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Added to natural number a discussion about the fact that constructively, the natural numbers may fail to be (order) complete, as highlighted by Andrej Bauer in a very nice blog post. I quite like this example, because by interpreting a related lemma in the internal language of a certain sheaf topos one obtains a well-known proposition in algebraic geometry almost for free (see entry); but please let me know if stuff like this is too localized for the nLab.
Thanks!
Don’t worry, nothing is too localized for the nLab, and certainly not what you just added, I’d say. There is unbounded room. If something becomes too specialized for some entry (or feels like it does) we can split it off as a separate entry and link to it.
One thing about type-setting: when you give names to propositions, you should put them in the third line, like this
+-- {: .num_prop}
###### Proposition
**(a Brouwerian counterexample)**
instead of at the end of the second line. Otherwise the numbering is moved too far to the right and can become hard to spot or else look awkward (I think). I have edited the entry accordingly, ever so slightly.
Thank you for the editing and the tip! Looks much better this way.
In fact the equivalence implies LEM. See here.
It depends on what you mean by well ordering; if you define this properly, then it is constructively equivalent to induction. We call the (simpler!) definition of well ordering involving minimal elements classical well ordering; the natural numbers are constructively well ordered (if you accept induction), but their classical well ordering implies LEM (by Ingo's argument).
Added to natural number a discussion about the stopping behaviour of weakly decreasing sequences of natural numbers.
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