d-finite | Asaf Karagila

Posts tagged d-finite

Question on Dually Dedekind-infinite sets

Jul 27 2022, 09:42

I got an email a few days ago from Lucas Polymeris, a Chilean student, who asked me a very nice question. I want to go through the journey I took from that question to the answer.

Before we get to the question, let's review some definitions.

Cohen's Oddity

Oct 20 2018, 12:05

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

May 26 2015, 10:44

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.

First Post

Oct 19 2012, 05:46

Well... This is my first post on this blog, and I have absolutely no idea how to start it.

Should I make it about myself? about my life? about my academic status? How I about I tell cool stories from my life, perhaps inebriated adventures? army experiences? Maybe I should write about mathematics. Perhaps some nice proof or some nice theorem?