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

quantamagazine.org

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

#161

Post by myhf »

BoiledCabbage wrote: post #28
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…

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…

> 2) Do a bunch of steps

It's not just a bunch of steps. It's an infinite number of steps. It requires the axiom of choice.

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

#162

Post by question000 »

_2d30 wrote: post #114
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.

Yeah of course I didn't read the article. It's literally was the same as an article saying "Cows are green! If you're smart enough it becomes math!" I can't feed the clickbait machine.

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

#163

Post by phkahler »

You can map all the real numbers to the interval 0 0.0 => 0

0.1 => 1

0.2 => 2

...

0.14159 => 95141

The argument against this is that there are real numbers with an infinite number of digits, which will not have a specific integer associated with them. Or do they? Can there be an "infinitely large integer?" or are there just infinitely many of them? I think this question gets to the core of the problem.

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

#164

Post by dang »

question000 wrote: post #159
dang wrote: post #123

Earlier quoted context omitted.

" Please don't post shallow dismissals, especially of other people's work. A good critical comment teaches us something. " https://news.ycombinator.com/newsguidelines.html

I can't believe you called this shallow and didn't even include an explanation as to why, ironic. There's nothing shallow about this at all, it goes to the heart of this fake intellectualism on this site.

It's shallow because (a) contentless denunciations of "soft science" are cliché; (b) reducing serious mathematical work to "obvious" is the worst sort of dismissal.

Please don't post like this to HN. We're trying for higher-quality discussion.

https://news.ycombinator.com/newsguidelines.html

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

#165

Post by question000 »

phkahler wrote: post #163

You can map all the real numbers to the interval 0 0.0 => 0 0.1 => 1 0.2 => 2 ... 0.14159 => 95141 The argument against this is that there are real numbers with an infinite number of digits, which will not have a specific integer associated with them. Or do they? Can there be an "infinitely large integer?" or are there just infinitely many of them? I think this question gets to the core of the problem.

"Can there be an infinitely large integer"

No there can't there's always a larger integer. Anyone who claims otherwise is just making stuff up. It's just mind blowing to me that anyone think otherwise, it's like asking what the final digit of pi is, you'd get laughed out of any college class for insisting there might be one, rightly so.

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

#166

Post by victorbojica »

phkahler wrote: post #163

You can map all the real numbers to the interval 0 0.0 => 0 0.1 => 1 0.2 => 2 ... 0.14159 => 95141 The argument against this is that there are real numbers with an infinite number of digits, which will not have a specific integer associated with them. Or do they? Can there be an "infinitely large integer?" or are there just infinitely many of them? I think this question gets to the core of the problem.

But if you assign them randomly, then it should work

1->5.85916

2->8.7599

...

Or even randomize the order for the integers

5->7.52256951

77->848.455

...

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

#167

Post by bmitc »

erichahn wrote: post #31
bmitc wrote: post #23

Earlier quoted context omitted.

> I see no reason to suspect this is the case Isn’t history enough reason?

No history actually tells you that only 1% of produced maths is useful. You only see the good stuff not the ugly stuff.

I don't know why people are throwing around percentages so much in these replies. Percentage of what? Plus, the body of work of mathematics continues to grow all the time.

The core of the original comment that started this chain was:

> Applications of pure math happen downstream decades or centuries later.

My argument was that history shows that a surprising amount of mathematics does eventually trickle down into applications. I don't think anyone is arguing all of mathematics eventually sees an application.

Again, my point was that history does indeed seem to show that "applications of pure math happen downstream decades or centuries later". As the original commenter said, it's hard to predict what will be the next thing to be applied.

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

#168

Post by gowld »

phkahler wrote: post #163

You can map all the real numbers to the interval 0 0.0 => 0 0.1 => 1 0.2 => 2 ... 0.14159 => 95141 The argument against this is that there are real numbers with an infinite number of digits, which will not have a specific integer associated with them. Or do they? Can there be an "infinitely large integer?" or are there just infinitely many of them? I think this question gets to the core of the problem.

> Can there be an "infinitely large integer?"

What could that even mean? Aren't integers all defined as Succ^n(x) for a finite n and a base x?

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

#169

Post by Layke1123 »

jerf wrote: post #38
BoiledCabbage wrote: post #28

Earlier 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…

Cantor's diagonalization proof was never a proof to begin with. You can't construct a number that isn't on an infinite list of all numbers because you can't list all the numbers, otherwise you'd eventually find your number. It's honestly kind of ridiculous that this proof hasn't been subjected to more rigor than just handwaving away the problems.

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

#170

Post by Sharlin »

keymone wrote: post #158
Sharlin wrote: post #128

Earlier quoted context omitted.

No, what the article is talking about is the question whether or not the cardinality of real numbers is the smallest uncountable infinity or some other, larger uncountable infinity. The only countable infinity is aleph-0, the cardinality of natural numbers, and Cantor showed that aleph-0 is too small to hold all reals. So reals must be uncountable, but there is an infinite hierarchy of uncountable infinities, and it…

unrelated: how do we know there are no alephs between 0 and 1?

If you assume Axiom of Choice, you can define cardinalities in terms of ordinal numbers (an extension of naturals that generalizes the notion of "counting", or "indexing"). And ordinal numbers are well-ordered, ie. every element has a unique "successor" element.
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