How many real numbers exist? New proof moves closer to an answer • Hacker News Board

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

#71

Post by myWindoonn » Thu, Jul 15, 2021, 5:10 PM UTC

sillysaurusx wrote: post #39

helen___keller wrote: post #30

Earlier quoted context omitted.

Others have addressed why it matters (or doesn't matter) when viewed from outside mathematics. But within mathematics, unsolved problems usually matter for two reasons: (1) When lots of really smart people spend lots of time trying to solve something, and fail to do so, it becomes even more interesting for other smart people. A well-known example is the Collatz Conjecture[0] which, by most accounts is a meaningless p…

This is an excellent answer. I have a mostly unrelated question. In the case of the Collatz conjecture, it seems all but proved: > If the conjecture is false, it can only be because there is some starting number which gives rise to a sequence that does not contain 1. Such a sequence would either enter a repeating cycle that excludes 1, or increase without bound. No such sequence has been found. There are lots of hist…

Some theoretical physicist (Aaronson?) once quipped something like, "If physicists had discovered P vs. NP, then they'd have said P != NP and given a Nobel for it. And then if it turned out P = NP, then they'd give another Nobel."

Another, more important reason is that we know that the Collatz Conjecture is a single slice of a Turing-complete question about dynamical systems. Trying to find a complete proof expands our knowledge about the bridge between dynamical systems and the natural numbers. Such explorations were essential to founding modern physics in terms of dynamics and conservation laws.

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

#72

Post by tgflynn » Thu, Jul 15, 2021, 5:10 PM UTC

rssoconnor wrote: post #65

pavpanchekha wrote: post #4

Great article. Since it looks like a lot of folks are interested in this article, some extra background. First, what is forcing? The article actually has a great description of ultrapowers (a key part of the construction) but it goes by a little fast, so you might like Tim Chow's "A beginner's guide to forcing" [1] which does a good job not only laying out the mathematical details at a high level, but also really cle…

> but at the same time there is no "right answer"—all of these axioms, after all, are independent of ZFC, and it's "ok" to add any of them. Minor quible: Just because a potential axiom is independent of ZF(C) doesn't make it necessarily "okay" to add. Potential axioms can be unsound, for example if they prove new / untrue Sigma_1 statements. As an example, even in the likely circumstance that ¬Con(ZFC) is independent…

> he resulting system is unsound (in the sense of Tarski) because it asserts the existence of natural numbers that have no "written form" (i.e. the existance of natural numbers that are larger than any term you can write to denote a natural number).

Isn't that basically the definition of the natural numbers, ie. if you write down any natural number (say n) I can always construct a natural number that is larger than it (like n+1) ?

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

#73

Post by gotostatement » Thu, Jul 15, 2021, 5:11 PM UTC

jmgao wrote: post #32

jostmey wrote: post #17

> "Not all infinities are equal" In other words, there are different categories of infinite, and it might be inappropriate to represent infinity with just one symbol! This article is about how many types of infinity might exist. I was taught there is countably and uncountably infinite. Integers are countably infinite because the number of integers between any two numbers if finite. Real numbers are uncountably infini…

> I was taught there is countably and uncountably infinite. Integers are countably infinite because the number of integers between any two numbers if finite. Real numbers are uncountably infinite because there are infinite numbers between any two real numbers. This is unsound: a set is countable if there's a mapping that assigns an integer to each element of that set. There are an infinite number of rational numbers…

I guess you could rephrase OP's argument in a sound way as: a set is countable if there's a way to order it such that between every two numbers there are finitely many numbers

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

#74

Post by mikepurvis » Thu, Jul 15, 2021, 5:13 PM UTC

jostmey wrote: post #17

> "Not all infinities are equal" In other words, there are different categories of infinite, and it might be inappropriate to represent infinity with just one symbol! This article is about how many types of infinity might exist. I was taught there is countably and uncountably infinite. Integers are countably infinite because the number of integers between any two numbers if finite. Real numbers are uncountably infini…

Veritasium did a really good video on this recently, based around Hilbert's Hotel:

https://www.youtube.com/watch?v=OxGsU8oIWjY

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

#75

Post by privatdozent » Thu, Jul 15, 2021, 5:14 PM UTC

Another great piece from Quanta. Such an important institution

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

#76

Post by sillysaurusx » Thu, Jul 15, 2021, 5:14 PM UTC

Ivoirians wrote: post #60

sillysaurusx wrote: post #39

Earlier quoted context omitted.

This is an excellent answer. I have a mostly unrelated question. In the case of the Collatz conjecture, it seems all but proved: > If the conjecture is false, it can only be because there is some starting number which gives rise to a sequence that does not contain 1. Such a sequence would either enter a repeating cycle that excludes 1, or increase without bound. No such sequence has been found. There are lots of hist…

They've verified that the Collatz conjecture holds for all numbers up to ~2^68, but that's precisely 0% of all the numbers that need to be checked. But more importantly, the goal of (pure) mathematics isn't to declare truths. If you had a machine from God himself that outputted True or False for theorems you put in, that wouldn't demotivate (pure) mathematicians from doing the work they're doing. Understanding the re…

> They've verified that the Collatz conjecture holds for all numbers up to ~2^68, but that's precisely 0% of all the numbers that need to be checked.

This is the crux of what has me looking like a fool to every mathematician in the thread, but I don't mind: why is 2^68 0% of the "numbers that need to be checked"? From a physicist standpoint, you can do a lot with numbers from 0 to 2^68. After all, 64-bit floats are quite useful. Is there 0% value in proving the Collatz conjecture for all possible numbers one might want to use in a normal programming language without big number libraries?

I know the question must sound pretty crude, but it's also a source of mystery. Mathematicians are so obsessed with exactness. Is there no room for empirical analysis in number theory?

In other words, number theory relies on certain assumptions. What if one of your assumptions is "a number system from 0 to 2^64"? Why is there no value in that?

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

#77

Post by schoen » Thu, Jul 15, 2021, 5:15 PM UTC

sillysaurusx wrote: post #69

svat wrote: post #57

Earlier quoted context omitted.

No amount of empirical data is ever "good enough" for the mathematical standard of proof. (It may be enough for mathematicians to "believe" something in some informal sense, but not enough to consider it "proved".) You can see some examples at the answers to these questions; maybe at least one of them will be interesting to you: - https://math.stackexchange.com/questions/514/conjectures-tha... - https://math.stackexc…

That was another question in the back of my mind -- famously, the four color theorem was proved by computers through exhaustive analysis (checking every possibility). At the time, it was controversial as a "proof" since it didn't really take the usual form of a proof. I've often wondered "Why can't we do something like that, but for all instances of things like the Collatz conjecture?" Of course, it's computationally…

The four-color theorem had the preliminary challenge of creating a method to identify all the relevant cases. That also required a mathematical theory. (I don't actually understand how it was done!)

If you didn't have that, you would have an infinite search for possible maps that violate the conjecture, because you wouldn't be able to divide them into a finite number of equivalence classes, or enumerate a finite number of possible counterexamples, or whatever.

For Collatz, I don't think we have any lemma that gives a path to saying "a counterexample must be one of these 2¹⁰⁰ integers" or the like. So without such a thing, checking every possibility would mean checking every integer. (Well, there are definitely lemmas that make it unnecessary to check large numbers of integers, so I should say instead that checking every possibility would still mean checking an infinite number of cases.)

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

#78

Post by helen___keller » Thu, Jul 15, 2021, 5:18 PM UTC

sillysaurusx wrote: post #69

svat wrote: post #57

Earlier quoted context omitted.

No amount of empirical data is ever "good enough" for the mathematical standard of proof. (It may be enough for mathematicians to "believe" something in some informal sense, but not enough to consider it "proved".) You can see some examples at the answers to these questions; maybe at least one of them will be interesting to you: - https://math.stackexchange.com/questions/514/conjectures-tha... - https://math.stackexc…

That was another question in the back of my mind -- famously, the four color theorem was proved by computers through exhaustive analysis (checking every possibility). At the time, it was controversial as a "proof" since it didn't really take the usual form of a proof. I've often wondered "Why can't we do something like that, but for all instances of things like the Collatz conjecture?" Of course, it's computationally…

It can be valuable, but it's not as valuable as covering all cases.

If I could prove collatz for 99% of numbers it would be great.

If i could additionally prove which 99% converge, let's say all numbers not divisible by 100, it would be enormous because even showing this much is likely to be a stepping stone to a full proof (what property is it about the number 100 that excludes all non-divisors from diverging?)

> I suppose what bugs me about math is, physicists are proved wrong all the time. Mathematicians are rarely proved mistaken, because they construct assumptions that you can't disagree with. There's no chance for empirical data to falsify one's assertions.

Right, and that's the whole point and beauty of it.

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

#79

Post by chongli » Thu, Jul 15, 2021, 5:18 PM UTC

tgflynn wrote: post #72

rssoconnor wrote: post #65

Earlier quoted context omitted.

> but at the same time there is no "right answer"—all of these axioms, after all, are independent of ZFC, and it's "ok" to add any of them. Minor quible: Just because a potential axiom is independent of ZF(C) doesn't make it necessarily "okay" to add. Potential axioms can be unsound, for example if they prove new / untrue Sigma_1 statements. As an example, even in the likely circumstance that ¬Con(ZFC) is independent…

> he resulting system is unsound (in the sense of Tarski) because it asserts the existence of natural numbers that have no "written form" (i.e. the existance of natural numbers that are larger than any term you can write to denote a natural number). Isn't that basically the definition of the natural numbers, ie. if you write down any natural number (say n) I can always construct a natural number that is larger than i…

If you can write down the natural number n, you can write down n+1.

Of course, you can't write down the entire set of natural numbers but that is not a natural number.

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

#80

Post by stonemetal12 » Thu, Jul 15, 2021, 5:20 PM UTC

sillysaurusx wrote: post #39

helen___keller wrote: post #30

Earlier quoted context omitted.

Others have addressed why it matters (or doesn't matter) when viewed from outside mathematics. But within mathematics, unsolved problems usually matter for two reasons: (1) When lots of really smart people spend lots of time trying to solve something, and fail to do so, it becomes even more interesting for other smart people. A well-known example is the Collatz Conjecture[0] which, by most accounts is a meaningless p…

This is an excellent answer. I have a mostly unrelated question. In the case of the Collatz conjecture, it seems all but proved: > If the conjecture is false, it can only be because there is some starting number which gives rise to a sequence that does not contain 1. Such a sequence would either enter a repeating cycle that excludes 1, or increase without bound. No such sequence has been found. There are lots of hist…

It is simple empirical data isn't proof. It is as if you were testing primes 2 yes, 3 yes, 4 no. Here is empirical data that only 2 primes exist. Should we believe it?

Sure for Collatz we have tested more than 3 numbers but we can't guaranty it is true until we test all infinity of them, or have a proof that doesn't rely on empirical data. To do otherwise would be to look like a fool when someone got around to testing N+1 and it no longer being true.