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Questions about infinity


Kriminal99

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what if we changed the constants in the relabeling scenario? Let's say we put in five and take out one to relabel as the next highest. Then we wouldn't simply be 'adding zeros', we'd be relabeling to the next number highest.

 

Great question. Probably if you did that, the jar would explode, annihilating you and all the balls.

 

The scenario, I take it, goes like this: You start with balls labeled with all the natural numbers except numbers of the form 5n. In the first round, ball #1 gets relabeled "5". Then ball #2 gets relabeled "10". And so on. In general, in round n, ball #n gets relabeled 5n. So the original ball #1 will get relabeled like this:

Round 1: "1" [math]\rightarrow [/math] "5"

Round 5: "5" [math] \rightarrow [/math] "25"

. . .

Round [math]5^n[/math]: "[math]5^n[/math]" [math] \rightarrow [/math] "[math]5^{n+1}[/math]"

. . .

 

Your question, then, is what is the end result of carrying out this process, according to this pattern, infinitely many times?

 

"Undefined" seems to be the only answer. The succeeding changes don't converge on any determinate result. It would be tempting to answer that ball #1 will end up labeled with the symbol that refers to [math]5^{\infty}[/math]. But there is no such symbol, or at least, there need be none as far as the scenario is concerned. Ball #1 only gets relabeled with natural-number symbols. We could suppose that the symbol "[math]\infty[/math]" doesn't even exist.

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Here's a simpler version of the last paradox discussed:

 

There's a device for counting natural numbers. It displays the natural numbers on a screen, in sequence: "1", "2", "3", and so on. It displays "1" for one second. Then it displays "2" for half a second. Then it displays "3" for a quarter second. And so on. Anything on the display remains there until it is changed by another number being displayed. At the end of 2 seconds, what is on the display?

 

Let me add this: the device has no such symbol as "[math]\infty[/math]". It only has the characters "0", "1", "2", "3", "4", "5", "6", "7", "8", and "9".

 

If this were a finite sequence, it would be displaying the last number in the sequence at the end. But there is no last member of the sequence of natural numbers. The only available answer seems to be, "This series is impossible."

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I'd say, rather, that the only available answer is: "This question makes no sense." This is because there is no last stage, just as in the case of the vase. It makes no more sense to ask what the set in the jar is, at noon; this is why I haven't told you any finite number in answer to your challenge but instead said that it needn't be answered. Certainly, at least, not if the symbol [math]\norm\infty[/math] isn't available.

 

I meant to say the set of natural numbers (integers greater than 0) is well-ordered. Anyone in set theory will tell you that, besides that it's intuitively obvious. This means that any subset of it has a least member. I don't mean any finite subset; I mean any subset.
<nitpicking>The set of natural numbers includes 0.</nitpicking> This does not make it necessary to answer the challenge, if the set you ask about can't be defined.

 

Hyperreals are generated in the relabeling case because each ball gets a "0" added to its label, infinitely many times. They're not good in the original case, because no ball ever gets its label changed in any way.
The new number on the label is exactly the same as the number on the label of the ball that gets put in according to the other scenario. Removing or relabelling, these two scenarios are numerically equivalent, at any n-th stage they have exactly the same set of numbers on the labels. Relabelling is equivalent to removing the number and putting in another one. As a mathematical problem it is a sequence of subsets of [math]\norm\mathbb{N}[/math], not sets of balls. Remove balls, or remove numbers... you cannot arrive at different conclusions.

 

Once the task is completed, there are no finitie numbers, and there is no highest value number, all one needs, to conceive of the results, is expressed in Gallileo's paradox and Cantor's diagonalisation. If the Ross-Littlewood result is incorrect, the concept of infinity needs to be revised.
I don't see where Cantor's diagonal argument comes in, as we are talking about countable sets. Apart from this, Galileo's paradox, not considered a paradox in modern mathematics but just a fundamental property of sets of infinite cardinality, doesn't go toward making the Ross argument conclusive. Would you say that [math]\norm\mathbb{Z}\backslash\mathbb{N}=\emptyset[/math] or that [math]\norm\mathbb{Q}\backslash\mathbb{N}=\emptyset[/math]? Certainly not, but in the ball and vase problem the only essential thing is the order of the labels and the set of numbers being put in might as well map onto [math]\norm\mathbb{Z}[/math]. Trickier to map it bijectively onto [math]\norm\mathbb{Q}[/math], I admit, but it's all to easy to say that the complement of the set of removed balls is the empty set. If at each step we put in 2 balls instead of 10, and remove 1, the mapping onto [math]\norm\mathbb{Z}[/math] is trivial. Of course if the balls you put in actually are numbered according to this or some other map, they aren't all natural numbers and it's a no-brainer. The numbers put in are all naturals, this makes it highly subtle but not conclusive.

 

In general, for N balls in and 1 out, one may think of the mapping onto multiples of N. At no finite n-th step is [math]\norm\cal{I}_n\backslash\cal{O}_n=\emptyset[/math] and, although both these sets asymptotically approach [math]\norm\mathbb{N}[/math], on what grounds does it make sense to conclude that [math]\norm\cal{I}_f\equiv\mathbb{N}\equiv\cal{O}_f[/math]? If we assume it makes sense at all, the argument is that every n has been put in and has been taken out, but it's like saying that [math]\norm\frac{1}{0}=\infty[/math] (equals, not a limit) and then concluding thence that [math]\norm\frac{1}{0}=0[/math].

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Qfwfq: I apologise if I'm not getting a handle on this, but your stated objection concerned an unfairness in the use of hyper-real numbers. Both tasks, while they are underway, have exactly the same number of marbles in the jar and exactly the same numbers written on those marbles.

When either task is completed one is reduced to considering the consequences of performing an infinite number of specified actions, and these actions are different in the two cases of Ross-Littlewood and Allis-Koetsier, the action of removing a marble is entirely different from the action of adding a zero to the number that is written on the marble, do you seriously suggest that two such different actions produce the same result?

I dont understand how your previous post or your extended proof in post 81 are meant to challenge the validity of this result. As you know, from previous threads, I am ignorant of mathematical jargon, please try to communicate your objection on some kind of middle ground. And, bear in mind that this is a "paradox of infitinity", the fact that you accept a different result doesn't necessarilly refute the present result, this is a paradox.

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Extra nitpicking: "Natural numbers" is sometimes used to refer to all positive integers, and sometimes to all nonnegative integers. I.e., sometimes "0" counts, sometimes not, but it's more common not to include "0". :) See, e.g.,

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Extra nitpicking: "Natural numbers" is sometimes used to refer to all positive integers, and sometimes to all nonnegative integers. I.e., sometimes "0" counts, sometimes not, but it's more common not to include "0". …
Owl’s nitpick agrees with my academic and professional experience. In all my texts, class and personal notes, and math correspondences, the naturals (N) have never been assumed to contain 0. That set is always written something like [math]N + \lbrace 0 \rbrace[/math] or [math]Z^+ + \lbrace 0 \rbrace[/math]. To the best of my knowledge, this convention stems from Peano’s 9 axioms, the first of which is “1 is a natural number”, so is essentially as old as modern number theory.
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Sorry but in my university memories, including many chaps that are nobody's fool, [math]\norm\mathbb{N}[/math] was always considered according to the more modern convention, as Craig's wiki says (especially to the purpose of Peano's arithmetic). I've never considered dictionaries to be of much use for these things.

 

I was only nitpicking about the explicit statement. Given that the problem is posed with labels starting from 1, it would be more exact to talk of [math]\norm\mathbb{N}_0[/math] but I haven't been bothering myself, where it doen't make an essential change in the arguments.

 

When either task is completed one is reduced to considering the consequences of performing an infinite number of specified actions, and these actions are different in the two cases of Ross-Littlewood and Allis-Koetsier, the action of removing a marble is entirely different from the action of adding a zero to the number that is written on the marble, do you seriously suggest that two such different actions produce the same result?
How could the two scenarios differ about the set of numbers after completion? The Ross argument can only conlude that no finite natural is in this set.

 

Relabelling the same ball from [math]\norm ab_n[/math] to [math]\norm ab_{n+1}[/math] is a re-mapping of balls to numbers which doesn't change it being bijective. It's perfectly equivalent to removing the ball with the first number and adding one with the second number (as well as the b - 1 others). Would you seriously suggest it makes a difference which ball has which number? In both cases, it is essential that there are no more nor less numbers than balls (same cardinality). If the set of balls can't be empty in the one case, neither can it in the other. If the Ross argument were conclusive about the set of numbers being empty, this would necessarily hold for both cases.

 

I dont understand how your previous post or your extended proof in post 81 are meant to challenge the validity of this result. As you know, from previous threads, I am ignorant of mathematical jargon, please try to communicate your objection on some kind of middle ground.
That post only meant to say that propositon 1 is no more conclusively consequent than proposition 2. Sorry if the notation troubled you.

 

And, bear in mind that this is a "paradox of infitinity", the fact that you accept a different result doesn't necessarilly refute the present result, this is a paradox.
Are you suggesting that two incompatible conclusions could both be true? Either a proposition is undecidable (shortage of axioms) or, if it can be proven true then you can't also prove something that contradicts it.
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Qfwfq: The situations are not the same. RL involves 10 balls at every action and the action concludes with the removal of a ball, AK involves 9 balls at every action and no ball is removed. These are not equivalent. As you say, the cardinalities are the same, so there is a time, at which any ball is removed, that corresponds to that ball. This is a standard convention concerning infinity, (hence my mention of Gallileo and Cantor), and no eccentricities of infinity are specified in the description of either RL or AK.

I'm not sure that it's important to decide which result is "true", as the scenarios are imaginary. However, the results have a bearing on and arise from the way we think about these things, and thus have consequences as to how we think about other, more or less related, things. It appears that various people hold differing views as to the result, and not just among this board's members, which view one accepts seems to be mainly a consequence of how one thinks about the problem.

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I think our misunderstanding here is that the ball I say is number 1736 is the one you say is number 58, which is the one I say is number 157 while that's the one you say is number 7192 and I say that one's number 314593 and...... I mean, look at what a difference it makes!

 

;)

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  • 1 month later...
Returning to the nit-pick, is the in- or exclusion of zero a question of which foundations of mathematics theoryist a person is?
I don’t think so.

 

Although the development of number theory I was taught began with the first of the Peano postulate being “1 is a number” (as does their description at wikipedia), others begin “0 is a number” (as does mathworld’s description). Though these differing postulates have an impact on the details of the theoretical development of various number systems, I don’t think it much affects their important implications.

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CraigD: Thanks for the reply. According to Peter Suber (Peter Suber, "Infinite Sets") set theorists include zero among the natural numbers.
Including 0 makes a lot of things handier, especially in further developments.
The impression I get is that it's most conventional now to define the Natural numbers as containing zero.

 

My math education is pretty old (1982), my textbook involving numbers systems, “Abstract Algebra” by Davidson and Gulick, even older (1976), and its treatment perfunctory (though I recall my instructors and classmates were enthusiastic about number theory). I suspect the tendency in those days, if not just local to my Math department, was in part an effort to be historical, in keeping with Peano’s writing (which was older still - 1889), and in part because doing so made for a neat equivalence between the Naturals and the Ordinals. In my student Math days, people tended to order elements of sets [math]a_1, a_2, a_3 …[/math], where now the convention [math]a_0, a_1, a_2 …[/math] seems more common. I’ve a suspicion that computers had something to do with this, as arrays in early and later programming languages usually began with a(0), while in math (eg: matrix notation), they still tend to begin [math]a_1[/math].

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Actually my math education also dates back to around '82 but it was already long common to use 0 based indices. I don't think mathematicians got it from computer programmers.

 

Historically, it was from the Arabs that we learned to consider 0 a number. In classical culture it isn't, nothing wasn't regarded as a quantity but instead as a lack of. This is typically considered a Roman shortcoming, but perhaps only because they most directly influenced medieval Europe; Greek and Hebrew notation didn't use a numeral for 0 either. Of course these systems explicitly distinguished units, tens and hundreds (and even included thousands in the Roman) so they didn't have such an obvious need for the placeholder (originally a dot which then became the closed line 0) while it was difficult to indicate as much as a million.

 

Those, such as Fibonacci, who brought the Arabic progress to Europe were quite before Peano but some folk just find 0 hard to get the hang of... I mean, when you're in a shop you ask for 1 of this and 2 of that without specifying 0 of each other thing in stock! :eek_big:

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