the infinity lamp ...[paradox two]

[Amorphos]

the infinity lamp
this puzzle is a lot simpler than the hotel one, and i think we can resolve a few issues with it.

quote:
Question. Consider a very durable ceiling lamp that has an on-off pull string. Say that the string is to be pulled at noon every day, for the rest of time. If the lamp starts out off, will it be on or off after an infinite number of days have passed?

Answer. The lamp could in fact be either on or off after infinitely many days. Information about its state after any finite number of days is not enough to enable us to extrapolate past infinity. What makes this question interesting is that it is possible give an argument that seems to indicate the lamp will be on, as well as an argument that seems to indicate the lamp will be off. On: “The light starts out off, and then we turn it on. Each time we turn it off again, we immediately turn it back on. Therefore it must ultimately be on.” Off.- “Each time we turn the light on, we immediately turn it back off. Therefore it must ultimately be off.” This type of lamp is called a Thompson Lamp.

my answer:
we never arrive at an end point, you can keep switching it on and off and no matter how many times you do this you never reach infinity i.e. arrive at and end of sequence - and it is a sequence of finite operations.

firstly we may ask; what exactly is an infinite number set? in maths surely we are actually talking about open number systems! it is presumed that you can have an infinite set of natural numbers, yet in truth what is visualised is a very large number without end. this means that you can add for instance; injective or bijective sets, to the original infinite number set - add infinitum. you can keep adding sets because infinity is thought of as without end, thus another set can be added without detracting from that space [the open end].
in short then you cannot have an infinite number set, nor add sets to it. you can have open ended number systems or potential infinities just as we can have theoretical geometric shapes which begin and end at infinity, yet these do not amount to an actual infinity nor an infinite number.
thus you cannot have a lamp that can be switched on or off infinitely, nor an infinite amount of days to perform this task in.

[Gary88]

WHY!!! why do you ask these things

[Chaigidel]

because thats how the feathered serpent with its head in jupiter and tail in mars ROLLS SON.

[Amorphos]

i just like messing with peoples heads - light up chaps!

[Simetrical]

Quetzalcoatl, while your commentary may appeal to you philosophically, it is mathematically nonsense. I forbear opinion on any philosophical infinities you may care to ponder, and give the mathematical response.

The answer given is mathematically correct, insofar as this represents a standard mathematical question in elementary analysis: what's the limit of (−1)ⁿ? The answer is, it has none. The sequence is divergent. Why? It follows, of course, from the definition of a limit. We define that the limit of s_n (as n goes to infinity) equals some finite number L if for all ε > 0, there exists an N so that if n > N, |s_n − L| < ε.

To show that the limit does not exist, we need only select an ε and show that no N exists satisfying the definition's requirements. But this is simple. Let ε = 1. Then suppose there were some N and L so if n > N, |(−1)ⁿ − L| < 1. Now, (−1)ⁿ can be either +1 or −1, so this is equivalent to saying |L − 1| < 1 and |L + 1| < 1. But this is absurd, because the first statement requires that L is between zero and two exclusive, while the second requires it to be between negative two and zero exclusive. Those cannot both be true, so the limit cannot be a real number.

(Of course, it is desirable to also show that the limit is not existent but infinite, but that is even simpler and I think I've made my point.)

[LSJ]

It is important to note that infinite is not a number but a dependant attribute. If the lamp string is pulled infinitely then it has no value. As soon as the lamp string is not pulled it is no longer infinite - it has a value, and thus the question is nonsensical.

[Amorphos]

sim

I forbear opinion on any philosophical infinities you may care to ponder, and give the mathematical response

http://en.wikipedia.org/wiki/Infinity_%28philosophy%29

infinity is as much a philosophical notion as it is mathematical! in real terms i will state categorically that it is more of a philosophical notion, due to the ‘fact’ that math cannot define infinity itself, it has only the capacity to explain complex operations comparable to, or involving infinity [e.g. as a base].

so you think arguing in gnome language [mathematical expression] is going to prove me wrong - an unfair approach may i say [because actual infinity is not a mathematical problem, we only enter the world of math to show this]!? i challenge you to put all that poo into English and i will show how it is incorrect - that is if it is telling me that you can have an infinite number sequence! that is in context to what i am saying about infinity. honestly science is worse than religion - hiding behind its own secret language and being unwilling to except arguments that show it to be incorrect.

The sequence is divergent. Why? It follows, of course, from the definition of a limit.

the limit here is quite simple; you have a light switch, you begin a sequence of turning it on and off , then it is postulated that you may keep doing this infinitely when it is quite clear that it is a finite operation. you can keep switching it off and on all you like, it makes no difference if it is 8 times or 9 trillion billion times, you will never be able to reach an infinite amount of times. similarly you can inject as many number systems as you like e.g. you have the natural number line 123456789 then inject a prime number sequence 1,3,5,7,9,11,13,17, this makes absolutely no difference, it just means that you can build quicker - however the very same person who wrote the above question and answer is probably a much more advance thinker and mathematician than you and he clearly states that ‘you cannot build up to infinity’!

Those cannot both be true, so the limit cannot be a real number

i agree with that - absolutely!

ok let us begin at the beginning:

as we are considering limits let us ask how we can form an infinite number system using only natural numbers.

we have all numbers apparently arising from 1 - x [where x is the largest number], the beginning point itself defines the sequence as limited i.e. there would still be an infinite amount of negative integers remaining. i can see your point that x is undefined, but what i am saying is that you are working with that undefined factor because you cannot find a true value for it. nevertheless what ever x is there would always be an infinity remaining - as we are talking about a finite sequence that is.

so tell me how you visualise an infinite number set? both in terms of a sequence as above, or as an imagined infinity of integers as if they are simply there somehow?

[DisgruntledGoat]

the limit here is quite simple; you have a light switch, you begin a sequence of turning it on and off , then it is postulated that you may keep doing this infinitely when it is quite clear that it is a finite operation. you can keep switching it off and on all you like, it makes no difference if it is 8 times or 9 trillion billion times, you will never be able to reach an infinite amount of times.

I think here is the problem (the same problem you hand with the hotel). You are presenting a mathematical argument (what is the limit of (-1)ⁿ) and trying to ground it in the physical. Then you identify the supertask (being able to carry on turning the lamp on and off till infinity) and say "You can't do this." By identifying an impossible supertask you then claim that an actual infinity does not exist. You've made connections that aren't there. Meanwhile all you have really done is reasoned that supertasks are not possible. Which coincidentally is the whole point of the Thompson Lamp, to propose the impossibility of a supertask and not disprove the possibility of a natural infinity. However there are those that disagree with Thompson and since I can't put it any other way I will quote wikipedia (I know but it puts the whole thing quite elegantly)

The state of the lamp at t = 1 need not be logically determined by the preceding states. Logical implication does not bar the lamp from being on, off, or vanishing completely to be replaced by a horse-drawn pumpkin. There are possible worlds in which Thomson's lamp finishes on, and worlds in which it finishes off not to mention countless others where weird and wonderful things happen at t = 1. The seeming arbitrariness arises from the fact that Thomson's experiment does not contain enough information to determine the state of the lamp at t = 1, rather like the way nothing can be found in Shakespeare's play to determine whether Hamlet was right- or left-handed. So what about the contradiction? Benacerraf showed that Thomson had committed a mistake. When he claimed that the lamp could not be on because it was never on without being turned off again — this applied only to instants of time strictly less than 1. It does not apply to 1 because 1 does not appear in the sequence {0, 1/2, 3/4, 7/8, …} whereas Thomson's experiment only specified the state of the lamp for times in this sequence.

Again, it should be stated that this really isn't an argument about infinity and comes down to finite tasks performed over an infinite amount of time (supertasks) which is more for philosophy than math and science to discuss. Not to mention it really doesn't apply at all to a natural infinity. Just because there are arguments for the impossibility of a supertask does not mean that a natural infinity does not exist. For example in your hotel thread you proposed because of the impossible supertask presented by the hotel the universe could not be infinite. However, what supertask governs the universe?

[Amorphos]

goaty

thanks for that, it makes a lot of sense! perhaps i should be asking ‘what is a natural or real infinity like’ rather than trying to show that math does not ultimately describe it.

[Simetrical]

infinity is as much a philosophical notion as it is mathematical!

But I find philosophical infinity completely uninteresting. Thus I gave the mathematical answer and ignored the philosophical answer, as I clearly said: "I forbear opinion on any philosophical infinities you may care to ponder, and give the mathematical response."

in real terms i will state categorically that it is more of a philosophical notion, due to the ‘fact’ that math cannot define infinity itself

You persist in making straightforwardly incorrect statements like this in your defense of philosophy/attacks on mathematics. Mathematics can and does define infinity itself. Fact, period, end of story. Infinity is defined in mathematics. That you dislike its definitions is beside the point, they're still definitions.

(There are, by the way, a handful of mathematicians who share your dislike of mathematical infinities. Some of them have tried to rewrite analysis and other infinity-heavy areas of mathematics to view them without infinity. But they will not deny the validity of the classical approach, just express dislike of it.)

so you think arguing in gnome language [mathematical expression] is going to prove me wrong

I merely gave a rigorous mathematical proof of the correctness of the provided answer for the mathematical analogue of the question. As I say, I made no comment as to the philosophical correctness (if such a thing exists) of your answer. I just gave an alternative approach.

i challenge you to put all that poo into English and i will show how it is incorrect - that is if it is telling me that you can have an infinite number sequence!

Infinite number sequences are defined to exist. Therefore, in mathematics, they exist. Do not tell me what does or does not exist in mathematics, or what is correct or incorrect. Everything I wrote is completely correct. It is not, in fact, very difficult to understand at all, if you have a high-school knowledge of math and are willing to carefully consider every statement in sequence and ask if you don't understand a particular point. Let me walk you through the definition of a limit:

We define that the limit of s_n (as n goes to infinity) equals some finite number L if for all ε > 0, there exists an N so that if n > N, |s_n − L| < ε.

So, we have a particular term we're defining: "the limit of s_n". That term will have a single value for any s_n, and I give a way that potentially allows you to check whether the limit exists and, if it does exist, what it equals.

But first, what is s_n? It is what is normally called an infinite sequence. The terminology doesn't matter to the correctness: you can call it a snazzlefloo if you like, and just replace "infinite sequence" with "snazzlefloo" everywhere in the proof, and it will remain correct. But what is an infinite sequence? Well, there are such things as real numbers, that I'm sure we accept. You can think of an infinite sequence as a rule, a function, that associates each natural number with a real number. One example of an infinite sequence is

(1, 2, 4, 8, 16, 32, ...)

We call the sequence something like (s_n). Once more, the letters chosen are arbitrary: we could make it (q_Γ) if we liked. But most often we choose (s_n). Now, I said this associates each natural number with a real number. How so? By convention, we consider 1 to be associated with the first term listed, 2 with the second term, and so forth. We can refer to a specific term like s₁, s₂, s₃, etc. So for instance, here we have

s₁ = 1
s₂ = 2
s₃ = 4
s₄ = 8

and so on.

But I said that I associate every natural number with a real number. I have only listed six real numbers. What about the rest? Well, you can see the pattern: it doubles each time. So s₁ = 1, s₂ = 1×2, s₃ = 1×2×2, and so on. In fact, keeping in mind that 2⁰ = 1, we can write that for any natural number n, s_n = 2^{n − 1}. Thus I have given you a real number for each natural number. That is what is called an infinite sequence.

So we have defined an infinite sequence. Now let us recall what we were doing: defining the limit of an infinite sequence. We write "the limit of (s_n)" as "lim s_n". I said that the limit of s_n equals some real number L if

for all ε > 0, there exists an N so that if n > N, |s_n − L| < ε.

This may seem a bit opaque. (In fact, it's not particularly so: a more concise way to write the definition would be ∀ ε > 0, ∃ N: n > N → |s_n − L| < ε.) So, let's look at it step by step, starting at the end.

"|s_n − L| < ε." This is simple: it says that the absolute value of s_n − L is less than ε. (You will recall from high school mathematics that the absolute value of a number is its "distance from zero": absolute value turns negative numbers into positive ones and leaves positive numbers unchanged.) This is just an algebraic statement. The question is, what do all the symbols mean? Well, L is a number, the limit of (s_n). s_n is another number, just a term in the sequence (s_n) (but we haven't said which term yet). So what we're saying is that they're "closer together" than ε: the distance between L and this particular term s_n is less than ε. If ε is, say, a million, this is not saying much; but if ε is 0.0000000000001, it says they are very close together. So what is ε, and which term of the sequence are we talking about?

Let's look at the rest of the statement: "for all ε > 0, there exists an N so that if n > N, . . .". Okay, so what does that mean ε and s_n are? Well, I started off with "for all ε > 0", so what follows is true for any positive ε, no matter how big or how small. ε could in fact be either a million or 0.0000000000001.

Then what's s_n? Well, I haven't said, exactly, but I did say something about what n is, and if we know n, of course we know s_n too. What did I say about n? Well, I introduced a third number first. Let's look at the third number, N (mathematics is case-sensitive!). I said that for every ε, "there exists an N . . .". What does that mean? It means N can be any number that satisfies the following condition: "if n > N, |s_n − L| < ε." Now, we already went over the bit at the end: it means s_n is close to L, closer than ε. But now we see which n we're talking about: any n at all, provided it's larger than N.

So to recap: if lim s_n = L, then that tells us roughly that for large n, the terms of the sequence get very close to L. In fact, no matter how small a distance ― ε ― you choose, we can get the distance between L and the nth term of (s_n) ― |s_n − L) ― less than that distance, if we just choose n large enough. In fact, if we choose N large enough, any term beyond the Nth will be as close as we want to ε.

I hope you understand a bit better now. Of course, to properly understand you need to do a bunch of exercises now, but that goes beyond what I have the inclination to offer, and anyway I doubt you'd do them.

honestly science is worse than religion - hiding behind its own secret language and being unwilling to except arguments that show it to be incorrect.

And you don't think they may be on to something, what with their ability to use their mysterious science to construct things like computers for you to use to disparage them? It is not our fault that we must use "secret languages" that require effort to learn. The universe's secrets are not so simple that they can be understood without effort and dedication. We can try to make them as accessible as possible, but there is a limit to how much you can learn without truly devoting a lot of time to it.

As for not accepting arguments, quite simply, mathematicians do not accept arguments that are mathematically wrong, and yours are. There is no disputing that for anyone who understands mathematics. Mathematics is based on formal logic, which is simple and mechanical and cannot be disputed. When you say things like that infinity is not defined in mathematics, you are as clearly and unarguably wrong as if you had said black is white, whether or not I can ever show you that.

the limit here is quite simple; you have a light switch

Well, no. I gave an approach to the problem that took a mathematical analogy. You do not, in fact, have a light switch. The problem is not a physical one. Physically it's most certainly nonsense.

however the very same person who wrote the above question and answer is probably a much more advance thinker and mathematician than you and he clearly states that ‘you cannot build up to infinity’!

Please provide an exact source for that quote. James Thomson, who as far as I can see is the author of this paradox, was evidently not a mathematician but a philosopher, from what I understand. I assure you that you cannot find me an actual mathematician who disagrees with the validity of classical mathematics, with its infinite sets and all; or in the unlikely event that you can, they will be outnumbered thousands or more to one.

as we are considering limits let us ask how we can form an infinite number system using only natural numbers.

Peano's axioms are a simple way to do it. They use recursion:

1. There is a natural number 1.
2. Each natural number n has a successor, denoted n + 1.
3. Two natural numbers are equal if and only if their successors are equal.
4. 1 is not the successor of any natural number.

This clearly defines an infinite number of distinct natural numbers. You can then go on to define nonpositive numbers and rational numbers easily.

so tell me how you visualise an infinite number set? both in terms of a sequence as above, or as an imagined infinity of integers as if they are simply there somehow?

Visualization is not mathematics, I should note, and mathematics is rife with things that cannot possibly be visualized. Normally I don't visualize something like the set of natural numbers. I would tend to manipulate it symbolically instead. Geometry and maybe topology are where most of the visualization comes in.

[Darkragnar]

sim
so you think arguing in gnome language [mathematical expression] is going to prove me wrong - an unfair approach may i say [because actual infinity is not a mathematical problem, we only enter the world of math to show this]!? i challenge you to put all that poo into English and i will show how it is incorrect - that is if it is telling me that you can have an infinite number sequence! that is in context to what i am saying about infinity. honestly science is worse than religion - hiding behind its own secret language and being unwilling to except arguments that show it to be incorrect.

Its not some secret language mate! its pretty basic stuff hes talking about ...kind of like the stuff you do in 10^th grade, it might look unknowable but thats because your reading it on a computer , if he put that same thing on paper it would be instantly recognizable.

[Gary88]

is the answer not just a schrodingers cat type thing? on and off at once

[chris_uk_83]

i challenge you to put all that poo into English and i will show how it is incorrect - that is if it is telling me that you can have an infinite number sequence!

You just had to challenge him didn't you Quetz

[chris_uk_83]

There's also nothing secret about mathematics. The entire language is there for anyone to learn if you bother to find out how. It does take time and effort to learn it but it's by no means secret, in fact it's actively promoted.

[Amorphos]

chris uk

You just had to challenge him didn’t you Quetz

haha yes - bloody hellfire!

sim

But I find philosophical infinity completely uninteresting

ha, yes you are right in that. for me it is more a contemplative thing, interesting in the way it avoids definition etc. the math is more fun - alas i am quite crap at it.

There are, by the way, a handful of mathematicians who share your dislike of mathematical infinities

sure, i don’t dislike them, i just feel they explain comparative infinity i.e. as the finite interacting with the infinite. either that or they explain potential infinities.

all of this is to show that one cannot have an infinite universe or an infinitely cyclic one - but i must say that i am having to reconsider this idea! the apparent paradox of an infinite amount of finite, may be exactly the basis of existence.

Infinite number sequences are defined to exist. Therefore, in mathematics, they exist. Do not tell me what does or does not exist in mathematics, or what is correct or incorrect.

my apologies good sir - perhaps a little of my own frustration with myself came out there.

(as n goes to infinity)

n = an infinite number i presume? this is the part i have difficulty with, i can see it as an unlimited number, or an open ended number but not as actually infinite. this is especially true when we consider it in real terms; if you kept adding objects e.g. lamps of all variety, you can keep adding them without arriving any closer to achieving an infinite amount of lamps.
thus the same would go for physical universes.
can you imagine the mess an infinity of objects would actually appear as lol, we would also have to add infinite variance so as to arrive at that infinity of objects.

here then i hope you can see why i have trouble with the math. in maths one doesn’t imagine an infinity of numbers as an infinity of objects - i am probably wrong to imagine integers as like objects or entities. my problem is in seeing numbers as finite amounts with defined limits, a beginning and an end, whereas we may see them as e.g. xyz rather than 123. i have considered this also and still have a problem thinking of even ‘x’ as not being defined in some way.

I have only listed six real numbers. What about the rest? Well, you can see the pattern: it doubles each time. So s1 = 1, s2 = 1×2, s3 = 1×2×2, and so on. In fact, keeping in mind that 20 = 1, we can write that for any natural number n, sn = 2n - 1. Thus I have given you a real number for each natural number. That is what is called an infinite sequence.

hang on a minute before we go jumping the gun...
So s1 = 1, s2 = 1×2, s3 = 1×2×2, and so on
yes and so on? so far you have described an evolving number set...

Thus I have given you a real number for each natural number. That is what is called an infinite sequence

.

indeed you have given me a real number for the naturals, then made an absolutely huge leap to calling that an infinite number sequence? all i see is a number evolution, this could be primes odds or even or what have you, but they only keep building i.e. are potentially infinite.

let us take it one step at a time eh!

sorry about the secret language stuff, i thought you were using it to put me down. i would never learn it as i havent even got past n yet.

[Simetrical]

n = an infinite number i presume?

No, it does not. Infinity is not a number. n, in that context, is a variable that can represent any natural number. So it's unlimited, in a sense, but definitely finite. By "as n goes to infinity" I mean "as we consider n's that are bigger and bigger and bigger without bound", basically.

hang on a minute before we go jumping the gun...
So s1 = 1, s2 = 1×2, s3 = 1×2×2, and so on
yes and so on? so far you have described an evolving number set...

Yes, pretty much exactly.

indeed you have given me a real number for the naturals, then made an absolutely huge leap to calling that an infinite number sequence? all i see is a number evolution, this could be primes odds or even or what have you, but they only keep building i.e. are potentially infinite.

You could call it that, but mathematicians do not. They call it an infinite sequence. You surely agree that the sequence is not finite. If it were finite, that would mean it would have to have a last element (since it's ordered), but it doesn't: it always has more and more and more elements. You can't pin a finite number on the quantity of numbers in the sequence, so it's non-finite, i.e., infinite.

[chris_uk_83]

indeed you have given me a real number for the naturals, then made an absolutely huge leap to calling that an infinite number sequence? all i see is a number evolution, this could be primes odds or even or what have you, but they only keep building i.e. are potentially infinite.

It's not really a huge leap, all you need to do to define an infinite number sequence is fail to define an end point. You don't actually have to do all the additions, multiplications etc. in order to define your set, you just say "this set does not end" and as long as the equation allows it that's fine.

I understand your difficulty with using maths to imagine infinite things though. You want to imagine an infinite number of objects that you can count, that's natural to someone who hasn't become intimite with the language of maths, you want to base your understanding on your day to day experiences. When you learn how to "talk maths" though, it's far easier to simply take the equations and manipulate them without really giving much thought to what that might mean in "real life". It's also pretty unnecessary to think in those terms, since the answer you arrive at can usually be interpreted in "real life" terms anyway.

Take quantum mechanics for example. You can spend a lifetime trying to figure out what all the maths that goes on actually means (particles being in two places simultaneously etc) or you can just calculate it and use the result to build yourself a transistor. The latter is far more useful, but the former is a little more fun

[Amorphos]

chris uk

It’s not really a huge leap, all you need to do to define an infinite number sequence is fail to define an end point. You don’t actually have to do all the additions, multiplications etc. in order to define your set, you just say “this set does not end” and as long as the equation allows it that’s fine.

exactly as i thought - you have just defined a potential infinity!

It’s also pretty unnecessary to think in those terms, since the answer you arrive at can usually be interpreted in “real life” terms anyway

so do you think there can be an infinite amount of universes? or an infinite amount of any objects or finite entities? you most definitely cannot! this has nothing to do with my limited knowledge of math [my logic is not so limited!], it has everything to do with huge suppositions that don’t add up. you cannot actually define an infinite number set, you can only assume that it is there.

if not then define one!

Take quantum mechanics for example. You can spend a lifetime trying to figure out what all the maths that goes on actually means (particles being in two places simultaneously etc) or you can just calculate it and use the result to build yourself a transistor. The latter is far more useful, but the former is a little more fun

absolutely! just because it works for such things doesn’t mean it works for true infinities as this is not what you are using it for! my argument is that when you do try to use it so, it fails!

[chris_uk_83]

It’s also pretty unnecessary to think in those terms, since the answer you arrive at can usually be interpreted in “real life” terms anyway

What I was talking about here was using maths to deal with infinities, and then coming up with a finite answer. Which I'm sure Sim will confirm can often be the case. You can then interpret the finite answer and make it work for you whilst forgetting about the nasty complicated infinities you met along the way that were only necessary for the calculation and didn't really mean anything in "real life".

if not then define one!

All numbers, carrying on forever. There you go, it's defined. That I haven't named each one individually makes no difference. I think we're arguing here over the definition of the word "define". In maths you can do what I've just done, I suppose your definition requires individual naming of each element; which is equally valid, just not required in a mathematical sense.

[Amorphos]

sim

You surely agree that the sequence is not finite. If it were finite, that would mean it would have to have a last element (since it’s ordered), but it doesn’t: it always has more and more and more elements. You can’t pin a finite number on the quantity of numbers in the sequence, so it’s non-finite, i.e., infinite.

ah i see both mine and maths problem now, its all in the definition; i would indeed agree that an open ended evolving number set is not strictly finite, yet it is also not infinite. it is like a flower that continually grows without limit, and as infinity is defined as being unlimited, then you can rightly call it infinity by that definition.

but it is still not infinity proper, it lies between the concepts and the actual entities. where i would refer to this as being defined and thus not ‘of’ infinity, and indeed as therefore a part of the ‘finite’ universe, in math you cannot exactly define it so it is considered infinite.

i think good sir we are both correct in context.

i have learned a lot -thank you! i hope you have learned something about a fully undefined infinity