How many real numbers exist? New proof moves closer to an answer

I just want to point out, because the HN crowd seems to generally not be mathematical Platonists, that this entire article is implicitly assuming a Platonist philosophical foundation. This may cause confusion for lay readers who are not mathematical Platonists.

In other words the article assumes that mathematical objects have an objective existence: they either exist or they do not. Hence every single logical axiom has a truth value. You do not get to arbitrarily choose what logical axioms you want. If you do, you can end up choosing the "wrong one." Therefore it is an important question to understand whether the Continuum Hypothesis is true or not, even if it's independent of ZFC (and hence requires ultimately philosophical rather than mathematical arguments).

If you aren't a Platonist and instead view logical axioms as having no inherent truth value, but rather foundations that you can pick and choose from as necessary (where in one case you may choose to use an axiom and in another you may choose its negation), then all that might sound very strange to you. In that case, you should mentally substitute every instance of "true" or "false" in the article with "agrees or disagrees with the meta model used to examine the semantics of a logical theory." In particular, whenever we talk about a formal treatment of the semantics (i.e. model) of a theory, whether that be something like ZFC or a programming language, we must always make those statements relative to a meta-model.

For example if we talk about the formal semantics of a language like C, we must first posit a meta-model which already contains notions of things like "integer" and "natural number" which can be used to give meaning to statements such as "performed an operation n times." If you're not a Platonist then you probably believe that there are multiple possible meta-models you could use.

https://news.ycombinator.com/newsguidelines.html

Problem 2: There's no natural number that maps to (say) 1. Even if you do allow 1/(2^aleph_0), there's no finite number n that would make n/(2^aleph_0) = 1. With any reasonable definitions of the operations involved here, n/(2^aleph_0) would always be infinitesimal, so it would never equal a non-infinitesimal.

Problem 3: You're still skipping over infinitely many numbers. If 1/(2^aleph_0) is a number (and again, this requires going beyond the real numbers) and 1.5 is a number, then 1.5 * 1/(2^aleph_0) = 1.5/(2^aleph_0) is also a number, but no natural number gets mapped to that.

2^{\aleph_0} is a cardinal number, which isn’t really a number in the sense of “an element of a field” or something like that. Dividing by it isn’t a well defined thing.

And, you certainly can’t just multiply any real number (or, any real number between 0 and 1) by 2^{\aleph_0} and get a different integer as a result.

(Now, if you work in the surreal numbers, you can define things like n/(2^{\aleph_0}) (identifying cardinals with the first ordinal of that cardinality), but these would not be real numbers. They would all be infinitesimal , smaller than 1/k for all positive integers k, and yet bigger than 0. Similarly in the surreal numbers, you could multiply real numbers between 0 and 1 by 2^{\aleph_0}, but you would get surreal numbers which are larger than every integer (in fact, larger than any countable ordinal))

Summary: What you wrote doesn’t define a mapping from the integers to the real numbers . (It can be interpreted as defining a map from integers to something else though.)