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Indicated where to find (in the nLab) a proof of the equivalence with the axiom of choice and with Zorn’s lemma (<– it’s there).
That seems slightly odd, but probably as long as it’s indicated where to find it it doesn’t matter.
Would it be OK to remove the discussion box on this page? Probably no need to copy it here; if it is announced on the nForum, we will be able to find it easily again if we ever needed to.
I agree that it seems a little weird, Mike.
Sure, the discussion can be removed.
This page claims that the well-ordering principle doesn’t imply excluded middle. But the way the paragraph is written is a bit ambiguous, and it may be interpreted as saying that the well-ordering principle doesn’t seem to imply excluded middle. If the former, does anybody know a reference for this?
I asked the same question in the Zulip HoTT chat. Andrew Swan’s answer is that the (inductive) well-ordering principle implies excluded middle (and hence choice, I believe): https://hott.zulipchat.com/#narrow/stream/228519-general/topic/inductive.20well-ordering.20gives.20excluded.20middle.3F/near/246881801
My student Tom de Jong formalized Swan’s answer in Agda, by the way (also taking into account a question I made after his answer).
Hi Martin, thanks for picking this up.
PS If you want urls to turn into links, you need to surround them with angle brackets, like <https://example.com>
, and have the Markdown option selected, to get https://example.com
Thanks for the angle-bracket hint. Done!
How do I reference this reference in nLab syntax?
Following the template here I’d do something like this:
Removed “if one admits the principle of excluded middle” from the beginning of the page, an assumption which is no longer needed after Andrew Swan.
Following up on #16:
In fact we already have a page John Lane Bell, so best to link to it, too.
Thanks, Urs, for both comments. I will add it later after I study the template.
Oh, I see that you already did it. Good.
By the way, I would re-structure that entry, regarding the material that was already there before you came in:
the section “Assessment” is hardly worth a section, this should be a \begin{remark} ... \end{remark}
and I find even that is unnecessary, as that assessment is at best besides the point (famous as that quote may be)
the section “History” should be moved down to right before the references, not to distract from genuine content
This way your material “In constructive mathematics” would be moved up to where it’s more relevant, and maybe it should be inside “statement and proof”.
I agree. I found the “assessment” a bit odd, not in its contents, which is interesting, but in its title.
Go ahead if you want to do it. It seems that you have an editing plan in your head already.
Actually, the “assessment” is better placed where it is, I think. This is because, in “In constructive mathematics” we are actually discussing the topics of the “assessment”. Namely that classically two things are equivalent to choice, about which in “In constructive mathematics” we learn that one is constructively equivalent to choice but the other isn’t. So perhaps I would keep things as they are, but changing the title “assessment”. Maybe “discussion”? Not perfect but perhaps less worse than “assessment”?
Looks good to me.
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