Is creating this infinite time step machine logically possible?

Question

I don't know much about philosophy, but this question has occupied my mind for hours, and I'm not even sure that it's a relatable question or not. Here is the question:

Imagine a machine that beeps for the first time after 0.5 seconds(t=0.5), it beeps for second time after 0.25 seconds(t=0.75), third beep after 0.125 seconds(t=0.875), and so on. Is creating this machine philosophically and logically possible?

I know that it's not possible to invent this thing physically, but it seems that we don't even need physics to reach to this point. Here is why: If this thing exists, it means that it has beeped infinity times after 1 second(t=1), which doesn't make sense. If the machine has a monitor on it that shows how many times it has beeped so far, there is no number that can be shown in t=1.
I have some questions about this:

1. Is my reasoning correct?
2. Does it make sense that a question that looks like a computer science problem, can be solved only by logic? This means that there are some more complicated questions like this.

Answer 1

This problem is analogous to Thomson's Lamp. It is related to Zeno's paradoxes. Thomson's paradox is even more intriguing because the question is whether the lamp is on or off after one second. Since it is either on or off, there must be an answer...? Thomson coined the term supertask which is a task that consists in infinitely many component steps, but which in some sense is completed in a finite amount of time.

EDIT: I am surprised with all the upvotes given that I failed to answer the question. To amend this, I will write a short follow-up to point to a sense in which the answer whether the machine is logically possible is 'yes'. For this I want to direct attention to an elegant extension of Thomson's description in chapter 1.2 of the SEP link to supertasks:

Suppose a metal ball bounces on a conductive plate, bouncing a little lower each time until it comes to a rest on the plate. Suppose the bounces follow the same geometric pattern as before. Namely, the ball is in the air for 1 minute after the first bounce, ½ minute after the second bounce, ¼ minute after the third, ⅛ minute after the fourth, and so on. Then the entire infinite sequence of bounces is a supertask.

Now suppose that the ball completes a circuit when it strikes the metal plate, thereby switching on a lamp. This is a physical system that implements Thomson’s lamp. In particular, the lamp is switched on and off infinitely many times over the course of a finite duration of 2 minutes.

What is the state of this lamp after 2 minutes? The ball will have come to rest on the plate, and so the lamp will be on. There is no mystery in this description of Thomson’s lamp.

Alternatively, we could arrange the ball so as to break the circuit when it makes contact with the plate. This gives rise to another implementation of Thomson’s lamp, but one that is off after 2 minutes when the ball comes to its final resting state.

The idea was proposed by one of the Modern Eleatics, P. Benacerraf, in 1962. Earman and Norton came up with the construction of the thought experiment in 1996.

These examples show that is possible to fill in the details of Thomson’s lamp in a way that either renders it definitely on after the supertask, or definitely off. For this reason, Earman and Norton conclude with Benacerraf that the Thomson lamp is not a matter of paradox but of an incomplete description.

So, there you are: there is no paradox, but just an incomplete description. If you complete the description you will find the lamp turned on or off depending on your mode of completion.

Thanks to these unsung heroes of experimental philosophy.

I hope I haven't broken any copyright with including the illustration of the SEP article. The reference for this article is: Manchak, JB and Bryan W. Roberts, "Supertasks", The Stanford Encyclopedia of Philosophy (Summer 2022 Edition), Edward N. Zalta (ed.)

Answer 2

If this thing exists, it means that it has beeped infinity times after 1 second(t=1),which doesnt make sense. If the machine has a monitor on it that shows how many times it has beeped so far, there is no number that can be shown in t=1.

It's true, that no monitor (imagined as a monitor with a finite number of digits on its display, a number that needs to be fixed in advance before the experiment starts and that cannot change afterwards) could ever show "how many beeps" there were when the (imagined) clock reaches the 1 second mark. But this is not essentially different from saying: It's impossible to count past 1000 and write down all the counted numbers (in decimal notation) using at most only 3 digits. Or: It's impossible to count past the number n (any number n) and write down n+1 using at most n digits (in unary notation). This is true, but doesn't seem very remarkable.

Even though the experiment you described could not be realized physically, we can make sense of "infinite number" of beeps by giving a precise definition of what we mean by that phrase. One way of doing so is by saying:

Given any natural number n, at some point in the experiment, but before the 1 sec mark has been reached, the number of emitted beeps will have become greater than n.

[Of course 'emitted beeps' here means imagined beeps. We're not talking about physical experiments or actual sounds. So having beeps or having a monitor representing numbers - that's all the same, and we'll consider anything to be possible in this realm except what is self-contradictory.]

In other words, it's true that no fixed-width monitor (or counter) will do for this (imagined) experiment. But we can say: Indeed, and that is exactly what we mean by "infinite" number! What's not clear? We will need a counter that can also keep growing wider indefinitely. Physically impossible, but perfectly imaginable -- conceivable without running into inconsistencies, at least, so it would seem, so far. We can count, cannot we? ("Conceivable without any derivable contradictions" is not a simple concept, however.)

Now, you may say: "Wait, but your definition only talks about before the 1 sec mark has been reached. I wanted to know what is the case when the 1 sec mark has been reached (or passed)."

I could dismiss that and say: "Well, we just agreed to imagine a counter that can grow indefinitely. In other words, one that can keep growing. At the 1 sec mark, it will have grown to infinite length. What's the problem?"

But according to some people there is a problem. There is a difference, they say, between a potential infinity (which is covered by the earlier definition) and actual infinity. The question is: Can we imagine an infinity as one completed whole? Is that a coherent concept? Can we conceive of that without running into contradictions? -- I have to confess that I never fully understood or saw this distinction. But see: Infinity.

[We are limited, because we exist as physical creatures. Does this mean what we can imagine -- in the broadest sense -- is also limited? We cannot picture or completely visualize an infinite sequence of 1's, of course, but does this imply that infinity itself is an inconceivable or incoherent concept? If it was, how would we know that we can not completely visualize it? "It" would not have a determinate meaning then.]

You could say: "Well, you just admitted that before the 1 sec has passed the counter will become greater than any fixed number n. This would imply that when the 1 sec mark is reached, we cannot have any particular number n showing in the counter. And I cannot imagine an actual infinite sequence of 1's." To which I could answer: "Ok. Let me try to tickle your imagination. Let's just imagine our counter resets at that point, and let it display a number shown as ω and which represents that countably infinite steps have been done." -- In other words, the relevant question, so far, just seems to be: Can we work with that? -- It turns out we can. Grep for "transfinite ordinals" in Early Set Theory. Quote:

The transfinite ordinals were introduced as new numbers in an important mathematico-philosophical paper of 1883, Grundlagen einer allgemeinen Mannigfaltigkeitslehre (...). Cantor defined them by means of two “generating principles”: the first (1) yields the successor a + 1 for any given number a, while the second (2) stipulates that there is a number b which follows immediately after any given sequence of numbers without a last element. Thus, after all the finite numbers comes, by (2), the first transfinite number, ω (read: omega); and this is followed by ω + 1, ω + 2, …, ω + ω = ω ⋅ 2, …, ω ⋅ n, ω ⋅ n + 1, …, ω², ω² + 1, …, ω^ω, … and so on and on. Whenever a sequence without last element appears, one can go on and, so to say, jump to a higher stage by (2).

---

UPDATE

Thomson's lamp could be seen as the problem of how to extend modular arithmetic to transfinite ordinals. If we look at the sequence, bottom-up as it were, the problem seems indeterminate. It's unclear what (if any) the answer might be. The pattern suggests it must be 0 or 1, but which of the two?

0 % 2 = 0
1 % 2 = 1
2 % 2 = 0
3 % 2 = 1
...
ω % 2 = ???

Could John Conway's surreal numbers come to the rescue? The reference given in @Philomath's answer suggests that there could be two valid answers to ω % 2 = ?, depending on how the switching is done. But this is a bit awkard. We cannot have ω % 2 = 0 and ω % 2 = 1, because then 0 = 1 would make our system crumble and flash out with a "ploof". But perhaps the "on" and "off" state at the end should not be seen as equivalent to the finite numbers 1 and 0?

In Conway's system ω/2 is well-defined. It turns out that

ω = {0, 1, 2, 3, ...|}
ω/2 = {0, 1, 2, 3, ... | ω, ω-1, ω-2, ω-3, ...}

So, whatever ω (mod 2) may be, if we can give it a determinate meaning, we must have

ω/2 ≡ ω (mod 2)

We could leave it at that and say that the lamp at the end is in the ω/2 state. But is that "on" or "off"?

But is the Earman and Norton's construction referenced in @Philomath's answer really more illuminating? With one method of switching we get "on", and with the other we get "off" as end state of the supertask. But that only seems to show what we also already knew:

ω + 1 ≡ ω (mod 2)

(if that is indeed a true modular equivalence; I didn't try to prove it).

I personally prefer ω/2 as answer, but don't know how to map that to the "on" or "off" (or both) demand. Also, it would seem that there are infinitely many solutions for a ≡ ω (mod 2), and a = ω/2 is merely the one born first...?

Answer 3

The machine you have in mind cannot be built, owing to physical limitations. However, your question is essentially a mathematical one, which is about whether you can count the individual elements in a converging sum. You should post it on maths SE. It is not a philosophical question.

Answer 4

From mathematics we know that infinite series of positive numbers may indeed have a finite value, e.g.,

1 +1/2+1/4+…+1/2^**n+ … = 2

But one cannot build a corresponding machine. Because the summands of this series approach zero, and one cannot build a device which operates within an arbitrary small time span. At least beyond the Planck-time the concept of time becomes questionable.

Aside: It is not clear what you mean by "philosophically possible".

Answer 5

I'm going to take answering your question in a reverse fashion.

Does it make sense that a question that looks like a computer science problem, can be solved only by logic?

Absolutely it makes sense. Computer science and the formalization of computation was invented by logicians such as Turing, Church, Kleene, and others. The original definitions of computation are very logical in their nature, and modern computer science has a great infatuation with mathematical logic. For instance, one can conceive of programming languages in terms of algebraic semantics or one can model hardware with a VHDL. Using formal systems and logical transformations on formal statements has been with computer science from the very start of the discipline.

You ask:

Is my reasoning correct?

in regards to some claims like:

If this thing exists, it means that it has beeped infinity times after 1 second(t=1), which doesn't make sense.

Dealing with infinity and infinitesimals often boggles our intuitions, and so we have to use logical possibility as a tool to deal with whether what feels impossible can actually be constructed and show to be consistent. Consider that in calculus, an infinitesimal is a non-zero value that is considered infinitely small. What does it mean to be infinitely small? As others have noted, there are a number of philosophical questions around actual and potential infinity that have been in circulation since the pre-Socratics. The important thing to remember is, just because something feels wrong about an idea doesn't mean it's logically inconsistent.

You ask:

Is creating this machine logically possible?

Do you have a full grasp of logical possibility? In some sense, what is logically possible is a function to your views on what logic is and how it obtains to a problem. Consider WP's article:

Logical possibility refers to a logical proposition that cannot be disproved, using the axioms and rules of a given system of logic. The logical possibility of a proposition will depend upon the system of logic being considered, rather than on the violation of any single rule. Some systems of logic restrict inferences from inconsistent propositions or even allow for true contradictions.

So, what are your views on logic and logic possibility? Are you ruling out dialetheia (SEP)? What are you thoughts on quantum logic and probability (SEP)? Do you believe that logic should adhere to classical conceptions? Without going into details, it's tough to say yes or no with a clear vision of what you mean by 'logically possible'. Ultimately, I would say that philomath's link to the supertask is a good starting place for you to continue your thinking.

Answer 6

A machine like yours, or Thompson's lamp, is not logically impossible without reference to the laws of physics. That is, there is a logically consistent model in which they exist.

Imagine a universe in which the timeline is represented by the half-open interval [0, 1). Then there is no difficulty in answering the question "is the lamp on or off at t = 1?" because the question itself can't be asked of that universe; for that universe, t = 1 isn't a point in time. The logical consistency of this model can also be seen by considering that it is topologically equivalent to a universe where time is represented by [0, ∞), and in which the machine beeps at a fixed interval (say, once every second). The transformation [0, 1) → [0, ∞) only fails to preserve the laws of physics.

Worse still, imagine a lamp which you turn on just once at time t = 0, and then you do not turn it off. Now ask if the lamp is on or off at t = 1. Even this much simpler question cannot be answered by logic alone, without reference to the laws of physics, because there is no logical law which says that a lamp switch at rest will remain at rest when no force acts on it. That is a physical law, specifically Newton's first law. Similarly, there is no logical law which says that a lamp will shine when its switch is in the "on" position, and not when it is in the "off" position. Logic alone really gets you very little.

So in order to reach a contradiction or paradox, you need at least some assumptions about the physical world in which the machine supposedly exists. For instance, a switch or transistor which oscillates at a frequency approaching infinity, would require a power source approaching infinity, since the power of an oscillator is proportional to the square of its frequency. Physically, the energy put into the system would collapse it into a black hole at some time t < 1.

Answer 7

If anyone is interested from a computational perspective:

The I/O is BY FAR the bottleneck. In other words, the simple act of printing a number to the screen will limit this so that you cannot print more than X numbers per second where X is relatively small compared to, say, the largest integer that can fit in computer memory, or even in a single register on the CPU.

You can check this out pretty easily. Copy and paste this into your browser terminal (not tested, I'm on my phone...):

function printWaitGeometric(n) {
    console.log(n);
    setTimeout(() => 
        printWaitGeometric(n+1), 
        1000*((1/2)**n)
    ); // Wait 0.5, 0.25, 0.125,...
}
printWaitGeometric(1);

What will happen is that once the time between print calls reaches a certain small size, it is dwarfed by the time needed to execute the print operation itself, so you actually end up calling print operations every tenth millisecond (not measured) or so. Since the time between print calls is not vanishingly small, then these add up to an infinite amount of time and the program runs forever.

This sort of thing is actually why Mathematics talks often about "limits" when dealing with infinity, because it is nice to be able to examine "infinite" operations in a finite context.

Of course, one could opt not to print the result, at which point memory access and CPU flops would become the bottleneck. At some point, though, the physical limitations of the computer will turn your "theoretically terminates in 1s" program into a "never terminates" program.

I realize that this is maybe not what the OP was exactly looking for, but possibly helps inform the problem.

Answer 8

Similarly to Him's answer, I want to emphasize that in each cycle your mechanism is doing something, specifically making a sound. Even if the sound could be arbitrarily short, its production would take finite, constant time: A (possibly electronic) switch must flip, electricity must flow along a wire, a switch must flip back. This finite, constant time of doing something will always result in your machine to run eternally.

This is the philosophical part, if you will: Doing something, the notion of a process, implies the passage of time: procedere, to go forward.

If, by contrast, we allow each act to degenerate to something that needs no time at all, the only possibility is that nothing at all happens. And yes, it is possible to construct a machine that repeatedly does absolutely nothing, between the intervals you describe. I can even construct it in my head. I have just let it run, and it ran for precisely one second.

Answer 9

In computer engineering perspective, the matter is about the master clock of the hardware computer. Think of the master clock as a super universe tick. It's the CPU clock speed like 3GHz.

So, no matter how your software or program (your own universe inside a super universe) uses an undelayed infinity loop in the computer with a variable++ counter to make your program fast. It's still bounded by the master universe.

The interesting fact is, your own universe clock tick perceives the super universe clock tick as a quantum realm.

If you ever do delta-time programming, you will notice that, when you fetch the timer from the master universe (what is the current nanoseconds since epoch?) and you fetch again, there is a non-deterministic difference.

If you deep diver, fetching timestamp counter from direct CPU in x86/64 assembly level, the rdstsc. Here, instead of returning physical unit such as nanoseconds, the instruction returning unsigned integer 64-bit of the clock tick unit since reboot, where the least significant bit (LSB) part has the highest clock tick rate. This instruction usually used for timestamp difference. So, if you fetch the instruction, the LSB might return 0 or 1 in unordered manner. Guess what, your program (universe) perceive that as uncertainty.

Your 1st universe:

• 1st fetch: 0
• 2nd fetch: 1
• 3rd fetch: 1
• 4th fetch: 0

Your 2nd universe:

• 1st fetch: 1
• 2nd fetch: 0
• 3rd fetch: 0
• 4th fetch: 1

Your 3rd universe:

• 1st fetch: 1
• 2nd fetch: 1
• 3rd fetch: 0
• 4th fetch: 0

Regardless, both program perceiving the instruction as probability of approximately 50% between state 0 and state 1. This is known as superposition in quantum mechanic, which means you don't know the current state of LSB until you fetch it.

Here, how you own program (your universe) fetch instruction from CPU (super universe) is analog to measurement in quantum mechanic.

So, to answer your question, the final answer is, the beep sound tick rate will be synchronized with super universe clock tick rate.

For the monitor, it will returning probability of the beep occured. It's depends how you measure, falling edge or rising edge.

Answer 10

[Assuming an infinitely fast computer, breaking electronic physics; but permitting auditory physics to apply]

Let us first define terms.

A "beep" is an audio tone played for non-zero time.

Let's say 1/1024th of a second.

That means that after only ten halvings of the time, the beeps will start to play over each other.

What happens next depends on what you expect to happen. Do two overlaid beeps simply turn into one slightly-longer beep? Or does the volume increase to the sum of the volumes of all played beeps?

1. If the former, then the machine plays ten beep sounds. The first to ninth are the same length, the tenth is twice as long, and ends at t = 1 + (1/1024) second. This version of the machine is entirely logically possible.

In the latter, you need to figure out how you "sum" volumes.

2a) If the volume asymptotically approached a maximum volume with each additional beep, then it would behave exactly as described above, except the last beep's volume would rise from one-bee's volume, to the maximum volume. This version of the machine is also entirely logically possible.

2b) If, with each additional beep, the volume increased linearly, then unfortunately, you can't have infinite volume, as this would require particles of the medium the sound passes through to move faster than light. More importantly, they would also need to move faster than sound. Since the criteria require a physical event (beep), we can assume that physics applies to the beep, in which case, not logically possible.

So, yes, unless you require infinite volume, this is entirely possible, assuming that the computer is either infinitely fast, or can extrapolate suitably to create the same output as if it were infinitely fast.

Answer 11

Well, if I were going to approach this philosophically, then I'd say we're talking about a system that is essentially a quantum superposition: both beeping and not beeping until we make an observation of it. We alternate between sampling a beep observation and sampling non-beep observations with ever-increasing frequency, until at time t=1 we are left with the superposition itself, as though the system returned to an unobserved state.

This question seems paradoxical because it couches itself in terms of action (making a sound), which is inherently time-dependent. Observations are momentary (time-independent), so — if we can accept superpositions — the paradox disappears.

Answer 12

The fastest you can do something is the planck frequency, 5 times 10 to the 41 times per second.

As a computer science problem it's completely legitimate, although it's really in the field of Number theory.

The paradox of a function reaching an infinite frequency in a finite time has a name, but I don't remember it.