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1. • CommentRowNumber1.
   • CommentAuthorIngoBlechschmidt
   • CommentTimeDec 9th 2013
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   Author: IngoBlechschmidt
   Format: MarkdownItexAdded 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](http://math.andrej.com/2009/09/08/constructive-stone-minima-of-sets-of-natural-numbers/). 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.

   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.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeDec 9th 2013
   • (edited Dec 9th 2013)
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   Author: Urs
   Format: MarkdownItexThanks! 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.

   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.

3. • CommentRowNumber3.
   • CommentAuthorIngoBlechschmidt
   • CommentTimeDec 9th 2013
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   Author: IngoBlechschmidt
   Format: MarkdownItexThank you for the editing and the tip! Looks much better this way.

   Thank you for the editing and the tip! Looks much better this way.

4. • CommentRowNumber4.
   • CommentAuthorColin Tan
   • CommentTimeDec 27th 2013
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   Author: Colin Tan
   Format: TextIf I am reading you correctly, then does this Brouwerian counter example demonstrate that the usual proof of the equivalence of the induction principle and well ordering principle uses the law of the excluded middle? That is to say,constructively, the natural numbers satisfy only the induction principle but not the well ordering principle.

   If I am reading you correctly, then does this Brouwerian counter example demonstrate that the usual proof of the equivalence of the induction principle and well ordering principle uses the law of the excluded middle? That is to say,constructively, the natural numbers satisfy only the induction principle but not the well ordering principle.
5. • CommentRowNumber5.
   • CommentAuthorZhen Lin
   • CommentTimeDec 27th 2013
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   Author: Zhen Lin
   Format: MarkdownItexIn fact the equivalence implies LEM. See [here](http://math.stackexchange.com/a/609788/5191).

   In fact the equivalence implies LEM. See here.

6. • CommentRowNumber6.
   • CommentAuthorTobyBartels
   • CommentTimeDec 28th 2013
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   Author: TobyBartels
   Format: MarkdownItexIt 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).

   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).

7. • CommentRowNumber7.
   • CommentAuthorIngoBlechschmidt
   • CommentTimeDec 30th 2013
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   Author: IngoBlechschmidt
   Format: MarkdownItexAdded to _[[natural number]]_ a discussion about the stopping behaviour of weakly decreasing sequences of natural numbers.

   Added to natural number a discussion about the stopping behaviour of weakly decreasing sequences of natural numbers.