"Hey Kevin. I think I accide..."


by Michael Cohen Jun 17 2016

Hey Kevin. I think I accidentally clicked ignore on your query. I'll look into that definition. I think potentially this definition isn't circular, but you might be right. I don't know quite how to put it, but it seems like this criticism could be directed at any algorithm that attempts to point towards all real numbers. The algorithm, it seems, would have to keep gesturing closer and closer to the real number it is trying to indicate. It's not as if the algorithm can take the set of real numbers for granted, and then say that we just need to find the number in the set of real numbers that satisfies the condition that we keep getting closer and closer to it, and then finally we add that to the set of real numbers. And yet, any algorithm that doesn't have a set in mind that it can pull the answer from, I think it will involve something along the lines of pointing towards a number that doesn't yet come from a well-defined spot. Otherwise, the algorithm can't create the well-defined spot. That might all be utter nonsense though.


Kevin Clancy

I understand what you're saying and I think it's a good point. The problem is that you're developing an algorithm (a non-terminating one) that finds real numbers rather than providing a definition of them. It turns out that providing a definition of real numbers is not a simple as it may at first seem. This presentation is somewhat similar constructive analysis, in which a real number is defined as regularly converging sequence of rational numbers; importantly, constructive analysis does not define real numbers as infinite sums of these sequences, because as I've said, that would be a circular definition.

If you want to learn more about rigorous foundations for real numbers and related topics, I think that the book Calculus by Michael Spivak is a very approachable and well respected introduction to the topic.

Michael Cohen

Is an infinite sum of rationals isomorphic with a regularly converging sequence of rationals (something along the lines of 1/2, 5/8, 11/16, etc.), where each rational in the sequence is the sum of all the addends up until then? I agree it is probably worth putting up a different definition anyway. I'm not sure I'll be able to do that for a little bit, since I haven't studied real analysis yet, but if you want to do that sooner, go for it. This is a fun conversation!