Comments on: Modelling the continuum http://www.unwantedcapture.org/2009/05/16/modelling-the-continuum/ logic, philosophy, mathematics, linguistics, ... Sun, 31 Jan 2010 04:59:17 +0000 hourly 1 http://wordpress.org/?v=3.1.3 By: Edward Dean http://www.unwantedcapture.org/2009/05/16/modelling-the-continuum/comment-page-1/#comment-8 Edward Dean Tue, 19 May 2009 02:43:36 +0000 http://www.unwantedcapture.org/?p=169#comment-8 Nice first post, Spencer. It is clear that your syntactic approach to the continuum differs from Nelson's pioneering internal set theory, since his shares with semantic approaches a main feature you wish to avoid: that of starting with a perfectly serviceable $$\mathbb{R}$$ and adding nonstandard reals on top. On the other hand, Jeremy Avigad's syntactic approach in <a href="http://www.andrew.cmu.edu/user/avigad/Papers/nonstandard2.pdf" rel="nofollow">Weak theories of nonstandard arithmetic and analysis</a> (15 ff.) already avoids this. Roughly, we have a theory whose variables are intended to range over $$\mathbb{N}^*$$; then $$\mathbb{Z}^*$$ and [math]\mathbb{Q}^*[/math] are defined as expected. From there, $$\mathbb{R}$$ is defined to be those members of [math]\mathbb{Q}^*[/math] which are <em>bounded</em>, i.e. which have standard ceilings, so avoiding any talk of some $$\mathbb{R}^*$$. Beauty is in the eye of the beholder, of course, but I find this way of doing things aesthetically pleasing. Now nothing about Jeremy's setup addresses the particular kind of aesthetic appeal you seem to be after when, say, making a very deliberate choice for your [math]1_{\mathbb{R}}[/math]; I look forward to seeing what beauty you intend to extract. I am wondering (while not knowing exactly what you want to get out of this) whether the effect you ultimately want could also be got while starting off in the manner just described, rather than building $$\mathbb{R}$$ as you do in the post? Nice first post, Spencer.

It is clear that your syntactic approach to the continuum differs from Nelson's pioneering internal set theory, since his shares with semantic approaches a main feature you wish to avoid: that of starting with a perfectly serviceable \mathbb{R} and adding nonstandard reals on top. On the other hand, Jeremy Avigad's syntactic approach in Weak theories of nonstandard arithmetic and analysis (15 ff.) already avoids this.

Roughly, we have a theory whose variables are intended to range over \mathbb{N}^*; then \mathbb{Z}^* and \mathbb{Q}^* are defined as expected. From there, \mathbb{R} is defined to be those members of \mathbb{Q}^* which are bounded, i.e. which have standard ceilings, so avoiding any talk of some \mathbb{R}^*. Beauty is in the eye of the beholder, of course, but I find this way of doing things aesthetically pleasing.

Now nothing about Jeremy's setup addresses the particular kind of aesthetic appeal you seem to be after when, say, making a very deliberate choice for your 1_{\mathbb{R}}; I look forward to seeing what beauty you intend to extract. I am wondering (while not knowing exactly what you want to get out of this) whether the effect you ultimately want could also be got while starting off in the manner just described, rather than building \mathbb{R} as you do in the post?

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