I'm not trying to be flippant, although it may come off that way: why does any of this matter?
Because this advances math. Same could be said for lambda-calculus and yet here we are with FP.
How many real numbers exist? New proof moves closer to an answer
21–30 of 359 posts
Re: How many real numbers exist? New proof moves closer to an answer
#22It 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.
You are confusing numbers with natural numbers.
"A number is a mathematical object used to count, measure, and label."
Re: How many real numbers exist? New proof moves closer to an answer
#23Earlier quoted context omitted.
> it doesn’t matter It doesn't matter yet . Applications of pure math happen downstream decades or centuries later. It's hard to predict the impact.
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?
Re: How many real numbers exist? New proof moves closer to an answer
#24I don’t get for Cantor’s diagonalization proof, why do we need to use the diagonal digits to form the new number? Would the proof work the same if we instead used the first digit of every number in the list?
You need the diagonal because you are making a new number in (0,1) that can't be in the list by just choosing the nth digit of said number different than the nth digit of the diagonal.
Re: How many real numbers exist? New proof moves closer to an answer
#25It 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.
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.
Re: How many real numbers exist? New proof moves closer to an answer
#26> "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.
So this isn't exactly right, although I suppose the argument for integers isn't exactly wrong. When we get to the rational numbers however, they are countable but there are an infinite number of rational numbers between any two rationals in the typical way of thinking about "between" numbers. Based on your argument above, this would mean that rationals are uncountable.
A better way to think about this is that the set of natural numbers is the first (and smallest) infinite set you can construct. This is the set of all counting numbers so we call it countable.
We then say two sets are the same size (cardinality) if you can create a one-to-one mapping between the two that covers both sets (a bijection). This way you exactly pair one element of one set with one element of another. You can do this with the natural numbers and integers by just alternating positive and negative (so 0 -> 0, 1 -> 1, 2 -> -1, 3 -> 2, etc.). All the sets that you can do this sort of mapping with the natural numbers are considered countable.
You can't do this sort of mapping from the natural numbers to the reals (see cantor's diagonalization argument) so since you can't "count" the reals, the set is "uncountable."
Re: How many real numbers exist? New proof moves closer to an answer
#27I'm not trying to be flippant, although it may come off that way: why does any of this matter?
We know that the cardinality of the natural numbers is less than the cardinality of the real numbers. The Continuum Hypothesis, which is a long unsolved problem, states that there are no sets with cardinality between the two. The posted article states that this new result strengthens the case against the hypothesis, that is that’s it’s probably false. All of this is nuanced but is important to mathematics and philoso…
Why does “the set containing the natural numbers and a sandwich” not have cardinality between the two?
Re: How many real numbers exist? New proof moves closer to an answer
#28Great 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 your initial assumption (that every real can map to a natural number) is flawed.
That all makes sense to me. Forcing seems broken though: 1) Capture of list/set of all real numbers 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 your initial assumption (that you can produce a list of every real number) is flawed.
Now you're left in an odd situation. Didn't you just conclude that it's impossible to have a set of all real numbers? Which isn't what you're trying to prove at all.
Re: How many real numbers exist? New proof moves closer to an answer
#29Earlier quoted context omitted.
> it doesn’t matter It doesn't matter yet . Applications of pure math happen downstream decades or centuries later. It's hard to predict the impact.
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 think it's more about being open minded to strange uses. Maybe 3% of strange pure math ends up being used for something bigger? But since nobody knows what it'll be, it's good to be curious about all sorts of nonsense. Riemannian geometry was OBVIOUSLY absurd useless pure math sophistry, but then it turned out we lived in it. I think a lot of the work on modular forms and elliptic curves has come across that way too, despite the crypto intersection. The general idea is that, even if it's been a century or two, it's too soon to tell if something is useless, that's solid. And formalization of infinity has been useful in some practical exploration of proof systems used by humans. It might be the future of programming, who knows?
Re: How many real numbers exist? New proof moves closer to an answer
#30I'm not trying to be flippant, although it may come off that way: why does any of this matter?
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 problem, but endlessly fascinates many for its renowned difficulty.
(2) In mathematics, you often stumble upon problems that are equivalent to other problems. If X is difficult to solve so you give up and go work on Y, only to find out that Y as a problem is basically the same to X, or they have some other intimate relationship, and it makes X that much more tantalizing. A common example for computer scientists is the whole class of NP complete problems[1] for which we don't know if there exists a polytime algorithm to decide. Every time we discover a new problem that lives in the realm of NP-complete, the P vs NP problem gets a little more tantalizing.