# ω1, the First Uncountable Ordinal

## Presenter(s)

Nick Kendall Gonano

## Description

This poster serves to highlight several of the properties of ω1, the first uncountable ordinal. Several of the more interesting properties are presented, including those of functions from the first uncountable ordinal into the real numbers. Key to this presentation is the definition of an ordinal: an ordinal n is the set of all numbers less than n, starting at 0. For example, the number 3 is an ordinal composed of the numbers {0, 1, 2}. Also important is the definition of countable: a countable set has the same number of elements as omega, which one should note is very different than ω1. An additional definition to note is that of a well-ordered set. These sets are nonempty and have a least element. Closed and unbounded subsets, hereafter referred to as cub sets, have the properties that all sequences in the set converge to an element of the set, and also that there is no upper bound to the set. Stationary subsets of ω1 are those that, intersected with every cub set in ω1, have a nonempty intersection. This poster will also cover some of the differences between ω1 and the real numbers.

4-22-2020

Capstone Project

Joe Don Mashburn