January 7th, 2025

How many real numbers exist? New proof moves closer to an answer (2021)

A recent proof by David Asperó and Ralf Schindler suggests the total number of real numbers may be knowable, challenging the continuum hypothesis and indicating an additional size of infinity.

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How many real numbers exist? New proof moves closer to an answer (2021)

A recent proof by mathematicians David Asperó and Ralf Schindler challenges the long-held belief that the total number of real numbers is unknowable. Their work, published in the Annals of Mathematics, connects two previously competing axioms in infinite mathematics, suggesting that both may be true and strengthening the argument against the continuum hypothesis. This hypothesis, proposed in 1878, posits that there are no sizes of infinity between the cardinality of natural numbers and that of real numbers. The new proof indicates that there is indeed an additional size of infinity, which has significant implications for the understanding of infinite sets. The continuum hypothesis has been a central question in mathematics since it was placed on David Hilbert's list of unsolved problems in 1900. Gödel and Cohen previously established that the hypothesis is independent of the standard axioms of mathematics, meaning it cannot be proven or disproven using them. The recent findings provide a coherent alternative to the continuum hypothesis and suggest that the structure of infinite sets may be more complex than previously thought. This development is seen as a pivotal moment in the field of mathematics, with potential for new axioms to emerge that could clarify the nature of infinity.

- A new proof suggests the total number of real numbers may be knowable.

- The proof connects two competing axioms in infinite mathematics.

- It strengthens the case against the continuum hypothesis, indicating an additional size of infinity.

- The continuum hypothesis has been a central unresolved question in mathematics for over a century.

- This development may lead to new axioms that clarify the structure of infinite sets.

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By @dang - about 1 month
Discussed at the time:

How many real numbers exist? New proof moves closer to an answer - https://news.ycombinator.com/item?id=27845576 - July 2021 (344 comments)

(Reposts are fine after a year or so! especially for a great article like this. Links to past threads are just to satisfy extra-curious readers)

By @JadeNB - about 1 month
The title:

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

and the subhed:

> For 50 years, mathematicians have believed that the total number of real numbers is unknowable. A new proof suggests otherwise.

both suggest a mystery that isn't there. Not that there isn't a mystery—there is!—but I think it's unreasonable to describe it as the title and subhed do.

As the rest of the article, which is at Quanta's usual excellent expository standard, explains, there is absolutely no debate about the number of real numbers; it is 2^{\aleph_0}. You don't have to know what that is, but it is important to know that it is a definite, precise answer. You can ask whether it's the same as \aleph_1, and that's something we don't know. But that doesn't mean that we don't know the answer to how many reals there are, only that we don't know whether some other potential answer is also the answer.

By @ajkjk - about 1 month
It's a trick! Despite the name, the real numbers aren't real at all. Most of them don't exist...

Now the computable reals, on the other hand... maybe.

By @trehalose - about 1 month
> But Cantor discovered that natural numbers can’t be put into one-to-one correspondence with the continuum of real numbers. For instance, try to pair 1 with 1.00000… and 2 with 1.00001…, and you’ll have skipped over infinitely many real numbers (like 1.000000001…). You can’t possibly count them all; their cardinality is greater than that of the natural numbers.

I take issue with this paragraph. It makes it sound like the infinitude of reals between 1 and 1.00001 is proof there are more reals than naturals. One might think this same fact proves that there are more rationals than naturals...

By @qnleigh - about 1 month
I wish I understood how mathematicians conceptualize the notion of 'right' and 'wrong' axioms in light of Gödel. I would've thought that the incompleteness theorems would be taken as evidence against Platonism, but clearly this didn't happen. What do people think is the source of 'ground truth'? Is it more than aesthetic appeal? Are there any articles or books that convey it?

I've talked to a friend with a doctorate in algebraic geometry about this several times, but all I've really gotten out of her is that if I studied a lot more math, I would gradually get the same sense.

(For reference, I have about as much mathematics background as a US student who minored in maths.)

By @dataflow - about 1 month
Meanwhile all I've ever wanted for Christmas was a notion of size that admits (A ⊆ B) ∧ (A ≠ B) ⇒ size(A) < size(B).
By @neuroelectron - about 1 month
tl;dr: Godel's incompleteness theorem. When you try to create completely self-consistent axioms, it just creates an infinite hierarchy of further unprofitable axioms. Numbers don't really exist, so that was your first mistake.

tl;dr^2: Mentally move the hyphen one word to the right.