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I want to prove some results like that every vector space has a base without the use of Zorn's lemma becuase I want to practice set theory and become confident with things like ordinals. I have some basic understanding of ordinals that I could find online and, if I didn't miss anything, online sources don't seem to go into much detail. So, if you can, please provide me with sources from which you got the knowledge for posible answers to my question. Also, forgive me for not specifying my sources, they are basically random websites and blogs.

My understanding of that every vector space has a base as an example:

So basically we can start picking vectors that are not a linear combination of vectors we already picked until we can not do it anymore. Okay, formally, we will define a mapping from the class of ordinals to vectors using transfinite recursion. Two confusions here: for transfinite recursion, they say, we need the axiom schema of replacement and what bothers me more: how do we know that there is enough ordinals to drain the whole vector space? We can't even mention classes as they don't exist in ZF and that's why (I think) we need the axiom schema of replacement to work with collection of formulas. Now to answer the other question we can introduce the axiom of choice (which we needed for this picking the next vector anyway) and remember that every set is well-orderable. With well-order on the set of vectors we can apply transfinite recursion directly on the set of vectors and avoid the use of ordinals completely. Then we would still need this axiom schema of replacement (I think), but no ordinals. But wait, when I search for the proof that every set can be well-ordered, they use ordinals to "pick the next element". Basically the same proof I am trying to use it for.

So, how exactly can we define ordinals in ZF and, using AC, how do we derive necessary properties of them for proving results like Zorn's lemma or that every set is well-orderable?

PLEASE NOTICE:
Of course, very short answers that basically link to some source are good enough since that is what I mainly failed to do: find a good source! Mentioning a sketch of the proof is also welcome.

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    $\begingroup$ You might want to take a gander in my lecture notes that are on my website from when I was teaching set theory. There's no proof of Zorn's lemma there (all my students had seen those before), but most of the other questions are answered, I hope. $\endgroup$
    – Asaf Karagila
    Nov 27, 2022 at 9:19
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    $\begingroup$ For all sets, there is an ordinal which cannot be injected into that set (the least of which being that set's Hartogs number). That ensures that at some point before reaching that step you'll run out of vectors to pick, lest you would end up with an injection from that ordinal into your set. $\endgroup$
    – Uro
    Nov 27, 2022 at 13:38
  • $\begingroup$ @AsafKaragila Good answer! I saw your name a lot around 'set-theory' flag, still didn't know about your website. Turns out that pointing to it is exactly what I wanted to hear! Excellent website, thank you! $\endgroup$
    – donaastor
    Nov 27, 2022 at 16:00
  • $\begingroup$ @Uro Indeed, Hartogs number was the milestone I needed to pass here (I eventually answered my own question after learning Hartogs original proof). I also used the principle that in weak enough theories like Zermelo's set-theory I should indeed define ordinals as equivalence classes instead of by von Neumann's approach. The ultimate result I wanted to prove was DC-w => every non-atomic measure maps to the whole interval. These 2 hints really helped me in understanding this result. Sorry for a little digression... $\endgroup$
    – donaastor
    Nov 27, 2022 at 16:06

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