# If something takes forever, will it be done in infinite time?

Agnishom: I am trying to post a problem but it is taking forever

Sandeep: (quoting Swami Vivekanand Ji) Have infinite patience and success is yours

Note by Calvin Lin
6 years, 1 month ago

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We have to define what "taking forever" means. Does rolling a $7$ on a six-sided die take forever? Or does getting $0$ by subtracting successive powers of $0.5$ from $1$ take forever? These two events will yield "not done" and "done", respectively.

But by defining "taking forever" rigorously as being accomplished after infinite time, we have just answered our own question, for "taking forever" and "done in infinite time" are equivalent statements.

- 6 years, 1 month ago

I'm not sure if having "infinite" patience, whatever the definition, ordinal, etc., is always a practical, or even healthy, state of mind. There comes a point where it's better to cut one's losses and just move on, or at the very least set the present task aside and focus on something different. Frustration can be counterproductive.

- 6 years, 1 month ago

Actually, "Have infinite patience and success is yours" is a quote by Swami Vivekanand Ji.

- 6 years, 1 month ago

Nope, if something takes forever then it will never be done.. Not even with infinite time..

- 6 years ago

As Daniel correctly points out, this question hinges on the meaning of "takes forever". We could draw different conclusions depending on the exact meaning or definition of "takes forever". For example, suppose we are tasked with counting the number of a particularly special sub-atomic particle, theoretically only just one, not only in our universe but in all the other universes in the multiverse, past or present. The theory is confirmed if we counted just one, so this would be "something that takes forever" to do. Can it be completed in infinite time? That depends on whether or not success is based on finding exactly one, or success is merely confirming or disproving the theory.

What's worse, many multiverse theories involve scenarios where time does not have any meaning, so even the concept of a task taking any time, finite or forever, becomes moot, thus rendering the question moot as well. We are already making a number of unstated assumptions in just speaking about any "infinite time".

If we restrict such tasks to more prosaic ones that can be defined more precisely in mathematical terms, such as "what is the last digit of the digit sum of the decimal form of $\pi?$", then we have a better chance of giving meaning to questions like, "will it be done in infinite time?" Then it could depend on whether or not finding that last digit of that digit sum has any meaning. Does it?

It should be pointed out that mathematics doesn't need to have any concept of time, unlike in science where the concept of time is useful (but merely useful and not necessarily a fact).

- 6 years, 1 month ago

Let the amount of things needed to get done at a time $t$ be $f(t)$ where $1$ is everything is not done and $0$ is everything is done.

No matter what finite time $t$ you choose, I can choose a positive real $\varepsilon$ such that $f(t) > \varepsilon$, so $\text{doneness of }f(t) = \left\{ \begin{array}{l} \text{not done}\quad \text{if }t \text{ is finite}\\ \text{done}\quad \text{if }t\text{ is not finite} \end{array}\right.$

So even though $\lim_{t\to\infty}f(t)=0$ we still have $\lim_{t\to\infty}\text{doneness of }f(t)=\text{not done}$

- 6 years, 1 month ago

But tending to infinity is not infinity.

Consider this statement:

The intersection of sets $\cap_{n=1} ^ N ( 0, \frac{1}{n} )$ is non-empty.

This statement is true for all finite N.

However, it is not true in the infinite case.

The intersection of sets $\cap_{n=1} ^ \infty ( 0, \frac{1}{n} )$ is empty.

IE the limiting of "non-empty" becomes "empty".

Staff - 6 years, 1 month ago

No, because forever itself is limited but its limitations is beyond our existence

- 6 years ago

We really need to have Brilliant discussion on this... Interesting topic.

- 6 years, 1 month ago

Well, it all depends on your definitions. What do we define as "time"? Are we talking about our, physical Universe, or a purely mathematical one? With the former I can conjure several answers, all quite simple. With the latter, well, I don't know. I guess more defining needs to be done. Also, is analysis confined to the fourth-dimensional system? I.e. - 3 spatial dimensions and one temporal one? Because if I take this to time-space instead of space-time this is a no-brainer.

- 6 years, 1 month ago

This note reminds me of Infinite monkey theorem. Now, how do we make the distinction between "almost surely" and "surely"?

On the other hand, this also reminds me of a scene in Spongebob Squarepants where the narrator says "One eternity later". Is there such a thing as an event after an infinite time?

- 6 years ago

This reminds me of the Cyndi Lauper song "Time After Time". I'm sure that she had the Gödel metric in mind when she wrote this piece, such that time is "caught up in circles". :)

- 6 years ago

This also reminds me of the "Is there a god?" note posted on brilliant. But I think that the discussion about anything being endless is endless (figuratively).

- 6 years ago

This is a truly interesting topic, the answer to this question depends on the definition of " infinite time " and " taking forever" as cases can be derived from that simple yet large difference in definitions. For instance, we name whatever goes out of our range "infinite" yet it can actually refer to very large numbers, and in this sense it will be considered finite in a way or another. If something takes " infinite time " it will still rely on the definition of " infinite", if it's something that no matter how much time you give it it will still need more that renders it to be impossible, if it's impossible then it won't be done even if you give it " infinite time ". If the thing itself can be done yet progresses very slowly( close to 0), then it will at some point be done if it's given "infinite "time to be done, for example: if we define infinity as never ending then at the end, if the thing itself CAN be done then it will be done.

- 6 years ago

How exactly are we defining "infinite time" here ?

- 6 years, 1 month ago

That is a good question. The problem takes forever (as opposed to infinite time, not sure if there is a difference), and we have infinite patience.

Does it matter what ordinal of infinite they are?

Staff - 6 years, 1 month ago

Frankly, I'm not entirely sure. The term "infinite time" is very misleading to me. Basically, the term "infinite" here needs to be more explicitly defined. Infinity is not a real value, hence what do we actually mean by "infinite time" ?

If we think of "infinite time" as a limiting case, we can use Daniel's argument (although I'm not sure if it's rigorous enough) to conclude that the work won't be done.

If we think of "infinite time" by its literal meaning, this question sounds like a paradoxical one to me. If a work is "done", the time in which it was done should be measurable / finite which contradicts "infinite time". Then again, if we have "infinite time", one might argue that there can be no work that won't be done in infinite time.

Also, regarding your last question about the ordinal, again, I'm not entirely sure if it matters.

My argument would be that no conclusion can be made without properly defining "infinite time".

This is why I sometimes hate mixture of maths and philosophy. It blows my mind. ;_;

- 6 years, 1 month ago

this yields to one result that the thing can't be done because as Agnishom said taking forever is that either we are talking about weeks or that his internet is down and here infinite would mean if connection is down thus problem wont be posted this is a practical approach

- 6 years ago

Infinity is something that's created by man. Just like 0. Nothing is truly absolute. But then again, what marks the duration of a second? Can you say with 100% certainty that so-and-so many transitions have taken place in a Cs$^{133}$ atom? No, you cannot. Also, because the definition is for a Cs$^{133}$ atom at 0 K, and because you can't reach 0 K in a finite number of steps, the duration of a second is not defined! So in a way, one second lasts for an infinite amount of time.

So let's not bother about something immeasurable and get back to solving and creating problems!

- 6 years ago

I'll let you know when I get there : D

- 6 years ago

Thinking about infinite has propelled me to ponder on the fact that the universe does not run on our terms and conditions, i.e. We infer our universe through the numbers which are actually immaterial to the universe itself. Here, infinite is the result of the discovery of zero, which again is delicately balanced over the development of our civilisation and its thinking...So,we can talk about infinite only if it is in the scope of Human Inference...which I think is not...

- 6 years ago

A quote to reflect on before seriously considering these types of problems: "IS PONDERING ABOUT INFINITY WORTH YOUR FINITE TIME?"

- 6 years ago

Infinity is a concept that plays a significant role in mathematics, so I think it's worth [our] finite time to think about it. But, unlike much of philosophy, in mathematics we can define for ourselves what infinity is, and what would be the properties or consequences of it, and, moreover, even have a number of different interpretations of it, making for slightly different branches of mathematics, much like non-Euclidean geometries. It is not a waste of time to think about infinity, it can be an useful tool. Like the concept of time, another useful tool, but not absolutely necessary to have either in mathematics or physics.

- 6 years ago

This note was supposed to be a joke, right? That's what I thought when I first read it. After all those years in school when the limit of a function 'tended' to infinity, and my teacher yelled, "IT DOES NOT EXIST", why this paradigm shift to seriously consider infinity? All books agree that even with inversion, infinity is not tangible. The trend can be observed and may even seem to be the same for really large numbers, but no one knows what goes on at infinity.

In the end, if you look at the comments here (as well those in "Is god real?") you may notice that the first thing argued and left unresolved or assumed and then left unresolved is the definition. "What exactly is the definition of .....?" is the first thing that I saw in replies to both posts. The thing is, to quote my teacher, "It is defined along with things that are measurable or countable, as something that's not measurable or countable." Naturally, my first question when I came across this post was about why we're questioning the definition of the intangibly defined or the contradictorily defined or the freely undefined?

Don't get me wrong; I too love pondering about things that even giants have failed to solve. And I too used to spend time thinking about such things, but I soon realised with dismay that I was not carpe-ing my diems. That's why I made this rule to remind me of reality whenever my head is stuck in the abstract clouds: "Philosophy can come after retirement."

- 6 years ago

May be this note was supposed to be a joke. After all when Something takes Forever, why interfere, let Something take Forever. May be Forever secretly wants to be taken by Something. We never know!

May be this note was supposed to be anything but a joke. The speed at which some people replied was impressive. Their reply was no joke and they could look beyond the "word-play". They could make out what the post really wanted to discuss about. And also found gaps without filling which it may be difficult to make any progress. Prasun's comment: "what do we actually mean by "infinite time" ?" indicates that infinite "time" is more troublesome than infinity. John's comment: "What do we define as "time"? Are we talking about our, physical Universe, or a purely mathematical one?" tells that definitions are context dependent. What is the context in this post? Physical? Mathematical?Psychological? (or worse) philosophical? We need to explicitly state it. Michael's comment: ..."We could draw different conclusions depending on the exact meaning or definition of "takes forever"..." makes things more colorful by providing some nice examples and elaborations. It was like really fast...What do these people eat in their breakfast? Intel?

My teacher also used to tell that "tends' to infinity"..."IT DOES NOT EXIST" or something like that (I never paid that much attention :) ). But you know, after reading your comment, I think George Cantor must have really wasted so much of his time. Only if he went to any of the schools in which we studied he could have saved so much of time by never bothering about infinity and cardinals and what not.

I also agree with you about intangible things...I think you have convinced me that I must not waste time by studying Mathematics. After all the whole Mathematics is so intangible! And it brags about it in its "abstractions". Leave infinity, Consider the number $2$. I have seen boys having $2$ girlfriends (god knows how they manage that), but I have never seen the number $2$ itself! I am so accustomed to its frequent use that I almost forget that $2$ is a mythical creature and not "real". Damn!

"what is reality? Something that we can perceive?"

"Well , it depends on its definition "

:)

- 6 years ago

Yeah, every boy has two hands (one left, one right). Hence, 2 girlfriends (if you know what I mean, huehuehue). :3

- 6 years ago

That's a very extreme special case you are talking about :D

- 6 years ago

I like that, "I've seen boys having $2$ girlfriends [...], but I've never seen the number $2$ itself!". Maybe $2$ is imaginary? Oh, wait, $2$ is a complex number, which, as we know, is imaginary. See my response to Vishnu

- 6 years ago

I Agree with shaping concepts to be useful and having more than one useful interpretation. We interpret same things differently even when we generalize from special cases.Complex numbers are good examples of shaped concepts.

I think we can't use 'infinite' as freely as 'infinity'. A set having infinite members makes sense. Can't say same about "infinite time" (as pointed out by Prasun)

Without infinity how would physics behave? I guess not abnormally :) But really what if we remove some Mathematical concepts from Physics...

- 6 years ago

One of the most fascinating things about mathematics is that we seem to have the power to shape concepts in it, and yet...and yet!....there are limits just how far we can go in shaping things to our desires. For example, after the great success of the development of the idea of complex numbers, mathematicians, quite naturally, wanted to extend the idea in three dimensional form. Alas! It took the better part of 19th century for mathematicians to discover that it's not possible, and the best that can be done is quaternions. Yes, quaternions are much like complex numbers, but nevertheless differ in a number of ways. Lots of interesting mathematics had to be developed, including group theory, before this was realized.

- 6 years ago

Hey! A COMPLEX NUMBER NEED NOT BE IMAGINARY!!! Humans thought of quantifying different objects and came up with numbers as the solution to the problem. By giving each number a symbol, they were materializing something that was in their heads! They had something in their minds and boom! it was there in front of them, on paper. That's why the first chapter in any course of elementary physics is Units and Dimensions. Simply saying, "I can bench 50" has no meaning. What units are you using to measure? Pounds, grams, kilograms or ounces?

Have you noticed how bigamy is taboo but bi-ami, having two "friends", is not? What's the difference? Two marriage contracts? Objectification is ruining an entire generation (sigh).

- 6 years ago

More than one time I can remember, a native sculptor working on a block has spoken of "freeing the [animal] form from the stone", as if his work is about removing the material that entombs his subject, rather than seeing it as an act of creation. Is mathematics about "freeing the form", or about creation? How much of mathematics is arbitrary creation? I argue that a lot of it is not arbitrary. Try to create a regular polyhedron that is not one of the 5 classic Euclidean regular polyhedra, and see how arbitrary creation can be in mathematics.

- 6 years ago

That's why the difference between a PhD in Theoretical Mathematics and a large pepperoni pizza is that the latter can feed a family of four! XD

- 6 years ago

My response about infinity being a concept was actually an effort to move the subject away from "retirement armchair philosophy", which seems to be also perpetually asking, "Is God Real?". I was arguing against wasting time on "finding out what is infinity", saying we should merely see the concept of infinity as an "useful tool", that we can more or less shape to our needs in mathematics. And there can be more than one useful interpretation of it.

Here's food for thought: Is there anything in physics that demands the existence of infinity? That is, let's imagine we have two universes that look alike, except that one is truly infinite in some way, and the other merely "very large". Can we ever detect the difference? To put it in another way, what if we eliminated the concept of infinity from the mathematics of physics, substituting in its place "a very large quantity". Besides the inconvenience of having to do the math in that way, would we find a difference on how physics behaves?

- 6 years ago

In my experience, whenever infinity comes up as a solution to a problem, it usually implies something else like "IT DOES NOT EXIST" or, "IT NEEDS TO BE AS BIG AS IT CAN", etc and it is that implication, which is usually finite, that is used in practical applications. I love the abstract things in mathematics and how the human mind continues to create something from nothing, but sometimes, those things can also have no actual coherence to reality.

I didn't get the message of 'infinity is not a retirement armchair philosophy' from your reply. Sorry about that. But like I said, the concept of infinity has never directly been analysed in detail, has it? Even in your comment about shaping new useful concepts, you mentioned quaternions and group theory. But thousands of years since its discovery, the concept of infinity is used indirectly as a marker for an implication. And as far as infinity is being considered, and NOT ALL ABSTRACT CONCEPTS IN MATHEMATICS, which is the case I've assumed in my argument about this note, I think that pondering about what comes after infinite time or other similar things, is highly analogous to trying to count up to it. I wrote the response with the quote because I felt that people were literally trying to think of something that comes after infinite time.

But a good debate all the same :D

- 6 years ago

What I am saying is that while, yes, people have wondered about "infinity" for thousands of years, it's only recently that mathematics have attempted to deal with it rigorously. That's what I meant by "moving the subject away from retirement armchair philosophy". But, in dealing with the concept of infinity "rigorously", we might discover that there are more than one kind of infinity. Cantor, you know?

- 6 years ago

A well defined collection of distinct objects is called set. What important factor in this definition is ensured by the term "well defined," collection? I mean what is meant ( mathematically) by saying well defined collection?

- 6 years ago

No. The term "Forever" may not necessarily mean never ending; it depends on how it is used. It can be applied when something is taking a long time and the person's patience it dwindling. E.g. 'Uploading this video is taking forever'

Infinite can also just represent an immense number. A calculator (A scientific one) can't display a number say 999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999......................................................................... and so on until there are 2000000 9s.

- 6 years ago

It will forever be approaching completion. :D

- 6 years ago

It maybe true sometimes, or may not be depending on the situation.

- 6 years ago

It depends on the type of process which refers to something. If it is a radioactive decay it may take infinite time but if it is doing something like solving a problem it can be done in someone's half life.

- 6 years ago

Time is a storm in which we all are lost

- 6 years ago

Yes, because both of the statements are equally possible and impossible. In fact we can say that beginning from "forever" is an infinite time of something and while we don't really know what's the end of "Infinite" we can say that the statement "Something that takes forever, it's done in infinite time" it's a valid, possible and correct assimilation.

- 6 years ago

I feel like the answer is "yes" since forever kind of counts as infinite time.

- 6 years ago

If something is done in infinite time, then we cannot measure it. We cannot measure the infinite, so therefore we cannot measure the completion of that which is measured in infinite time.

- 6 years ago

Yes

- 6 years ago

Assuming "takes forever" is describing the duration an action will take to accomplish an end, then no it will not be accomplished, even in an infinite amount of time.

- 6 years ago

fine but what can be its answer ans . is no.... infinite time= forever they will be crossed and the work wont be done but nothing is impossible I M Possible

- 6 years ago

taking forever means takes until time ends. when you introduce infinite time to this, you have no "end of time", so takes forever becomes meaningless

- 6 years ago

No, It will never be done. It takes forever.

Infinite is not yet defined, and we know that nothing takes forever, even loading a ticket on IRCTC with BSNL as a network.... so,no, of anything takes forever,which I am sure it doesn't, won't take infinite time.

- 6 years ago

It's already done, it just hasn't happened yet(under a solar clock cycle) is very similar to, but not the same as, we've already been here before , you just haven't remembered how to get there yet, again(cadence of Plato's Cave).

- 1 year ago

It's just my assumption... I think that there are some quantities like infinity, division by zero, etc. which are kind of intrinsic and for know are difficult to explain. Taking forever, to me, is not the same as infinity. We are in the same condition as the people were in earlier times where they jumped to conclusions about things they didn't know. I have come across many who use such intrinsic quantities without knowing them. We have to wait till some prodigy or so comes up and clarifies our misconceptions about these things. Until then we can not say that forever is infinity and never is zero. In addition to this, the indeterminate operations on zero and infinity support the above mentioned. (before any further criticisms, do rethink on the issue and take into consideration my limited knowledge)

What if it GIVES forever? Then the question becomes whether or not altruism > selfishness. We've all heard the theorem given by Christmas television programs, "It's better to give than to receive." This suggests that altruism is, in fact, greater than selfishness, thus giving "taking forever" a lesser value than "giving forever." My answer is "No," it would not be done in infinite time.

- 6 years ago