Comments on: Constructive gem: double exponentials http://math.andrej.com/2009/10/12/constructive-gem-double-exponentials/ Mathematics for computers Mon, 30 Jan 2012 07:14:48 +0000 hourly 1 http://wordpress.org/?v=3.2-beta2-18055 By: Andrej Bauer http://math.andrej.com/2009/10/12/constructive-gem-double-exponentials/comment-page-1/#comment-12717 Andrej Bauer Wed, 21 Oct 2009 13:13:50 +0000 http://math.andrej.com/?p=338#comment-12717 <a href="http://ai.ijs.si/France/" rel="nofollow">France Dacar</a> asked me to post <a href="/wp-content/uploads/2009/10/22n-iso-n-classic.pdf" rel="nofollow">this classical proof (PDF)</a> that $2^{2^\mathbb{N}} = \mathbb{N}$ in the category of topological spaces. It should help "normal" mathematicians see what is going on. Thank you, France! France Dacar asked me to post this classical proof (PDF) that $2^{2^\mathbb{N}} = \mathbb{N}$ in the category of topological spaces. It should help “normal” mathematicians see what is going on. Thank you, France!

]]> By: Andrej Bauer http://math.andrej.com/2009/10/12/constructive-gem-double-exponentials/comment-page-1/#comment-12716 Andrej Bauer Wed, 21 Oct 2009 13:10:15 +0000 http://math.andrej.com/?p=338#comment-12716 I gave a talk about this at the local <a href="http://www.fmf.uni-lj.si/en/news/agregator/seminar-osnove/" rel="nofollow">Seminar for foundations (of mathematics and theoretical computer science)</a>. <a href="http://www.fmf.uni-lj.si/~petkovsek/" rel="nofollow">Marko Petkovšek</a> pointed out that classically $2^{2^\mathbb{N}} = \mathbb{N}$ if we interpret exponentiation as ordinal arithmetic. That is, for ordinals we have $2^{2^\omega} = \omega$. Thank you for a nice observation, Marko! I gave a talk about this at the local Seminar for foundations (of mathematics and theoretical computer science). Marko Petkovšek pointed out that classically $2^{2^\mathbb{N}} = \mathbb{N}$ if we interpret exponentiation as ordinal arithmetic. That is, for ordinals we have $2^{2^\omega} = \omega$. Thank you for a nice observation, Marko!

]]> By: Bas http://math.andrej.com/2009/10/12/constructive-gem-double-exponentials/comment-page-1/#comment-12685 Bas Mon, 12 Oct 2009 08:04:24 +0000 http://math.andrej.com/?p=338#comment-12685 Nice post! Regarding references to the intuitionistic axioms. You could try the following two text books: A. S. Troelstra and D. van Dalen. Constructivism in mathematics, vols. 1 and 2. Not online, but <a href="http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.ndjfl/1093635756" rel="nofollow">Beeson's review</a> is. Bridges and Richman - <a href="http://books.google.com/books?id=oN5nsPkXhhsC&printsec=frontcover&dq=bridges+richman+constructive+mathematics#v=onepage&q=&f=false" rel="nofollow"> Varieties of Constructive Mathematics</a> (google books has some of the relevant pages online). There are many other references, I tried to give my own view on them in the first two pages of this <a href="http://arxiv.org/abs/math/0703561" rel="nofollow">paper</a>. Nice post!

Regarding references to the intuitionistic axioms. You could try the following two text books:
A. S. Troelstra and D. van Dalen. Constructivism in mathematics, vols. 1 and 2.
Not online, but Beeson’s review is.
Bridges and Richman – Varieties of Constructive Mathematics (google books has some of the relevant pages online).

There are many other references, I tried to give my own view on them in the first two pages of this paper.

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