There is a good video about that on Numberfile[1]
How many real numbers exist? New proof moves closer to an answer
51–60 of 359 posts
Re: How many real numbers exist? New proof moves closer to an answer
#52Great 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…
Why does forcing work? To me it seems flawed (which obviously means I don't understand it fully). For diagonalization argument: 1) Assume every real can be assigned a natural number. 2) Do a bunch of steps that essentially find a new real that differs from any real you have listed from step 1. 3) Conclude that either your steps are flawed, or your initial assumption is wrong 4). Because your steps aren't flawed then…
In a forcing-based consistency proof of a proposition P, you start with a model of ZFC (or a sufficiently large subset of ZFC, depending on the exact formalism being used), approximate a witness to P (e.g. a bijection between a set of reals and aleph_2) within the starting model, and then apply the forcing machinery to construct a new model of ZFC where P holds. Unlike the diagonalization argument, it's inherently metamathematical.
Re: How many real numbers exist? New proof moves closer to an answer
#53Earlier quoted context omitted.
I don’t like this framing because it implies any piece of pure math becoming useful is just a matter of time, and I see no reason to suspect this is the case.
> I see no reason to suspect this is the case Isn’t history enough reason?
In addition to what the other commenter said, it’s important to understand the scale. Cutting edge research 600 years ago was solving cubic equations. Today, cutting edge research requires potentially years of advanced study to understand the objects involved.
There are more mathematicians alive today than at any point in human history, and their work has become specialized to a tremendous degree. Gone are the days where one mathematician could do substantial new work in a dozen diverse areas.
So, in short, the fact that historically lots of pure math has not found use, the specialized nature of modern research math, the volume of work being out out, and an irreverence for applications(again, this is _pure_ math) leads me to be doubtful that even a moderate amount of modern pure math research will ever be useful in any practical sense.
Re: How many real numbers exist? New proof moves closer to an answer
#54Earlier quoted context omitted.
Why does forcing work? To me it seems flawed (which obviously means I don't understand it fully). For diagonalization argument: 1) Assume every real can be assigned a natural number. 2) Do a bunch of steps that essentially find a new real that differs from any real you have listed from step 1. 3) Conclude that either your steps are flawed, or your initial assumption is wrong 4). Because your steps aren't flawed then…
"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…
> aleph-1, which is the size of the power set of aleph-0
In ZFC, aleph-1 is not the size of the powerset of aleph-0. Instead, aleph-1 is the next larger cardinal number after aleph-0. The size of the powerset of aleph-0 is called continuum or beth-1. In ZFC, we can show that the size of the set of real numbers equals continuum, but it is not possible to relate continuum to a specific aleph-k (though some can be ruled out using Easton's theorem).
Now, the statement that aleph-1 is the size of the powerset of aleph-0 is known as the Continuum Hypothesis (CH) and is independent of the axioms of ZFC. Therefore, your claim that aleph-1 is the size of the powerset of aleph-0 cannot be made in ZFC. The statement can be made in ZFC+CH, but then the question which aleph-k is the size of the real numbers has a straightforward answer: aleph-1.
Re: How many real numbers exist? New proof moves closer to an answer
#55I'm not trying to be flippant, although it may come off that way: why does any of this matter?
Let me venture a possible application, since there are already many responses celebrating pure mathematics.
As far as we understand, the natural numbers are not sufficient for modeling physical phenomena. The reals/complex while immensely useful also occasionally turn out to be “too complicated” to give theoretical guarantees/proofs of models working well. On the practical side, that means that these models/algorithms can’t be guaranteed to not give junk results, while working on the domain of real numbers.
Now, purely speculatively, since the naturals/rationals are aleph0 in size and the reals are aleph2, that means there likely exists a set of numbers of size aleph1 sitting in between the two. What if we could use that set of numbers to construct our physical models? Could we somehow guarantee better behavior… Eg: in quantum mechanics, or field theory, or chaos, etc?! :-)
Re: How many real numbers exist? New proof moves closer to an answer
#56Earlier quoted context omitted.
You are confusing numbers with natural numbers. "A number is a mathematical object used to count, measure, and label."
You can't count reals or complex numbers. Real numbers by definition describe objects with infinite precision. In the real world infinite precision cannot exist therefore real numbers are the limit of an arbitrary precision measuring process not real themselves. Meanwhile natural numbers like "two" most certainly do exist as a quantitative attribute of a set. Show me how you can count with reals.
In practical terms, "counting and measuring" means well-behaved arithmetic operations like addition, subtraction, multiplication, division, roots etc. (and often specific algebraic structures like rings, fields etc.)
Rational and real numbers represent the most intuitive concept of quantity with different cardinality; natural numbers are more basic in theory but a restricted special case in most application (they can only count "countable" objects and they aren't dense); complex numbers compensate unnatural weirdness with important applications; then there are other very number-like niches (e.g. p-adic numbers, quaternions, algebraic extension fields...) and number-based arbitrary constructions (multidimensional vector spaces, matrices and tensors...).
> Show me how you can count with reals.
Like you count with natural numbers, plus many other possibilities because there are more numbers to count with. The same applies to rational numbers (which exist more than real numbers), complex numbers, etc.
Re: How many real numbers exist? New proof moves closer to an answer
#57Earlier 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…
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.stackexchange.com/questions/111440/examples-of-... (and maybe https://mathoverflow.net/questions/11517/computer-algebra-er... )
- https://mathoverflow.net/questions/15444/examples-of-eventua...
(There are over a hundred examples at those questions; I guess this counts as a lot of empirical data that empirical data is not enough! However, sometimes empirical data can be enough for a mathematical proof; for instance if you know somehow that a polynomial f is of degree less than n and you have shown that f(x)=g(x) at n distinct points, you can indeed conclude that f=g and so on — see the "Proofs by Example?" section of the first "Proof Machines" chapter of the book "A=B" available online https://www2.math.upenn.edu/~wilf/AeqB.html .)
Re: How many real numbers exist? New proof moves closer to an answer
#58It depends on what you define as a number. For example reals and complex have different properties from integers. Is there a mathematical reason why both are considered numbers? You can count (with) integers but not with reals. Thus one appears to be a number while the other appears to be a measure.
"Number" is not a mathematical term.
"Natural number", "real number", "complex number" are.
The fact that they all contain the string "number" is completely irrelevant.
Re: How many real numbers exist? New proof moves closer to an answer
#59Earlier 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…
>My question is, why is this empirical data not "good enough" for mathematicians? There are two reasons: firstly, because mathematics is not an empirical discipline (well, unless you're a number theorist...), so it is possible to be certain of mathematical truth, unlike the inherent uncertainty of physical truth; secondly, because every finite bound on the natural numbers may as well be 0 when compared to the numbers…
https://math.stackexchange.com/questions/514/conjectures-tha... was excellent. Thank you.
> 𝑛^17+9 and (𝑛+1)^17+9 are relatively prime
> The first counterexample is 𝑛=8424432925592889329288197322308900672459420460792433
I think this helped me appreciate the difficulty of being satisfied with the Collatz conjecture.
Re: How many real numbers exist? New proof moves closer to an answer
#60Earlier 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…
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 reason things are the way they are (and being able to share those understandings) is the purpose of math. I'd be willing to wager that almost every mathematician would rather have a proof that Collatz holds for all numbers divisible by 17 rather than a definitive yes/no answer to whether it's true or not, because the former would lend much more illumination to the secrets behind the problem, and would lead to new, more interesting mathematical methods and disciplines. The latter would be a fun fact to share at parties.