Asaf Karagila
I don't have much choice...

## Cohen's Oddity

We all know and love Cohen's first model where the axiom of choice fails. It is the O.G. symmetric extension. But Cohen didn't invent the idea on his own, he used Fraenkel's ideas from his work on set theory with atoms and permutation models. The two results, however, are significantly different.

Fraenkel's construction does not affect sets of ordinals, in particular the real numbers can still be well-ordered in his models. Cohen's work, however, directly breaks that. The Dedekind-finite set added is a set of reals. In particular, the reals cannot be well-ordered no more.

## When the box means nothing

When assuming the axiom of choice the product topology and box the topology are quite different when considering infinite products. For example the Tychonoff product of countably many sets of three elements is compact, metrizable an all in all a very nice space. On the other hand, the box product is not separable or second countable at all.

But without the axiom of choice the world is indeed a strange place. This was posted as answer on math.SE earlier today.