12/26/2007, 09:27 PM
Andrew, methinks that infinitesimals would necessarily have density measured with 2^C, using a Cantorian system. C is 2^omega or aleph-null or whatever. The reals are infinitely more dense than the rationals, but that's not really enough to describe them. Between any two rationals, there are an infinite number of rationals. Thus, there are an infinite number of rationals between 0 and any rational. There are an infinite number of reals between 0 and any rational. Yet somehow the reals are more dense.
In real analysis, reals are as dense as the number line gets. I'm assuming that with this non-standard analysis, the infinitesimals would create a field that is as dense relative to the reals as the reals are to the rationals. But I don't particularly see how a coherent system could be derived from this. Of course, I'm not a very good topologist, and at any rate, I never bought the distinction in density (perhaps cardinality is the right word?) between rationals and reals, so I'm not really in a position to be convinced of yet another layer of density.
In real analysis, reals are as dense as the number line gets. I'm assuming that with this non-standard analysis, the infinitesimals would create a field that is as dense relative to the reals as the reals are to the rationals. But I don't particularly see how a coherent system could be derived from this. Of course, I'm not a very good topologist, and at any rate, I never bought the distinction in density (perhaps cardinality is the right word?) between rationals and reals, so I'm not really in a position to be convinced of yet another layer of density.
~ Jay Daniel Fox