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

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

#341

Post by chmod775 »

skoodge wrote: post #298
chmod775 wrote: post #270

Earlier quoted context omitted.

You've convinced me that R without all numbers that are not expressible in typical languages is countable at least. Though I'm not convinced that numbers that are not expressible don't exist. I could for instance say "the length of this line", which may very well have a length that is not expressible in a language that uses a finite set of symbols (hah, that's why your expressions are countable, of course!). Consider…

That's an interesting thought experiment, though I'm not sure how using "sounds of the appropriate length" or "lines of proportional length" would get you more than the rational numbers, which are already countable and thus fully captured by any Turing-complete language. To say that there are inexpressible real numbers is to say that there are numbers that are not rational but which can never be practically used or a…

> though I'm not sure how using "sounds of the appropriate length" or "lines of proportional length" would get you more than the rational numbers, which are already countable and thus fully captured by any Turing-complete language

Consider the canonical form of Pi. You can't express it accurately in English, but you can have a distance of Pi length in the physical world, so you could express Pi by that distance.

Now we can refer to Pi in English because it is tied to a concept which we can describe and because we can assign a name to it.

If I picked two arbitrary points in the physical realm, then due to the distribution of real numbers there's a good chance you wouldn't be able to express the distance accurately in a language that uses a finite set of symbols/sounds to express numbers.

I'm convinced such 'numbers' exist however, Pi exists after all, and it happens to be just one we assigned a name to because it is of note.

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

#342

Post by chmod775 »

ummonk wrote: post #326
chmod775 wrote: post #135

Earlier quoted context omitted.

> 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.

"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" I can give you a function that will never output the same natural number on any two real numbers unless those two real numbers are equal. My function outputs the binary value of the U…

The thread already went down that line of argument, you can take this comment as an response to yours: https://news.ycombinator.com/item?id=27851171

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

#343

Post by civilized »

sharpener wrote: post #336
sharpener wrote: post #184

The mileage of others may vary, but it has been my experience that there is no cogent solid proof of uncountability that can withstand concerted critique.[0] Being charitable one might argue that the meanings of terminology had been lost in translation over time and that perhaps Cantor was trying to create non-standard analysis, but then the diagonal argument seems to represent nothing more than the truism that finit…

For the sake of HN archival history... The executive summary of my paper is that provably there are fatal inconsistencies in Cantor's Diagonal Argument (CDA). They take a few forms: (1) Application of basic classical analysis tools reveals that the contradiction sought by CDA does not hold if those tools are used. E.g. Assume we have the table of all unique infinite length binary strings {0,1}* (where * represents th…

Your argument has some gaps. Here are the two basic problems:

1. f(k) is not defined in your notation, only f(k,k) is. My guess is that f(k) is supposed to represent the real number defined by the k^th row of the table.

2. Whatever f(k) is supposed to be in your notation, you have not shown that it has a limit as k approaches infinity, let alone that z = f.

In fact, f(k) cannot possibly have a limit f. If it did, then by the definition of limit, all but finitely many numbers in [0,1] would have to be within epsilon of f, for any epsilon > 0. We could then show that all but finitely many numbers in [0,1] have the exact same first trillion digits of their decimal expansion. Obviously this is absurd.

This is what always happens with Cantor diagonal argument critiques. The critique invariably ignores the logically rigorous argument of the CDA itself, sets up its own new version or extension of the diagonalization construction, and then makes whatever false assumptions are needed for the new construction reach a conclusion contrary to the CDA.

It's like walking into a sturdy bridge with a few pebbles in your pocket, arranging the pebbles into a flimsy tower, knocking the tower over and declaring that you've destroyed the bridge.

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

#344

Post by DangitBobby »

plus wrote: post #302
DangitBobby wrote: post #299

Earlier quoted context omitted.

I took well-defined to mean "if it exists, we know what it is." So really I guess what I want to know is why it matters that we can't actually calculate the mapping.

Any finite number of randomly chosen integers can be sorted in increasing order. The quantities invented for the proposed mapping do not share this property, and thus cannot possibly be integers.

Again, you don't know how to compute the function which creates the mapping, so you don't know the results, so you don't know how to sort the results. I don't follow the logic which claims your inability to compute something is now an intrinsic property to the resulting set. I do understand that sortability is a property intrinsic to integers.

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

#345

Post by ummonk »

chmod775 wrote: post #342
ummonk wrote: post #326

Earlier quoted context omitted.

"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" I can give you a function that will never output the same natural number on any two real numbers unless those two real numbers are equal. My function outputs the binary value of the U…

The thread already went down that line of argument, you can take this comment as an response to yours: https://news.ycombinator.com/item?id=27851171

Upthread you said: "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."

That is false - you're actually unable to name these numbers for which the rule doesn't work. Instead you're just insisting we accept on faith the existence of these unnamable numbers.

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

#346

Post by ummonk »

chmod775 wrote: post #341
skoodge wrote: post #298

Earlier quoted context omitted.

That's an interesting thought experiment, though I'm not sure how using "sounds of the appropriate length" or "lines of proportional length" would get you more than the rational numbers, which are already countable and thus fully captured by any Turing-complete language. To say that there are inexpressible real numbers is to say that there are numbers that are not rational but which can never be practically used or a…

> though I'm not sure how using "sounds of the appropriate length" or "lines of proportional length" would get you more than the rational numbers, which are already countable and thus fully captured by any Turing-complete language Consider the canonical form of Pi. You can't express it accurately in English, but you can have a distance of Pi length in the physical world, so you could express Pi by that distance. Now…

First of all, Pi is not an example of such a number, since it can be defined constructively without requiring physical measurements.

Second, even your proposed mechanism of picking out points would be capable of identification (albeit not computation) in a Turing-complete language. Quite simply, to be able to pick out two points you'd need to create some kind of stable structure that identifies those two points - this could be a metal bar as with the old meter bar, or some other kind of structure / apparatus which has two points in space (at a stable distance) locked in. Crucially though, atomic / quantum configuration of this stabilized apparatus would be encodable in theory, once again providing us with a way to count numbers.

There are of course numbers which are specific to our universe, such as the fine structure constant, that don't have any objective definition without reference to the world. But there are finitely many of these, and they're nameable, just not computable.

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

#347

Post by chmod775 »

ummonk wrote: post #346
chmod775 wrote: post #341

Earlier quoted context omitted.

> though I'm not sure how using "sounds of the appropriate length" or "lines of proportional length" would get you more than the rational numbers, which are already countable and thus fully captured by any Turing-complete language Consider the canonical form of Pi. You can't express it accurately in English, but you can have a distance of Pi length in the physical world, so you could express Pi by that distance. Now…

First of all, Pi is not an example of such a number, since it can be defined constructively without requiring physical measurements. Second, even your proposed mechanism of picking out points would be capable of identification (albeit not computation) in a Turing-complete language. Quite simply, to be able to pick out two points you'd need to create some kind of stable structure that identifies those two points - thi…

> First of all, Pi is not an example of such a number

It is in its canonical form (i.e. as a numbers, not something that represents it), which is what I said.

> Quite simply, to be able to pick out two points you'd need to create some kind of stable structure that identifies those two points.

No I don't. I could pick two atoms floating through space somewhere and say "This distance, right now."

I could also say: "The distance traveled by this particle in a second."

Either number is likely to be irrational. Either number could then only be approximated in binary or whichever system your prefer - unless it just so happens to be expressible using some named irrational constants.

> There are of course numbers which are specific to our universe, such as the fine structure constant, that don't have any objective definition without reference to the world. But there are finitely many of these, and they're nameable.

I guess we'll have to agree to disagree that there are finitely many of these. I don't believe there's even countably many. And if they're not countable, they're not nameable in any common language, since in those names are countable.

It would be quite surprising to me if the universe was nice enough to limit itself to things that are neatly expressible using using the somewhat limited systems humans use, even though it has already shown that it does contain things that will never be expressible using a finite set of symbols, such as numbers or an alphabet.

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

#348

Post by a1369209993 »

rssoconnor wrote: post #325
ummonk wrote: post #313

Earlier quoted context omitted.

Why is it wrong to say that the machine halts?

ZFC+¬Con(ZFC) definitely proves that the machine that searches for an inconsistency in ZFC halts. That is a direct consequnce of ¬Con(ZFC). Keep in mind that by "proves" I'm only saying that there exists a logical deduction from the axioms. I'm not saying that it is true or false. Remember that logical deductions are "truth preserving", but we can only conclude that the conclusions are true if the axioms at the start…

> if the axioms at the start are all true.

Axioms are true by definition; that's what "axiom" means. (They might also be false, but (if and) only if your system is inconsistent.)

> Case (1) that machine doesn't really halt

This is not a state of affairs: all you know at any given time is that the machine hasn't halted yet (after some specific finite number of steps), or that it has aquired some termination-precluding invariant like a infinite loop (which you can detect only strictly less than all of).

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

#349

Post by Tainnor »

kmill wrote: post #91
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…

This surprisingly doesn't mean repeatedly adding 1 will exhaust all natural numbers -- there are models for the natural numbers with elements that can't be reached this way! The ultrafilter construction gives one such model. You take the set of all sequences of natural numbers (n1, n2, n3, ...) then use an ultrafilter to decide which of these sequences are considered to be equal. The usual operations of natural numbe…

> This surprisingly doesn't mean repeatedly adding 1 will exhaust all natural numbers -- there are models for the natural numbers with elements that can't be reached this way!

Isn't that just true for models of a first-order axiomatisation of the natural numbers? As far as I know, if you write down the Peano axioms in second order logic (as is required), you do get the natural numbers uniquely, and applying the successor function over and over again does exhaust all the natural numbers eventually.

The reason why you get non-standard models of arithmetic in first-order logic is the Löwenheim-Skolem theorem, but that isn't available for second-order logic.

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

#350

Post by btilly »

chmod775 wrote: post #347
ummonk wrote: post #346

Earlier quoted context omitted.

First of all, Pi is not an example of such a number, since it can be defined constructively without requiring physical measurements. Second, even your proposed mechanism of picking out points would be capable of identification (albeit not computation) in a Turing-complete language. Quite simply, to be able to pick out two points you'd need to create some kind of stable structure that identifies those two points - thi…

> First of all, Pi is not an example of such a number It is in its canonical form (i.e. as a numbers, not something that represents it), which is what I said. > Quite simply, to be able to pick out two points you'd need to create some kind of stable structure that identifies those two points. No I don't. I could pick two atoms floating through space somewhere and say "This distance, right now." I could also say: "The…

No I don't. I could pick two atoms floating through space somewhere and say "This distance, right now."

I guess that you've never heard of the Heisenberg Uncertainty Principle?

You can say that. But it isn't actually well-defined.

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