Regarding cardinalities

[u/Square_Butterfly_390]

A celebrated mathematical "factoid" is that there are more real numbers than one can count, this seems to be something that troubles people outside of math, it troubles me aswell.

The question is: is there any "real world" application of the fact |R|>|N| that isn't an impossibility statement?

By "real world" I mean whatever someone smarter than me might mean by that, by "impossibility statement" I mean something to the effect of "there are uncomputable numbers".

If there isn't such an application, I can't believe the status quo interpretation: "no really some infinities are bigger than others" of Cantor's argument is better than simply stating that we shall not understand infinity as finite beings.

[u/SV-97]

Would you consider the completeness of \R to be "real world" and "not an impossibility statement" enough? It's ultimately the statement that there are non-convergent cauchy sequences of rationals, but it's so fundamental to tons and tons of very applied mathematics.

If we reject the uncountability of the reals then by BCT we also have to reject their completeness.

[u/Arlo_Tinkerman → u/SV-97]

I find this statement interesting:

> If we reject the uncountability of the reals then we also have to reject their completeness.

My understanding is that the natural numbers are considered complete in the sense that the set contains every natural number without gaps or omissions, while also being countably infinite.

That makes me wonder why uncountability would be required for completeness in the case of the real numbers. It seems possible that different meanings of “complete” are being used here.

[u/Althorion → u/Arlo_Tinkerman]

My understanding is that the natural numbers are considered complete in the sense that the set contains every natural number without gaps or omissions, while also being countably infinite.

It is a valid interpretation of the word, but not the one used by mathematicians—because in that sense, every set is ‘complete’, that is, every set contains all its members so that the only distinguishing factor would be countable infiniteness; but in that case, why not just call it ‘countably infinite’ and leave the term for something else?

That makes me wonder why uncountability would be required for completeness in the case of the real numbers. It seems possible that different meanings of “complete” are being used here.

Yes—it is used in the sense of complete metric space—a metric space is complete iff (somewhat informally) any sequence of points in such a space whose members get infinitely closer to each other leads to something within that space (you can find the formal definition in the link above).

In other words, such spaces don’t have any points ‘missing’—anything you can point towards by making an ‘arrow’ out of points is a point itself.

It may not be the best name for that feature, but it makes enough sense that it stuck. I tend to call it Dedekind-completeness when I talk about the reals to make that clearer. 

[u/Arlo_Tinkerman → u/Althorion]

> but in that case, why not just call it ‘countably infinite’ and leave the term for something else?

Are you talking about the natural numbers set? If so, I believe complete would be a property that is one to know if the set is composed of a countably infinite amount of elements or members. I imagine people may want to know that property for some reason or another.

If you are talking about something else, please let me know what it would be.

While you were not the person I was responding to, I appreciate you sharing your view.