Refuting the Anti-Cantor Cranks

dcpo

As an aside, you don't need to phrase the diagonal argument as a contradiction, you can just use it to show that any function from the naturals to the reals must fail to be onto.

dcpo

S0, even if you can define it at each step, which you may not be able to do, will have an infinite number of digits before the decimal point, and so will not be a real number at all.

HallsofIvy

In order that the result be an integer, it has to have only a finite number of digits. Your method does not guarentee that the "number" created by the diagonal argument does not just keep going and have an infinite number of digits.

viraltux

Oh I see... thank you HallsofIvy and dcpo, so the problem is that Integer numbers must be finite and so it goes to the left side of a real number. I understand. Yet, it seems now more a definition problem rather than anything else.

But that raises one interesting question I think, what kind of number would be the number Pi without the decimal dot?? 314159......

Because I can construct this number yet it does not fit the definition of Integer or Real.

jgens

It is worth noting that the "controversy" over Cantor's argument presented in that article actually has nothing to do with controversy about the validity of his argument; the controversy lies in the axioms needed to make the argument work. So the title of that page is kind of a misnomer.

Back when Cantor first presented his argument for the uncountability of the real numbers, the axiomatic framework for mathematics was not well-developed, so mathematicians had varying opinions on whether or not you could talk about things like the collection of all natural numbers, the collection of all real numbers, etc. In the mathematics of today, the axiomatic framework is fairly well-developed, and using the usual axioms of set theory, Cantor's argument is completely valid.

jgens

It does not belong to any conventional number systems and is probably meaningless to discuss. You could try to define specific properties that the infinite sequence of digits might have, but I doubt the theory you could derive would be particularly fruitful.

In any case, this is not relevant for Cantor's argument. The diagonalization argument is usually applied to the real numbers between 0 and 1. This prevents having to worry about the placement of the decimal.

viraltux

Thank you jgens for your answer, but as much as I'd like to believe that I just discovered a new kind of numbers I think other possibilities are more likely.

I am surprised though you say is probably meaningless to discuss that kind of numbers, particularly considering all the arcane concepts mathematicians end up working with, but anyway.

jgens

You are free to feel however you like and if you want to research these sorts of things, then you are more than welcome. I can say with a reasonable amount of certainty, however, that most members of the forum will probably agree with my position on the matter.

Think about it this way: We already do this with decimal representations of real numbers and with p-adics, so presumably you plan on endowing them with some structure which makes them distinct from either of these; otherwise, the theory of your new numbers will be no different than the theory of the real numbers between 0 and 1 essentially. The problem is that this idea is nowhere near novel. I am pretty sure I had a similar idea back in grade school when I was thinking about decimal representations of numbers and I am sure many mathematicians have had similar ideas over the past several hundred years. Don't you think it is likely that if they couldn't work out a structure on infinite strings of digits that is entirely distinct from the theory of real numbers between 0 and 1, it would be a well researched field by now?

viraltux

I am an statistician and this is far away from my field/interest of research.

Well, actually when I said "I think other possibilities are more likely." I was referring exactly to the kind you mention now. I find hard to believe there is no a theory already in place that fully explain this kind of numbers which, obviously, not me nor you are aware of.

dcpo

Well, as it stands these new 'numbers' are not properly defined, so it cannot be said whether or not existing theory fully explains them. My guess is that any way they could be defined would fit esily into existing theory, and likely wouldn't yield anything particularly useful. You could get something similar, for example, by considering the interval [itex][0,1) \subset \mathbb{R}^+[/itex] and taking [itex][0,1) \times[0,1)[/itex] equipped with a kind lexicographic ordering, so [itex](x,y)\leq(x',y')\iff \text{ either } x< x' \text{ or }(x=x' \text{ and } y\leq y')[/itex]. Here the left element of the pair gives you something you might like to describe as an 'infinite digit whole number', and you can interpret the right element as the decimal part. This will be a lower bounded dense total order, so elementarily equivalent to [itex] \mathbb{R}^+[/itex] (as an order at least), and it will have the same cardinality. Completeness too will be inherited from the completeness of [itex]\mathbb{R}[/itex], so this structure will be in many ways similar to [itex]\mathbb{R}^+ [/itex], though I can't say off the top of my head if the two will be order isomorphic. You could likely define arithmetic operations without too much trouble too.

viraltux

I think I see what you mean dcpo but I maybe disagree with them having the same cardinality than [itex]\mathbb{R}[/itex]. Think about this, imaging you have the π number without the decimal dot: 314159.... now you can decide to place the dot in any position, which means that for every number of this kind you have infinite real numbers. Wouldn't this show they have both different cardinalities?

dcpo

Well, the way I've defined it the base set is [itex][0,1)\times[0,1) [/itex], and since [itex][0,1)[/itex] has the same cardinality as [itex]\mathbb{R} [/itex], that the cardinality of [itex][0,1)\times[0,1) [/itex] is the same as that of [itex]\mathbb{R} [/itex] follows from the fact that the cardinality of the product of two infinite cardinals will be the max cardinality of the pair.

ETA: The rule is, when dealing with infinite quantities cardinality gets a bit weird. The first 'weird' result is Cantor's theorem itself, and it only gets worse.

viraltux

That's why I think the definition you give does not quite fit the numbers we're talking about, but anyway, I'm no expert on this so thank you for your explanations and patience!!!

jgens

Nope. They will have the same cardinality.

Explanation: It is easy to show that all sequences consisting of just 0s and 1s have the same cardinality as the real numbers. Since these are just a subset of your numbers, it follows that the cardinality of your numbers is at least as great as the cardinality of the reals. On the other hand, clearly each one of your numbers corresponds in a natural way to a unique real number (just put the decimal before the first number). This proves that the cardinality of the reals is equal to the cardinality of your numbers.

viraltux

Well in fact it is not a subset but the same set, the fact that I used base 10 instead base 2 to write the numbers does no make one subset of the other.

Very interesting that all sequences of 0's and 1's have the same cardinality than Real numbers, yet, here we are not talking about all sequences but only about the infinite ones, any finite sequence of 0's and 1's is not in the set. Could this alone change it's cardinality to be smaller than the Reals? After all the correspondence you do with the Real numbers is one among the infinite you can do; you could place the dot after the second number, the third...

micromass

Numbers which are infinite to the left are certainly useful, but in a surprising way. Check out the p-adic numbers: http://en.wikipedia.org/wiki/P-adic_number

It gives rise to surprising identities such as

[tex]...999999999 = -1[/tex]

jgens

I actually mentioned the p-adics in post 59 of this thread :)

The point of my comment was that if the poster intends to give his infinite sequences of digits some sort of meaning apart from the p-adics or reals, then he/she will have a difficult time doing so. A lot of the meaningful ways of dealing with infinite sequences of digits is captured by the real numbers and by the p-adics, so IMO it would be rather difficult to find an entirely new structure on infinite sequences of digits that proves to be particularly fruitful.