Refuting the Anti-Cantor Cranks

[lugita15]

Cantor forms this very list that can't be formed and then a step *later* brings us to a contradiction?

No, Cantor is saying "suppose there is a complete list of real numbers". Then he is using the diagonal construction to show that there is a real number not on the list. Thus, he is concluding that the assumption that there is a complete list of real numbers is wrong.

 

[Antiphon]

No, Cantor is saying "suppose there is a complete list of real numbers". Then he is using the diagonal construction to show that there is a real number not on the list. Thus, he is concluding that the assumption that there is a complete list of real numbers is wrong.

I'm not trying to split the hair that finely. Nothing I'm saying changes if you say "suppose there is a complete list of real numbers". I think we're ok here as long as everyone agrees that the phrase in quotes is the obvious contradiction and that the diagonal construction of a new real is unnecessary.

 

[lugita15]

I'm not trying to split the hair that finely. Nothing I'm saying changes if you say "suppose there is a complete list of real numbers". I think we're ok here as long as everyone agrees that the phrase in quotes is the obvious contradiction and that the diagonal construction of a new real is unnecessary.

No, it is not such an obvious fact that you get a contradiction if you assume that there exists a complete list of real numbers. It takes a nontrivial proof like Cantor's.

 

[DonAntonio]

I was trying to lead the horse to water. The infinite limit I presented is an algorithmic construction that in principle is no different from the way one might sum an infinite series. The point of it was to show that one simply cannot arrive at Cantor's result without starting off with unbounded sets as you pointed out. That you cannot approach the result the way you might show that .999... is the same as 1.0. But let's move on because this doesn't invalidate the Cantor result.

A couple of posts back I did exactly give you as requested "any list of real numbers". It was in fact the first three entries of an infinite countable list exactly as you specified, and between 0 and 1 to keep things from getting messy.

But that's not good enough apparently. So it looks like you need to tighten up your specs before we can go to the next step.

Is your objection to my list the fact that it excluded 0.05? Because perhaps that was the very Cantor number that would be added by diagonolization.

Next you're going to say that I'm not getting it, that the first entry corresponding to natural number 1 is all zeros (say) and the second entry which corresponds to natural number 2 is "the next real number after 0.000..." so that I haven't skipped any.

Which brings us full circle to my very first post. It seems the point of the constructed proof is to arrive at the contradiction that by adding one more real, I wasn't able to enumerate them all after all. But one doesn't need to add the diagonalized real to arrive at this conclusion. You arrived at it YOURSELF when you kept saying that my finite, truncated, first few entries of the countably infinite list I provided aren't good enough because why? Because they didn't contain all the real numbers!

That's like saying sin(x) does not equal x+x^3/6+... because I didn't start the series with all the terms already there.

Check this: as already noted, by "list" one has to understand "elements of an infinite countable set".

Next: nobody adds anything to anything. Upon getting any list, one can construct a number which is not in that specific list.

Next: perhaps you think you've arrived to a great insight when you use exclamation marks, but I honestly can't see it. Your example

of a finite list with three elements shows nothing as I can construct easily a number not in it without even using the diagonal trick.

Next: I don't care, nor any other mathematician would, what's the first element in such a list. It can be 0, 1, 0.5 or whatever. Anyway, after

you're done with your list, I can always construct a number not in it.

DonAntonio

 

[DonAntonio]

Let me further add (because I don't want to fly off on a tangent here) that I understand the result but disagree so far with the soundness of the diagonalization proof as its been presented.

I get it if you say that the terms in a series like sin(x) are given by x^n/n! and clearly you can't "shoehorn in" a new term between the nth and n+1th. Cantor does exactly this, shoehorns in another term thus showing there is not the one to one association as in the series for sin(x). I get it.

The gist of what I'm trying to say (as a non-mathematician) is this: why is it that I can enumerate the first 3 natural numbers in an ordered set of them beginning with 0, but the same cannot be done with the reals beyond the first entry?

This is what Cantor's theorem states: the set of real numbers is infinite yet uncountable, i.e. it cannot be ennumerated. That's why.

If Cantor or anyone wants to start there and work toward the idea these sets are fundamentally different in nature, there'd be no Cantor Cranks.

Cantor forms this very list that can't be formed and then a step *later* brings us to a contradiction?

Either you're addressing something else or you're completely lost in this matter: neither Cantor nor anyone else trying to prove this theorem

"forms" any list at all. We assume such a list exists AND THEN we prove that there's always a real number not belonging to the list.

Sorry but that's a little too much like Morpheous jumping from building to building. Telling me to "free my mind" isn't going to cut it.

I would never tell you to free your mind, but I' definitely tell you to learn some mathematics. If you want to understand this stuff, that is.

DonAntonio

 

[SteveL27]

Fascinating to see the thread degeneration here. Any discussion of anti-Cantor cranks draws anti-Cantor cranks.

Purely from a behavioral point of view, the Cantor deniers and the Cantor denier refutors seem equally obsessive. The fact that one group is mathematically correct and the other not, is irrelevant. Because after all, most people manage to avoid these discussions altogether. And now you can see why. They always, always, always end up in exactly the same place.

What I used to like about sci.math on Usenet was that at least there, you could toss in gratuitous personal insults. Here you can't do that. So it's much less fun to tease and torment the deniers.

 

[mbs]

Let me further add (because I don't want to fly off on a tangent here) that I understand the result but disagree so far with the soundness of the diagonalization proof as its been presented.

I get it if you say that the terms in a series like sin(x) are given by x^n/n! and clearly you can't "shoehorn in" a new term between the nth and n+1th. Cantor does exactly this, shoehorns in another term thus showing there is not the one to one association as in the series for sin(x). I get it.

He doesn't do that at all. Plus, you can replace any term in that series and it will still converge to some real number. It just will no longer be sin(x).

Similarly any decimal expansion 0.a1a2a3... where a1, a2, a3,... are digits 0-9 is short for the series a1*10^-1 + a2*10^-2 + ... which converges to a real number between zero and one no matter what the digits are chosen to be.

The gist of what I'm trying to say (as a non-mathematician) is this: why is it that I can enumerate the first 3 natural numbers in an ordered set of them beginning with 0, but the same cannot be done with the reals beyond the first entry? If Cantor or anyone wants to start there and work toward the idea these sets are fundamentally different in nature, there'd be no Cantor Cranks.

Because well-ordered-ness is a completely separate issue from cardinality. The rational numbers are also not well ordered, yet the rational numbers can be put into 1-1 correspondence with the natural numbers and thus the Cantor diagonal argument fails. The existence of irrational numbers with non-terminating decimal expansions is necessary for the Cantor diagonal argument to work.

Cantor forms this very list that can't be formed and then a step *later* brings us to a contradiction?

Sorry but that's a little too much like Morpheous jumping from building to building. Telling me to "free my mind" isn't going to cut it.

The list is hypothetical. That's the essence of proof by contradiction. Do you have a problem with other proofs by contradiction? What about the proof that the square root of two is irrational? It's normal to assume there are some natural numbers n and m such that n^2 / m^2 = 2 and then show that the existence of such numbers leads to an absurdity. Do you also question the proof that the square root of 2 is irrational? If not, then what is fundamentally different about Cantor's diagonal argument?

 

[DonAntonio]

Fascinating to see the thread degeneration here. Any discussion of anti-Cantor cranks draws anti-Cantor cranks.

Purely from a behavioral point of view, the Cantor deniers and the Cantor denier refutors seem equally obsessive. The fact that one group is mathematically correct and the other not, is irrelevant. Because after all, most people manage to avoid these discussions altogether. And now you can see why. They always, always, always end up in exactly the same place.

What I used to like about sci.math on Usenet was that at least there, you could toss in gratuitous personal insults. Here you can't do that. So it's much less fun to tease and torment the deniers.

Indeed. For the time being the thread hasn't degenerated that much, imo. I won't get into any crank-bashing or

crank-educating rant here. For that we have sci.math. So far, though, Antiphon is not a crank but someone with

some doubts and some rather deserted areas in his/her mathematical education...for

now. If and when he, or anyone else, slip into crankhood I, for one, shall bail out of the thread.

DonAntonio

 

[mbs]

Indeed. For the time being the thread hasn't degenerated that much, imo. I won't get into any crank-bashing or

crank-educating rant here. For that we have sci.math. So far, though, Antiphon is not a crank but someone with

some doubts and some rather deserted areas in his/her mathematical education...for

now. If and when he, or anyone else, slip into crankhood I, for one, shall bail out of the thread.

DonAntonio

I agree Antiphon does not appear to be a "crank". I certainly hope he's not scared off by the implications and condescension being tossed around in this thread. Also if someone is honestly questioning at least give them credit for having the curiosity to want to learn rather than knocking them on their lack of current education.

 

[SteveL27]

I agree Antiphon does not appear to be a "crank". I certainly hope he's not scared off by the implications and condescension being tossed around in this thread. Also if someone is honestly questioning at least give them credit for having the curiosity to want to learn rather than knocking them on their lack of current education.

Sorry. I officially apologize for implying that doubters are the same as cranks. Antiphon, come back.

 

[Antiphon]

I'm still here. Been traveling coast to coast.

I'm quite sure I'll never become a crank on this issue. As a non-mathematician though I can see how someone would become one. Maybe if this issue can be zoomed in on, many future cranks-in-waiting might be saved. Please lead me down the reasoning path.

I'll do my best to explain using the language I have available.

Proofs by contradiction make sense. You make an assertion or assumption that may or may not be true, then you follow up with some valid deductions based on the assumption. If your subsequent deductions are valid but you arrive at a contradiction or falsehood, then the original assumption was false. This is proof by contradiction as I understand it.

For example (and I'm making this up on the fly) let's suppose that division by zero were legitimate arithmetic. I can probably form some simple algebraic expessions which would result in a statement like 1=2. Nobody should have a problem with such a proof.

But if you start a proof with 1=2 and then proceed to do valid algebra with it, the contradiction doesn't arise from the proof but is built in at the beginning.

I can't speak for any Anti-Cantor cranks but for me this is an issue.

A few posts back MBS says that the proof of the irrationality of sqrt(2) can begin by assuming the existence of two integers m and n such that n^2/m^2=2. You then perform valid reasoning on this and arrive at absurd conclusions. That's great. I don't have trouble with that because the expression above is legitimate algebra, it just so happens that no two integers will satisfy it.

But I'd bet it's not legitimate logic to start with the absurd result and reason your way backward to the expression above.

What am I missing?

 

[chiro]

I'm still here. Been traveling coast to coast.

I'm quite sure I'll never become a crank on this issue. As a non-mathematician though I can see how someone would become one. Maybe if this issue can be zoomed in on, many future cranks-in-waiting might be saved. Please lead me down the reasoning path.

I'll do my best to explain using the language I have available.

Proofs by contradiction make sense. You make an assertion or assumption that may or may not be true, then you follow up with some valid deductions based on the assumption. If your subsequent deductions are valid but you arrive at a contradiction or falsehood, then the original assumption was false. This is proof by contradiction as I understand it.

For example (and I'm making this up on the fly) let's suppose that division by zero were legitimate arithmetic. I can probably form some simple algebraic expessions which would result in a statement like 1=2. Nobody should have a problem with such a proof.

But if you start a proof with 1=2 and then proceed to do valid algebra with it, the contradiction doesn't arise from the proof but is built in at the beginning.

I can't speak for any Anti-Cantor cranks but for me this is an issue.

A few posts back MBS says that the proof of the irrationality of sqrt(2) can begin by assuming the existence of two integers m and n such that n^2/m^2=2. You then perform valid reasoning on this and arrive at absurd conclusions. That's great. I don't have trouble with that because the expression above is legitimate algebra, it just so happens that no two integers will satisfy it.

But I'd bet it's not legitimate logic to start with the absurd result and reason your way backward to the expression above.

What am I missing?

That is the best way to prove something IMO because you in doing so, you are accepting the premise that what you are trying to disprove is in fact true and from this a contradiction (if it's not true under some axiomatic system) is found and thus you have shown it's false.

This is important not just mathematically but also psychologically because when most people start off disproving something, in the back of their mind they assume that what they are proving is undoubtedly true which ends up screwing up their analysis, proof and way of thinking whereas the above method psychologically says "OK this is what you said, let's go along with this and see what happens" which is a much better approach because mentally you are saying "I'm going to disregard my own prejudices for the moment and I'm going to assume that you are right". It's very subtle, but it's so important as a logical tool and I'm afraid it's not used as much as it could (and should) be.

You definitely have the right approach and mindset for analyzing things not only mathematically, but in general situations overall.

 

[jgens]

But I'd bet it's not legitimate logic to start with the absurd result and reason your way backward to the expression above.

Whenever you do a proof by contradiction you are starting with an absurd result (by definition), so I don't really understand your objection here.

What am I missing?

Assuming that the real numbers are countable is exactly akin to assuming that $\sqrt{2} = \frac{p}{q}$ where $p,q \in \mathbb{Z}$ are relatively prime. Constructing a real number not on the list is exactly like showing that 2 divides both $p$ and $q$. The type of reasoning is identical.

 

[viraltux]

Is this a closed subject in the mathematical world? I ask because I have found this

http://en.wikipedia.org/wiki/Controversy_over_Cantor's_theory

And sure everyone is entitled to have an opinion but I'd like to know if experts logicians have reached an agreement on this.

 

[DonAntonio]

Is this a closed subject in the mathematical world? I ask because I have found this

http://en.wikipedia.org/wiki/Controversy_over_Cantor's_theory

And sure everyone is entitled to have an opinion but I'd like to know if experts logicians have reached an agreement on this.

This is one of the instances where Wiki, a very good source of immediate though generally not deep and sometimes

even unreliable knowledge, can mislead. This article begins with the following:

"In mathematical logic, the theory of infinite sets was first developed by Georg Cantor. Although this work has found some acceptance

in the mathematics community, it has been criticized in several areas by mathematicians and philosophers."

The words "some acceptance" are unduly and unjustly misleading: infinite sets, in this or that acception, are widel accepted by an

overwhelming majority of mathematicians. Period.

Now, the controversy exists within very narrow and, if may I add, unimportant frames and individuals, and it surely isn't something

that, as far as we know right now, would affect in some dramatic way neither the development of most of mathematics nor

most of its applications to other sciences, technology and/or the "real" world, whatever that is.

Back again with cranks: these persons are characterised by an inner and utterly unjustified certainty that they are right and ALL

the others are wrong, even when they are NOT mathematicians (99% of the cases) and the others are.

DonAntonio

 

[viraltux]

OK, I just thought up this counter-argument

Let's imaging it does exist a list of all real numbers such that they have only zeros in the decimal part. The list will look something like:

S
1 xxxxxxxxxxxxxxxxxxx.00000000000000000
2 yyyyyyyyyyyyyyyyyyy.00000000000000000
3 xyxyxyxyxyxyxyxyxyx.00000000000000000
4 yxyxyxyxyxyxyxyxyxy.00000000000000000

Then we try to construct S0 using a different digit from the Diagonal

yxyy... .00000000000000000

Then S0 cannot possibly be in that list, therefore that list cannot exist... but since the list have only zeros in the decimal places that list is equivalent to the Natural numbers which we know we can count, therefore the diagonal argument makes no sense.

What is it that I did wrong?

 

[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]

OK, I just thought up this counter-argument

Let's imaging it does exist a list of all real numbers such that they have only zeros in the decimal part. The list will look something like:

S
1 xxxxxxxxxxxxxxxxxxx.00000000000000000
2 yyyyyyyyyyyyyyyyyyy.00000000000000000
3 xyxyxyxyxyxyxyxyxyx.00000000000000000
4 yxyxyxyxyxyxyxyxyxy.00000000000000000

Then we try to construct S0 using a different digit from the Diagonal

yxyy... .00000000000000000

Then S0 cannot possibly be in that list, therefore that list cannot exist... but since the list have only zeros in the decimal places that list is equivalent to the Natural numbers which we know we can count, therefore the diagonal argument makes no sense.

What is it that I did wrong?

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]

OK, I just thought up this counter-argument

Let's imaging it does exist a list of all real numbers such that they have only zeros in the decimal part. The list will look something like:

S
1 xxxxxxxxxxxxxxxxxxx.00000000000000000
2 yyyyyyyyyyyyyyyyyyy.00000000000000000
3 xyxyxyxyxyxyxyxyxyx.00000000000000000
4 yxyxyxyxyxyxyxyxyxy.00000000000000000

Then we try to construct S0 using a different digit from the Diagonal

yxyy... .00000000000000000

Then S0 cannot possibly be in that list, therefore that list cannot exist... but since the list have only zeros in the decimal places that list is equivalent to the Natural numbers which we know we can count, therefore the diagonal argument makes no sense.

What is it that I did wrong?

In order that the result be an integer, it has to have only a finite number of digits. Your method does not guarantee 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]

Is this a closed subject in the mathematical world? I ask because I have found this

http://en.wikipedia.org/wiki/Controversy_over_Cantor's_theory

And sure everyone is entitled to have an opinion but I'd like to know if experts logicians have reached an agreement on this.

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]

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.

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]

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.

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]

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.

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.

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.

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]

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.

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

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?

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]

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.

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 $[0,1) \subset \mathbb{R}^+$ and taking $[0,1) \times[0,1)$ equipped with a kind lexicographic ordering, so $(x,y)\leq(x',y')\iff \text{ either } x< x' \text{ or }(x=x' \text{ and } y\leq y')$. 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 $\mathbb{R}^+$ (as an order at least), and it will have the same cardinality. Completeness too will be inherited from the completeness of $\mathbb{R}$, so this structure will be in many ways similar to $\mathbb{R}^+$, 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]

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 $[0,1) \subset \mathbb{R}^+$ and taking $[0,1) \times[0,1)$ equipped with a kind lexicographic ordering, so $(x,y)\leq(x',y')\iff \text{ either } x< x' \text{ or }(x=x' \text{ and } y\leq y')$. 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 $\mathbb{R}^+$, and it will have the same cardinality. Completeness too will be inherited from the completeness of $\mathbb{R}$, so this structure will be in many ways similar to $\mathbb{R}^+$, 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.

I think I see what you mean dcpo but I maybe disagree with them having the same cardinality than $\mathbb{R}$. 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]

I think I see what you mean dcpo but I maybe disagree with them having the same cardinality than $\mathbb{R}$. 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?

Well, the way I've defined it the base set is $[0,1)\times[0,1)$, and since $[0,1)$ has the same cardinality as $\mathbb{R}$, that the cardinality of $[0,1)\times[0,1)$ is the same as that of $\mathbb{R}$ 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]

Well, the way I've defined it the base set is $[0,1)\times[0,1)$, and since $[0,1)$ has the same cardinality as $\mathbb{R}$, that the cardinality of $[0,1)\times[0,1)$ is the same as that of $\mathbb{R}$ 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.

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]

I think I see what you mean dcpo but I maybe disagree with them having the same cardinality than $\mathbb{R}$. 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?

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]

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.

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]

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.

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

$$...999999999 = -1$$

 

[jgens]

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

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.

 

[jgens]

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.

If X is a set, then X is a subset of itself. So my original statement was entirely correct. What you mean is that the sequences of 0s and 1s is not a proper subset. In any case, formally your claim may/may not be true depending on what we mean by "infinite sequences of digits". We can talk about real numbers in base 2 versus base 10 because they can be defined independent of a particular representation. So if you define your numbers in a representation invariant way, then you can do the same thing here. On the other hand, in this case I would tend to define an infinite sequence of digits in terms of its representation in a particular base, and in this case the two are not equal.

Edit: It also worth noting the following. The set of all real numbers in [0,1] whose decimal expansion consists entirely of 0s and 1s is not equal to the set of all real numbers in [0,1]. The set of all real numbers in [0,1] whose binary expansion consists entirely of 0s and 1s is equal to the set of all real numbers in [0,1]. So we can reinterpret strings in a particular way that makes this true, but once we have fixed an interpretation (for example, using the characters 0,1,2,3,4,5,6,7,8,9 as you did in your example indicates at least base 10) we have to stay consistent with that. In my previous post, to stay consistent with the convention of using the symbols 0,1,2,3,4,5,6,7,8,9, the collection of all infinite sequences of 0s and 1s is a proper subset of your numbers.

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...

No. The cardinality is still the same. The infinite sequences of 0s and 1s have the same cardinality as the reals. In fact, if you take all infinite sequences of 0s and 1s that begin with 1, you get a set with the same cardinality as the reals. Or if you take the set of all infinite sequences of 0s and 1s that start with 101110110 this set has the same cardinality as the reals.

 

[viraltux]

aaaaaaand outta here. nice week end everyone :)