Category: Set theory

Long well-ordered chains of functions under eventual domination

\(\newcommand{\N}{\mathbb N}\)Consider the set \(\N^\N\) of all functions from \(\N\) to itself. We impose a partial order of eventual domination on this set: we say that \(f\) eventually dominates \(g\) if for large enough \(n\) we have \(f(n)< g(n)\). In this poset, we consider chains, in particular the well-ordered ones. One question we can ask is, how long can such chains be? Here we establish a lower bound \(\omega_2\) for the supremum of possible lengths.

All background necessary for this post is some basic knowledge about ordinals, in particular cofinalities.

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Finiteness of subsets of arbitrary finite sets

The goal of this blog post is to provide a novel proof of the following result:

Proposition: Let \(A\) be a finite set and let \(B\) be a subset of \(A\). Then \(B\) is finite.

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