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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

#131

Post by [deleted] »

finnh wrote: post #120
BoiledCabbage wrote: post #45

Earlier quoted context omitted.

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.

[deleted]

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

#132

Post by gooberwonder »

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.

45,000,000,001?

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

#133

Post by db48x »

Hamlet42 wrote: post #106
jaywalk wrote: post #8

I'm not trying to be flippant, although it may come off that way: why does any of this matter?

Some math problems seem uselees when first encountered until they become useful some dacades later. An example of this is knot theory and medicine[1] [1] https://science.sciencemag.org/content/255/5043/403

Also certain of the more recondite aspects of number theory that deal with prime numbers: completely useless right up until RSA was invented; now we use them to protect every webpage we view, every email we send, and every financial transaction we make.

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

#134

Post by howderek »

pelorat wrote: post #129

The existence of addition implies that the answer is infinity. Too simple an explanation for mathematicians obviously.

They know it is infinite. They want to understand the cardinality. Any two sets (including infinite sets) A, B have the same cardinality if there exists a bijection from A|->B

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

#135

Post by chmod775 »

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…

> there are only a countable number of real numbers

Then you should be able to come up with a function that assigns a natural number uniquely to each real number.

Of course if you tried that I could immediately name you a real number, or a pair of them, for which your rule doesn't work.

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

#136

Post by dexwiz »

I'm no mathematician, but I have always found it strange infinities are talked about as physical states (towers of tall towers, etc), and not functions.

Integer number counting is essentially a successor function; take N, add 1, output N+1. One input, one output.

Real number counting is a bit more loose. To find the numbers between .1 and .2, you find all fractionals of a given size, normally 1/10. So we get .10, .11, .12, etc. This function gives more outputs than inputs, thus an increase in cardinality. Forcing just seems like changing the function to increase the cardinality.

In my mind, all infinities are equal, it's how you generate them that matters.

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

#137

Post by IncRnd »

dexwiz wrote: post #136

I'm no mathematician, but I have always found it strange infinities are talked about as physical states (towers of tall towers, etc), and not functions. Integer number counting is essentially a successor function; take N, add 1, output N+1. One input, one output. Real number counting is a bit more loose. To find the numbers between .1 and .2, you find all fractionals of a given size, normally 1/10. So we get .10, .11…

I'm not a mathematician, either, but I believe your comment shows why not all infinities are equal.

When you attempt to enumerate .1, .2, .3, .4, etc. you get an infinite count of numbers. However, when you perform that same operation on .05, .15, .2, .25, and so on, you will get "twice as many numbers" as the first time around.

In this way, the second infinite sequence has "twice as many numbers" as the first infinite sequence.

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

#139

Post by LudwigNagasena »

btilly wrote: post #118
scapp wrote: post #105

Earlier quoted context omitted.

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…

What’s the problem with it being “artificial”? Is your problem purely linguistic? You just dislike the word “exists” being used in this context?

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

#140

Post by TameAntelope »

mmmBacon wrote: post #109

Maybe I misunderstood the article but if the set of real numbers is finite then it should be countable. But I can easily prove that the set of real numbers or any subset of real numbers is not countable. Been a really long time since I’ve thought about this but wondering what I’m missing.

How can you easily prove that the set of real numbers is not countable? I don't think it's as easy as you claim, but I'm kind of a dummy so it's quite probably I'm wrong.
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