we have que f(z 0) = f(z