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Equivalence of Dedekind cuts and Dedekind left sets
I am currently working on the book "Classic Set Theory" by Goldrei. Goldrei is using Dedekind left cuts or left sets, i.e. the subset $L$ of a Dedekind cut. He gives the following definition ...
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Partition of positive reals with each part closed under addition without choice
It is an easy exercise using transfinite recursion to prove the following (in ZFC):
There exists sets $S,T$ that partition $\mathbb{R}_{>0}$ such that each of $S$ and $T$ is closed under addition.
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