The aim of this blog post is to prove that if $\kappa$ and $\gamma$ are two infinite cardinals then $\kappa$ + $\gamma =\max(\kappa,\gamma)$.
It should be pretty clear that it suffices that we prove $\kappa+\kappa=\kappa$ for any infinite cardinal $\kappa$.
We can proceed using Zorn’s lemma: For an infinite set $X$ with cardinality $\kappa$, consider the set of all pairs $(M,f)$, where $M$ is a subset of $X$ and $f$ is a bijection from $2\times M\to M$. Now we can partially order this set by extention on both coordinates. Further it is pretty clear that every chain has an upper bound and so Zorns lemma appiles and we have a maximal element. The first coordinate of the maximal element, $(Z,g)$, must have the same cardinality as $X$, because if it doesn’t then we can extract a subset,$L$ from $X/Z$ with cardinality $\aleph_{0}$, and then $(L\cup Z, r)$ is a bigger element(where $r$ is defined in the obvious manner.) From here the conclusion rapidly follows.
I found the explanation to be brief yet well-organized.
I appreciate how you applied Zorn’s lemma to prove the sum of two infinite cardinals.
use of partial ordering and the concept of a maximal element was clear and effective
Provides a succinct and logical proof of the sum of infinite cardinals. I found your explanation of Zorn’s lemma and the concept of maximal elements to be particularly well-presented.