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

#111

Post by hvocode » Thu, Jul 15, 2021, 6:01 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…

I think others addressed the "why proof not empirical data" question well, but one additional point. A logical proof is only as solid as its weakest component. In mathematics any result that has been proven may be used in a component of another proof. If you have a result that is based on empirical evidence, then every result proven based on that also inherits that empirical evidence as part of its foundation. The reason we don't do this is that if, by some weird chance, the original result that used empirical data instead of proof is shown to be false, then every result built upon it is invalidated and needs to be revisited. That would be a mess. Sticking to requiring formal proofs at least reduces that possibility - although it is still entirely possible for a proof to have a subtle mistake as well, which would have a similar cascading effect on results built upon it.

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

#112

Post by thom » Thu, Jul 15, 2021, 6:03 PM UTC

bmitc wrote: post #47

thom wrote: post #27

Earlier quoted context omitted.

Why does “the set containing the natural numbers and a sandwich” not have cardinality between the two?

The non-sandwich analogy is called Hilbert’s hotel. Saying that two sets have the same cardinality is equivalent to them having a bijection between them. So the claim is that the natural numbers and the natural numbers plus a sandwich have the same cardinality. This can be proved by the bijection: 0 -> sandwich 1 -> 0 2 -> 1 3 -> 2 . . . n -> n-1 . . . There is actually more though! If you had an infinite but countab…

Because everyone gave me such good and well meaning answers perhaps you’ll permit me a follow up.

As I understand it we can say the cardinality of the reals is 2^aleph_0. Why is it cheating to create a bijection thusly:

    0 -> 0
    1 -> 1/(2^aleph_0)
    2 -> 2/(2^aleph_0)

etc?

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

#113

Post by klysm » Thu, Jul 15, 2021, 6:03 PM UTC

btilly wrote: post #88

As a constructivist I'll be over in the corner that says that there are only a countable number of real numbers, and the unimaginable number of unimaginable infinities that classical mathematics insiste exists is all made up nonsense. That, in fact, things that can't ever be named, even in principle, don't actually exist. What is interesting is that as shocking as constructivism may be, there is no logical flaw in it…

Does constructivism rule out uncountability because induction is always indexable?

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

#114

Post by _2d30 » Thu, Jul 15, 2021, 6:04 PM UTC

question000 wrote: post #84

The only thing this proves is that mathematics is a soft science, where concepts like "number" and "infinite" are subjective. There are obviously infinite numbers, if you think there's a finite number of numbers, take that number and add one to that. QED

1) Mathematics is in no way a science.

2) I think you didn't read the article at all.

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

#115

Post by Sharlin » Thu, Jul 15, 2021, 6:09 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…

> There are lots of histograms and empirical data supporting the conjecture.

There's a very interesting implicit question there: why should counterexamples be small? [1][2] Certainly, counterexamples to many conjectures about infinite sets can be found with a brute-force search, even if the problem is merely semidecidable. But isn't that simply an example of selection bias?

There is a certain subjective beauty in having counterexamples like 9 or 341 or even 23338590792. But is that just anthropocentrism? After all, no matter how many cases we check, we have made absolutely no progress in exhausting the whole set of natural numbers! We can never reach even reasonably easily constructable numbers like 3↑↑↑3 (using Knuth arrow notation [3]), and still almost all[4] natural numbers are bigger than that.

In physics, there's an (often implicitly made) assumption that more evidence in support of a hypothesis makes it more likely that the hypothesis is supported by any future evidence as well. But why should we be able to make that assumption? We do, because it seems to work, but why should it still work tomorrow? This is, of course, the famous philosophical problem of induction [5]. But math is basically what happens when you explicitly reject inductive reasoning and then start to explore the space of things that can still be reached, using purely deductive reasoning!

[1] https://math.stackexchange.com/questions/449886/the-largest-...

[2] https://math.stackexchange.com/questions/111440/examples-of-...

[3] https://en.wikipedia.org/wiki/Knuth%27s_up-arrow_notation

[4] https://en.wikipedia.org/wiki/Almost_all

[5] https://en.wikipedia.org/wiki/Problem_of_induction

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

#116

Post by dwohnitmok » Thu, Jul 15, 2021, 6:13 PM UTC

hypersoar wrote: post #93

tgflynn wrote: post #72

Earlier quoted context omitted.

> 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…

You can find a natural number that is bigger than any one natural number, but you can't write one down that's bigger than every natural number in the ZFC sense.

> that's bigger than every natural number in the ZFC sense.

I'm not sure what you mean by this.

Even for nonstandard models of the natural numbers there will never be a single number larger than all other numbers, since that violate the Peano axioms.

Did you mean by "in the ZFC sense" "the standard model?"

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

#117

Post by acchow » Thu, Jul 15, 2021, 6:13 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…

"famously, the four color theorem was proved by computers through exhaustive analysis (checking every possibility)"

This is impossible. The plane can be arbitrarily large with an arbitrary number of regions. It can be exhaustively checked for n number of regions up to a certain n. But not for arbitrary n.

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

#118

Post by btilly » Thu, Jul 15, 2021, 6:14 PM UTC

scapp wrote: post #105

btilly wrote: post #88

As a constructivist I'll be over in the corner that says that there are only a countable number of real numbers, and the unimaginable number of unimaginable infinities that classical mathematics insiste exists is all made up nonsense. That, in fact, things that can't ever be named, even in principle, don't actually exist. What is interesting is that as shocking as constructivism may be, there is no logical flaw in it…

You sound like you need to read this [0] answer to the question "Are real numbers countable in constructive mathematics?". > You are using the word "constructive" in an unusual way. It is true that, in ZFC, the set of computable real numbers is countable, but that is not directly a statement about constructive mathematics. > Not every school of constructive mathematics identifies real numbers with algorithms; that's…

Yes, there are multiple constructivist approaches possible. However since my objection to classical approaches is that I want "X exists" to be meaningful, I like mathematical objects that can be written down with a finite number of symbols in a finite space. Which means that I'm only interested in a countable universe of possible mathematical things.

If you say "exists" about anything else, I'll understand you - I do have advanced degrees in math. But I'll think that you're using the word "exists" in a deeply artificial way.

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

#119

Post by btilly » Thu, Jul 15, 2021, 6:22 PM UTC

klysm wrote: post #113

btilly wrote: post #88

As a constructivist I'll be over in the corner that says that there are only a countable number of real numbers, and the unimaginable number of unimaginable infinities that classical mathematics insiste exists is all made up nonsense. That, in fact, things that can't ever be named, even in principle, don't actually exist. What is interesting is that as shocking as constructivism may be, there is no logical flaw in it…

Does constructivism rule out uncountability because induction is always indexable?

There are different schools of constructivism. But if you insist on only dealing with mathematical objects that admit of finite descriptions with finite symbols, there is a surjection from a countable set onto the entire universe of possible mathematical objects.

That said, an enumeration of all mathematical objects is not possible. That's because we may not be able to resolve the question of whether 2 descriptions of a mathematical object actually refer to the same object or not.

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

#120

Post by finnh » Thu, Jul 15, 2021, 6:32 PM UTC

BoiledCabbage wrote: post #45

jerf wrote: post #38

Earlier quoted context omitted.

"Didn't you just conclude that it's impossible to have a set of all real numbers?" Cantor's diagonalization proof proves that it's impossible to list all the real numbers with a list of size aleph-0, which is the cardinality of the set of natural numbers. The forcing proof is an attempt to prove you also can't do it with the a list of the size aleph-1, which is the size of the power set of aleph-0. It purports to pro…

Thanks for the reply. Question though. in Cantor's argument we explicitly mapped the reals to aleph-0 so it makes sense that our conclusion decides that mapping to aleph-0 is too small so it's size must be larger. Where in the forcing process do we even "use" aleph-1? If we used aleph-1 then it could see the parallels and the argument would make sense - but all I see in the forcing process is "start with a set of all…

Great question, I have no answer.

The article explained forcing in such a way as to simply restate what I thought we already knew: given a real, there is no "next" real. (ie, there are a non-countable-infinite number of reals between any two reals).

I don't see the newness that forcing brings to this.