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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

#171

Post by bryan0 »

rssoconnor wrote: post #102
lupire wrote: post #90

Earlier quoted context omitted.

why is that "unsound"? what's wrong with an unwritable natural? Almost all reals are unwritable.

Mostly because it implies that some Turing machines halt that actually do not halt. That is unless you are willing to accept that a Turing machine can halt in some number of steps that is beyond any number that can be written. And I don't mean can't be written in the sense that we don't have enough paper. Just cannot be written in principle at all by our notation for numbers.

But isn’t this what the busy beaver numbers are? Numbers that we cannot write for arbitrary n but they do exist?

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

#172

Post by question000 »

dang wrote: post #164
question000 wrote: post #159

Earlier quoted context omitted.

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

Seriously fuck off, everyday I read the same bullshit like this and it never gets removed. Read your own comments! This is laughable.

Some things are obvious, and I'm allowed to say that to these self-proclaimed mathematicians.

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

#173

Post by munk-a »

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.

Do you mean that given finite resources (like time and matter) we would be unable to express the number? Or are you talking about a sort of recursive thing where writing down a number which is a sum of all natural numbers up to and including itself is impossible since that number is bigger each time you look at it?

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

#174

Post by daxfohl »

A related thing that occurred to me the other day: there's some number in [0, 1] that encodes every state of every possible Turing machine (the number of Turing machines is countable, the duration of its run is countable, and the states at each step are countable, so you can diagonalize that and make a real number out of it). I'm pretty sure you can take that further and show that all possible mathematical proofs, including the Godel unsolvable ones (I think...they're just proofs that are infinitely long but countably so), can also be encoded in a single real number. That makes it feel like the reals are pretty big.

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

#175

Post by red_trumpet »

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.

> You can map all the real numbers to the interval 0 Even easier use a well-established function such as arctan to map the reals bijectively to the interval -π/2 < x < π/2, and then scale and shift the interval to 0 < x < 1.

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

#176

Post by scrubs »

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…

@pavpanchekha: thank you indeed for this great post and great references. I took ML and ran into CH, Ultrafilters but never really got my head around it. Reading with interest!

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

#177

Post by red_trumpet »

dwohnitmok wrote: post #121

Man there's a lot of juicy stuff in this article (Woodin's Ultimate L program gets briefly alluded to at the end of the article). I just want to point out, because the HN crowd seems to generally not be mathematical Platonists, that this entire article is implicitly assuming a Platonist philosophical foundation. This may cause confusion for lay readers who are not mathematical Platonists. In other words the article a…

Thanks for that perspective! Searching around I found this article https://plato.stanford.edu/entries/platonism-mathematics/#Tr... which even distinguishes mathematical Platonism and truth-value realism. Interesting stuff!

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

#178

Post by kachnuv_ocasek »

dwohnitmok wrote: post #121

Man there's a lot of juicy stuff in this article (Woodin's Ultimate L program gets briefly alluded to at the end of the article). I just want to point out, because the HN crowd seems to generally not be mathematical Platonists, that this entire article is implicitly assuming a Platonist philosophical foundation. This may cause confusion for lay readers who are not mathematical Platonists. In other words the article a…

Thanks for the clarification. I was indeed a bit confused by some of the implicit assumptions as an intuitionist myself.

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

#179

Post by gowld »

TuringTest wrote: post #100
HelloNurse wrote: post #56

Earlier quoted context omitted.

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…

> The same applies to rational numbers (which exist more than real numbers) Wait, did you say there are more rationals than reals? Isn't that the other way around? I don't know if that's a slip of the tongue or I'm missing something in my recall of basic math lessons

They are "more existent" ("less fictional"), not "more in cardinality".

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

#180

Post by gowld »

croes wrote: post #94
morpheos137 wrote: post #25

Earlier quoted context omitted.

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.

I don't need to, I can measure and label with real numbers. That's enough for being a number.

You'll only ever use 0% of the real numbers for that. Unless you say that you count with "the reals", You don't need and won't use "the reals" for measurement and labelling, 100% of which are indescribable.
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