# Talk:Kleene–Brouwer order

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Mathematics rating:
 Start Class
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Field:  Foundations, logic, and set theory

The Kleene-Brouwer order is not only of interest in descriptive set theory. It carries the names of Kleene and Brouwer (and also, sometimes, if I remember correctly, Kolmogorov) and yet the only reference is to Moschovakis. Ideally, both Kleene and Brouwer should be cited, but I do not know where a reference for Brouwer can be found. Kleene gives a classical proof of the well-foundedness theorem that the article says is so important (in Introduction to Metamathematics?). The Kleene-Brouwer order can be defined over trees over any set ${\displaystyle X}$ which carries a partial order (the order does not have to be total).

I once had a professional interest in this order as I was trying to give a constructive proof of well-foundedness, using the well-ordering principle for the set ${\displaystyle X}$ and the well-ordering principle for the tree over ${\displaystyle X}$. I conjecture that this is equivalent to the fan theorem when the set ${\displaystyle X}$ is N but I never understood this well enough to attempt a proof.

I'm happy to be corrected on any of the above; and I'd dearly love to find someone who understands it enough to correctly formulate and prove the conjecture above.

btw, I'm pretty new to Wikipedia...

Fairflow (talk) 17:12, 17 October 2010 (UTC)

Well, I think the question is, what would you cite Kleene and Brouwer as saying, that Moschovakis doesn't say? We don't generally strain to use sources just because they're closer to the historical origins of something. However, if there's an important point that's left out, but that you can find in Kleene or Brouwer, then by all means add it and cite them. --Trovatore (talk) 04:46, 18 October 2010 (UTC)
Indeed. I added the Moschovakis reference awhile ago because the article had no citations or references and it was the best resource I had on hand that defined the KB ordering. Not to mention it's a pretty nice and canonical descriptive set theory text that people could go to if they wanted to look more into the topic. If you know a better reference, then by all means add it, but for the content currently on the article, Moschovakis is fine. Thanks for fixing the definition by the way. Wgunther (talk) 17:46, 18 October 2010 (UTC)

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