In addition to what Jess Mazer asks, it would also be of interest to know just what is known or believed about the infinite sets of those objects which mathematics does deal with. I've read in E.T. Bell that the infinity of curves or functions is greater than the infinity of the reals. An information scientist at another forum told me that the infinity of the hyperreal numbers is larger than that of the reals, & that the infinity of the surreals is larger than that of the hyperreals. Is this true? And does the infinity of curves or functions (of the standard or "archimedean" numbers) correspond to either the infinity of the hyperreals or that of the surreals? Is it known whether one could possibly define a larger set of numbers than the surreals? One also hears that anything that can be done with nonstandard numbers can be done with standard numbers, as long as it doesn't pertain to the difference between.them. Is the difference between them still regarded as not leading to an! ything of interest?
- Ben Udell >Does anyone know, are there versions of philosophy-of-mathematics that would allow no >distinctions in infinities beyond countable and uncountable? I know intuitionism is >more restrictive about infinities than traditional mathematics, but it's way *too* >restrictive for my tastes, I wouldn't want to throw out the law of the excluded >middle. >Jesse Mazer