Paradox!

[Deepak]

At 11:00 Maxwell's Daemon starts counting positive integers from 1 and reaches the next perfect square in half the time to noon (11:30). The Daemon continues counting up the integers successively reaching the next perfect square in half the time to noon (11:45).

If the Daemon continues to halve the time to the next perfect square, at noon has it counted more perfect squares or integers which are not perfect squares?

 

[Speakpigeon]

At infinity?!
You are aware that infinity just means "without bound", "not bounded". So, saying "at infinity" is a bit like saying "inside nothingness".

Annnndddd... we'e back to pre-calculus. You are aware that there are many different kinds of infinity? Some larger than others?

Sure, so what?

We found or didn't find schemes to count elements in infinite sets and we can or cannot compare them to each other. It's not that we're going to find any particular figure for how many rationals or reals there are but we can compare natural numbers and rationals, and maybe reals with other sets. Good. And?

That there are standard number systems with infinity as a point?

I'm not sure what you mean by standard.

There are many wierder and wierder logical systems being designed and investigated but that does not change the laws of straightforward deductive logic.

The point is, people looking at your so called paradox are under no obligation fto follow terms you have never specified.

Me I'm looking at this problem according to common sense, which as good as any. Are you going to prove that there is such a thing the number "infinity"? According to common sense? If not, why should I accept your rules? Rules you haven't even bothered to specify!

How people's intuition could possibly encompass the concept of "at infinity"?

Apparently, with great difficulty.

May I remind you that when Cantor came up with his scheme to count rationals in the XIXe century, and proceed from there to compare types of infinities, nearly all of his fellow mathematitians rejected his proposal.

You think you would have understood straight away, do you?

Rather, you specified a process, clearly enough, which just keeps adding 9 balls at every step before 12:00am. So it's not like it's difficult to do the maths! And that's just what people do.
Let's simplify the problem though. adding 9 balls may well be a tad too hard. Let's add 1 ball at a time. So let's have the function:

f(0)=0 and f = f(n - 1) + 1 for n > 0

Easy enough, yes?

So we have:

f(0) = 0
f(1) = 1
f(2) = 2
f(3) = 3
...
f(10) = 10
...
f(20) = 20
etc.

So, basically, without being a maths genius, we can see that the resulting set is N itself, yes? So f = n.

Wrong. You're regressing here. If I tell you that f(x) = x for all x < 1, you cannot say that f(1) = 1.

Wrong? What is wrong exactly?

You think you can get away with a reasoning by analogy?

Let's also assume that each f step occurs at the time specified in the OP. So, now, what is the value of f at 12:00am?

Me, I say the same thing as before, we just don't know because the question is meaningless because there's no step occurring at 12:00am as per your specification.

Nonsense, there's no reason to require that there be a step at 12.

I'm not saying that there's a reason to require that there be a step at 12:00am. I'm saying that I don't see any good reason to assert a value for the number of balls left in the vase at 12:00am.

Now, if you reply "infinity", let's hear your argument (or demonstration).

In fact, the question does not give enough information to determine \(f(\infty)\), which is why there are multiple possible (and simultaneously correct) values.

Sure, when we don't know x then we accept that x may take any particular value if there is a reason that it should take a particular value. But absent any reason for x to have a particular value, the question of what value it has is just meaningless.

In the material world, we assume unknown things to have particular characteristics but you accepted that we're not talking about the material world so there is no reason that the number of balls left over after the completion of each step of the process you specified should take a particular value and therefore the question of what value it has is just meaningless.
EB

Wrong. The same question with a labeling scheme specified is completely tame, and has a well-defined answer.

Apparently you just don't understand where the problem is.

You can label the balls all you want, the point is that there is no last step in the process as per your specifications. There is really nothing else to say.

Oh come on, you didn't specify f(a), that is, f(12:00am)! Nobody is assuming anything about f(12:00am) because you didn't specify f(12:00am).

The question asks (essentially) for \(f(\infty}\), and people assume that that is equal to \(\lim_{x \to \infty} f(x)\). Since, this is not true we have a problem.

The problem would be the same without infinity to make the maths hard to intuite.

In fact, the question does not give enough information to determine \(f(\infty)\).

The OP does not give enough information to determine f(12:00am). Infinity is irrelevant here. That's it. There's nothing else to say about it! It's not that f(12:00am) has a value we don't know or that we have vox populi delusions about it. It's that the question of the value of f(12:00am) is meaningless to start with.
EB

You're right that the statement does not give enough information to define f(12am). You're wrong in claiming meaninglessness of f(12am). It's completely reasonable to ask for the value of a function at a point for which it's defined. Meaningful too.

So, what is the definition of the number of balls at 12:00am exactly?
EB

 

[fromderinside]

At 11:00 Maxwell's Daemon starts counting positive integers from 1 and reaches the next perfect square in half the time to noon (11:30). The Daemon continues counting up the integers successively reaching the next perfect square in half the time to noon (11:45).

If the Daemon continues to halve the time to the next perfect square, at noon has it counted more perfect squares or integers which are not perfect squares?

Don't know since noon never is reached.......

 

[Deepak]

At 11:00 Maxwell's Daemon starts counting positive integers from 1 and reaches the next perfect square in half the time to noon (11:30). The Daemon continues counting up the integers successively reaching the next perfect square in half the time to noon (11:45).

If the Daemon continues to halve the time to the next perfect square, at noon has it counted more perfect squares or integers which are not perfect squares?

Don't know since noon never is reached.......

Ok, so if we take time out of the problem what do you think? Say the Daemon is chunking the positive integers in groups of 50. For each group we can see that the amount of perfect-squares << the amount of non-perfect-squares. In fact, we can quickly see that the number of perfect-squares quickly becomes less numerous than the amount of chunks we've iterated.

But looking at things a different way, I can define a set A which corresponds to the positive integers, and I can define a set B which for each element (say x) of A contains a number defined as x*x

Do these sets contain the same amount of numbers?

For any infinite set we could say that some arbitrarily high number is not reached - which corresponds exactly to noon in the OP. The time component is actually a canard, and it's entirely irrelevant to say that noon is never reached.

So was Cantor wrong?

 

[steve_bnk]

Set A all numbers from 1 to 2.
Set B all numbers from 3 to 20.

A calculus can be devised that says B is a larger infinity than A, and that A can added to B.

But that does not get you anywhere with Zeno type paradoxes.

 

[Artemus]

Set A all numbers from 1 to 2.
Set B all numbers from 3 to 20.

A calculus can be devised that says B is a larger infinity than A, and that A can added to B.

Every number in [1,2] can be mapped to a unique number in [3,20] through y = (17x - 11)/2 while every number in [3,20] can be mapped to a unique number in [1,2] through x = (2y +11)/17. Since every number in each set has a exact one-to-one correspondence with a unique number in the other set, neither set can be said to be larger than the other. The infinities are the same.

 

[Artemus]

Don't know since noon never is reached.......

Change the problem statement so that we first wait 30 minutes while doing nothing at all, then wait 15 minutes doing nothing, then wait 7.5 minutes, etc. Would anyone still argue that noon is never reached?

The only thing from this problem that does correspond to the real world is that noon will be reached.

 

[steve_bnk]

Set A all numbers from 1 to 2.
Set B all numbers from 3 to 20.

A calculus can be devised that says B is a larger infinity than A, and that A can added to B.

Every number in [1,2] can be mapped to a unique number in [3,20] through y = (17x - 11)/2 while every number in [3,20] can be mapped to a unique number in [1,2] through x = (2y +11)/17. Since every number in each set has a exact one-to-one correspondence with a unique number in the other set, neither set can be said to be larger than the other. The infinities are the same.

Interesting response. I am not a mathematician.

In this case A and B are not domain and range mappings of a function, they are defined as infinite sets. It is a question of definitions.

There was a lengthy thread on FRDB as to whether or or not one infinity can be greater than another.

I rejected the idea until I looked at infinite sets.

http://en.wikipedia.org/wiki/Cardinal_number

Adding

Not all functions are 1 to 1.

y = x^2

For every y there will be 2 in x. One infinite set > another?

 

[Artemus]

Interesting response. I am not a mathematician.

In this case A and B are not domain and range mappings of a function, they are defined as infinite sets. It is a question of definitions.

There was a lengthy thread on FRDB as to whether or or not one infinity can be greater than another.

I rejected the idea until I looked at infinite sets.

http://en.wikipedia.org/wiki/Cardinal_number

Adding

Not all functions are 1 to 1.

y = x^2

For every y there will be 2 in x. One infinite set > another?

The function itself is not what is important. I just picked the linear transform since it was simplest to show a one-to-one correspondence can be established, thereby showing that the cardinalities are the same. There are limitless ways to do it. Your function could be used by setting x = f, where f is the principal square root of y.

 

[Speakpigeon]

At 11:00 Maxwell's Daemon starts counting positive integers from 1 and reaches the next perfect square in half the time to noon (11:30). The Daemon continues counting up the integers successively reaching the next perfect square in half the time to noon (11:45).

If the Daemon continues to halve the time to the next perfect square, at noon has it counted more perfect squares or integers which are not perfect squares?

For number n² there are n² numbers and just n perfect squares, i.e. n², (n-1)², (n-2)² ..., 9, 4, 1.

So although there is an infinite number of square numbers in N, the proportion is that there is n perfect square numbers for n² numbers, or n/n², which tends towards zero when n tends towards the infinite (like 1/n).

So you can say that the proportion of perfect squares which are less than n tends towards zero as n tends towards the infinite or as we get nearer 12:00. So, the limit is 0.

And that's all you can say. In particular, you can't say what is the proportion of perfect squares for all n or at 12:00, or anything.

Nothing special in that.
EB

 

[Speakpigeon]

Don't know since noon never is reached.......

Change the problem statement so that we first wait 30 minutes while doing nothing at all, then wait 15 minutes doing nothing, then wait 7.5 minutes, etc. Would anyone still argue that noon is never reached?

The only thing from this problem that does correspond to the real world is that noon will be reached.

The response to that has already been given, several times over. You have to consider the problem either as a physically realistic or non-realistic one.

If realistic then there is no such a thing as counting perfect squares faster and faster with no upper limit to speed before you get to 12:00. So you have to admit the problem is not physically realistic.

In this case, you can't claim that noon is necessarily reached. This has also been explained already. The counting is done faster and faster so that you never get to a final count at 12:00. You just don't because there is no final count at all. That's your specification of the problem.

Sure, there is finite limit to the proportion of perfect squares, which is zero as shown in my previous post. So, this proportion tends towards zero but the limit at 12:00 doesn't not tell you the proportion before 12:00 and the counting is all done before twelve so that the proportion has no definite value at 12:00.
EB

 

[Artemus]

Change the problem statement so that we first wait 30 minutes while doing nothing at all, then wait 15 minutes doing nothing, then wait 7.5 minutes, etc. Would anyone still argue that noon is never reached?

The only thing from this problem that does correspond to the real world is that noon will be reached.

The response to that has already been given, several times over. You have to consider the problem either as a physically realistic or non-realistic one.

If realistic then there is no such a thing as counting perfect squares faster and faster with no upper limit to speed before you get to 12:00. So you have to admit the problem is not physically realistic.

In this case, you can't claim that noon is necessarily reached. This has also been explained already. The counting is done faster and faster so that you never get to a final count at 12:00. You just don't because there is no final count at all. That's your specification of the problem.

Sure, there is finite limit to the proportion of perfect squares, which is zero as shown in my previous post. So, this proportion tends towards zero but the limit at 12:00 doesn't not tell you the proportion before 12:00 and the counting is all done before twelve so that the proportion has no definite value at 12:00.
EB

The passage of time was not modified by the OP problem statement. The interval continually halves and Zeno's paradox will be resolved exactly the same way it is resolved in the real world. Now, if you were to argue that the problem is physically ill-posed since it requires multiple violations of the laws of physics (not the least of which is infinite work) to complete the assigned tasks by the time noon is reached and leave it at that, I certainly wouldn't argue. But noon will be reached.

 

[Speakpigeon]

The passage of time was not modified by the OP problem statement. The interval continually halves and Zeno's paradox will be resolved exactly the same way it is resolved in the real world.

Zeno's paradox is not resolved as in the real world. We don't even know the nature of time, space and matter to the point that we could explain exactly how Achilles does manage to catch up with the turtle. We don't even know whether these things are for real.

Now, if you were to argue that the problem is physically ill-posed since it requires multiple violations of the laws of physics (not the least of which is infinite work) to complete the assigned tasks by the time noon is reached and leave it at that, I certainly wouldn't argue. But noon will be reached.

I already explained why not.

Maybe you could sart by telling me which it is: a physically realistic problem or a not physically realistic problem?
EB

 

[Artemus]

Zeno's paradox is not resolved as in the real world. We don't even know the nature of time, space and matter to the point that we could explain exactly how Achilles does manage to catch up with the turtle. We don't even know whether these things are for real.

The problem statements all clearly state that time passes exactly as it does in the real world, that is the intervals are halved until noon is reached. If you are going to claim that noon is not reached you must show how the passage of time in the problems differ from the passage of time in the real world. Repeating myself "The only thing from this problem that does correspond to the real world is that noon will be reached."

I already explained why not.

Maybe you could sart by telling me which it is: a physically realistic problem or a not physically realistic problem?
EB

I've said multiple times that it is not. ("Violations of the laws of physics," "physically ill-posed", "the OP requires you to do something in an infinitesimal amount of time that in reality requires a finite amount of time.")

 

[Juma]

Zeno's paradox is not resolved as in the real world.

Zenos paradox is solved by fact that sums with infinite number of terms can be finite. Has nothing to do with reality.

 

[Speakpigeon]

Zeno's paradox is not resolved as in the real world.

Zenos paradox is solved by fact that sums with infinite number of terms can be finite. Has nothing to do with reality.

Exactly what I said.

Thanks for clarifying my thoughts.
EB

 

[Speakpigeon]

The problem statements all clearly state that time passes exactly as it does in the real world, that is the intervals are halved until noon is reached.

We don't know how time goes in the real world. We may have a very nice intuition about it but we don't know. Our physics model works but that does not tell us how time goes in actual fact.

However, what physicists seem to accept is that we can't do something like halving again and again, ad infinitum, the time left to 12:00. We just don't know how to do that.

If you are going to claim that noon is not reached you must show how the passage of time in the problems differ from the passage of time in the real world. Repeating myself "The only thing from this problem that does correspond to the real world is that noon will be reached."

I'm saying that your specification of the problem results in the ratio squares/numbers having no specified value at 12:00, which me and others have described as "12:00 is not reached".

I've said multiple times that it is not. ("Violations of the laws of physics," "physically ill-posed", "the OP requires you to do something in an infinitesimal amount of time that in reality requires a finite amount of time.")

Good, so to say "12:00 is not reached" shouldn't be misunderstood.
EB

 

[Artemus]

We don't know how time goes in the real world. We may have a very nice intuition about it but we don't know. Our physics model works but that does not tell us how time goes in actual fact.

However, what physicists seem to accept is that we can't do something like halving again and again, ad infinitum, the time left to 12:00. We just don't know how to do that.

Time is viewed as continuous in modern physics theories (see  Chronon#Early_work), but I'll agree that my statement that time in the problem statement passes "exactly as is does in the real world" is imprecise as there is some possibility (but no evidence) that time may be quantized. (Juma noted my imprecise statement as well.) I'll modify it to say that time passes in the problem statement as it does under modern physics theory. So, we repeatedly wait half the remaining interval. Do we ultimately reach noon?

(It is not relevant if it is possible to actually do anything within a Plank time, just how time passes within the context of the problem statement.)

 

[Speakpigeon]

Time is viewed as continuous in modern physics theories (see  Chronon#Early_work), but I'll agree that my statement that time in the problem statement passes "exactly as is does in the real world" is imprecise as there is some possibility (but no evidence) that time may be quantized. (Juma noted my imprecise statement as well.) I'll modify it to say that time passes in the problem statement as it does under modern physics theory. So, we repeatedly wait half the remaining interval.

Ok, that's much better.

I'm Ok with the idea of continuous time and even the possibility of doing something over and over again, faster and faster, without limits. Once we are clear about the problem being non-realistic we can better look at the logic of it.

Do we ultimately reach noon?

No.

What, in the context of this problem, could it possibly mean to ask if we ultimately reach noon? I suppose it's OK to say that if there were abstract observers looking at the counting process unfolding the same observers would still be there at noon. However, you say "we repeatedly wait half the remaining interval". So the question is whether "we", doing the counting by waiting half the remaining interval again and again, ad infinitum, will be counting anything at noon. I say no. This for the same reason that the sequence of times, as defined, does not reach 12:00. It gets closer and closer, as close as you want, but without reaching 12:00. Thus, what counting is done at 12:00 is not specified so we don't know what is the proportion of perfect squares at 12:00, except to say that before 12:00 it unambiguously tends towards zero as we get ever closer to 12:00.

I have to say this is also the maths I remember learning, or at least what I understood at the time, or remember understanding (and it was all theoretical maths, not applied). An infinite sequence of values (in French: séries) may have a limit but the limit is not part of the sequence. So 12:00 is not part of the sequence of times obtained by always adding half of the remaining time to 12:00.

In physics, it's reasonable to infer that the value at the limit is the limit of the values because we can assume that in fact the physical process concerned will actually stop at some point in time before the theoretical limit, say 12:00, but sufficiently close to the limit that the value at that point will be small enough to be assumed null, but this of course is no longer an infinite process and wouldn't apply here.

I also remember some of what we learned about Cantor, in particular his solution to pair up N and Q numbers ad infinitum. But, I don't see how that could change my reasoning here so maybe you could try to explain, in plain English, how Cantor's findings affect your reasoning.
EB

 

[Artemus]

Do we ultimately reach noon?

No.

What, in the context of this problem, could it possibly mean to ask if we ultimately reach noon? I suppose it's OK to say that if there were abstract observers looking at the counting process unfolding the same observers would still be there at noon. However, you say "we repeatedly wait half the remaining interval". So the question is whether "we", doing the counting by waiting half the remaining interval again and again, ad infinitum, will be counting anything at noon. I say no. This for the same reason that the sequence of times, as defined, does not reach 12:00. It gets closer and closer, as close as you want, but without reaching 12:00. Thus, what counting is done at 12:00 is not specified so we don't know what is the proportion of perfect squares at 12:00, except to say that before 12:00 it unambiguously tends towards zero as we get ever closer to 12:00.

I have to say this is also the maths I remember learning, or at least what I understood at the time, or remember understanding (and it was all theoretical maths, not applied). An infinite sequence of values (in French: séries) may have a limit but the limit is not part of the sequence. So 12:00 is not part of the sequence of times obtained by always adding half of the remaining time to 12:00.

As Juma pointed out, we are dealing with a series, not a sequence, which is a sum of the terms of a sequence. A convergent infinite series does not get closer and closer to a value, it is exactly equal to the value. Wikipedia  Series_(mathematics)#Convergent_series gives a simple proof for (almost) this very series. Note that it is an exact equality, not a limit. (The Wiki series starts with 1 instead of 1/2 so it is equal to 2 and not 1. Just subtract off the initial 1 to get our series which is equal to 1.) Noon will be reached in exactly 1 hour.