# Propositions Equivalent to the Continuum Hypothesis

5-1968

Report

## Degree Name

Master of Science (MS)

## Department

Mathematics and Statistics

E. E. Underwood

E. E. Underwood

## Abstract

Two sets A and B are said to have the same power if there exists a one-to-one correspondence between them. All sets which have the same power as the natural numbers are called countable and have power N0. All sets which have the same power as the real numbers are said to have the power of the continuum will be denoted by 2N0, since 2N0 can be shown to be equal to c as will be indicated in the preliminary results.

Given an element a of a well-ordered set B, the set of all elements of B which procede a is called a segment of B. Every uncountable well-ordered set, all of whose segments are either finite or countable, is said to have power N1. The Continuum Hypothesis is the hypothesis that the power of the continuum is N1 that is 2N0 = N1. In the sequel, this equality will be called hypothesis H.

## Share

COinS

#### DOI

https://doi.org/10.26076/0148-ebaf