I think that every metric space is dense in itself. If X is a metric space, then a set E is dense in X whenever every element of X is either a limit point of E or an element of E (or both).
I thought so too when I wrote it up; I put it there as a placeholder for a Wikipedia-style initial definition once we find one that's more suitable, because I'm having a hard time thinking of one.
This definition of the real numbers has a bigger problem with it than just circular logic — it also runs into the 0.9999... = 1 paradox. The sets N∖{1,2,3,4,5} and the set 5 both encode the number 1/8.
Normally the real numbers are defined using either Dedekind cuts or Cauchy sequences of rational numbers. Could we please use one of those definitions instead, as they're the standard ones used by most mathematicians?
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.
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!
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.
I'm pretty tired right now, but this definition seems kind of circular to me. It involves an infinite sum, and infinite sums are defined in terms of limits. But a limit of rational numbers is defined in terms of the set of real numbers. Maybe it would be better to present the definition of real numbers that one would find in a real analysis text.
Approved, but the summary could do with a bit of improvement, make it something that a non-mathematician will get something out of. Give examples of things that are and are not real numbers.
I think that every metric space is dense in itself. If X is a metric space, then a set E is dense in X whenever every element of X is either a limit point of E or an element of E (or both).
This is not a very good summary, since it relies on the reader understanding what a "complete number line" is.
I thought so too when I wrote it up; I put it there as a placeholder for a Wikipedia-style initial definition once we find one that's more suitable, because I'm having a hard time thinking of one.
The title mentions Cauchy sequences, but the body does not. Doesn't this definition consider classes of non-converging sequences as real numbers?
You're right; I was sloppy. I'll fix it, thanks.
This definition of the real numbers has a bigger problem with it than just circular logic — it also runs into the 0.9999... = 1 paradox. The sets N∖{1,2,3,4,5} and the set 5 both encode the number 1/8.
Normally the real numbers are defined using either Dedekind cuts or Cauchy sequences of rational numbers. Could we please use one of those definitions instead, as they're the standard ones used by most mathematicians?
Sorry about my inactivity on Arbital, and thanks for going ahead and fixing it!
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.
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!
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.
I'm pretty tired right now, but this definition seems kind of circular to me. It involves an infinite sum, and infinite sums are defined in terms of limits. But a limit of rational numbers is defined in terms of the set of real numbers. Maybe it would be better to present the definition of real numbers that one would find in a real analysis text.
Approved, but the summary could do with a bit of improvement, make it something that a non-mathematician will get something out of. Give examples of things that are and are not real numbers.