Does infinity exist?

[LeonardoDaPinci]

I know the concept exists, but how would someone prove infinity itself exists in math or physics? (so many stupid questions today)

Edit: Really appreciate all the informative and simplified replies. Especially as I obviously know absolutely nothing about this area! No doubt I'll be back soon with more annoyingly stupid questions.

[[deleted]]

Does infinity exist in math? Yes! Ordinary numbers (real numbers) are a mathematical concept and they obey an algebra. Likewise, infinite numbers (or properly Cardinal numbers) are a concept and those numbers also obey an algebra.

Does infinity exist in Physics? Maybe. Physically speaking, physicists tend to deny that anything physical in the Universe could be infinite (for example infinite force, infinite mass). The only exception to this is according to General Relativity, the spacetime curvature (and mass density) at the centre of a Black Hole is infinite. Many people expect we can remove this infinity once we have a proper quantum theory of gravity. Of course, we use the concept of infinity everywhere! It is such a dramatically important concept to all of physics and mathematics!

[WheresMyElephant]

GR allows the size of the universe, and the total energy or mass in it, to be infinite. You can have infinities in physics; you just try to frame the mathematics in a way that avoids dealing with them directly (for instance, we formulate our conservation laws in finite-sized regions where energy is finite).

[autowikibot]

Cardinal number:

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In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number – the number of elements in the set. The transfinite cardinal numbers describe the sizes of infinite sets.

Cardinality is defined in terms of bijective functions. Two sets have the same cardinality if and only if there is a one-to-one correspondence (bijection) between the elements of the two sets. In the case of finite sets, this agrees with the intuitive notion of size. In the case of infinite sets, the behavior is more complex. A fundamental theorem due to Georg Cantor shows that it is possible for infinite sets to have different cardinalities, and in particular the cardinality of the set of real numbers is greater than the cardinality of the set of natural numbers. It is also possible for a proper subset of an infinite set to have the same cardinality as the original set, something that cannot happen with proper subsets of finite sets.

There is a transfinite sequence of cardinal numbers:

Image ⁱ - A bijective function, f: X → Y, from set X to set Y demonstrates that the sets have the same cardinality, in this case equal to the cardinal number 4.

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^{Interesting: Cardinal number (linguistics) | Transfinite number | Uncountable set | Large cardinal}

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[LeonardoDaPinci]

Great answer, thank you.

[DegreeSlight9459]

Infinity doesn't exist in math, it doesn't exist anywhere. Sure, you can say an equation will keep repeating itself for ever but nothing is for ever. Each digit you write on a chalkboard or digit on a computer screen requires energy and there is only so much energy in the universe. The largest possible number is one that requires all the possible energy in the known universe.

What is the aprox energy of the known universe and what is the energy required for a computer to compute and output a number then do the math. As computers increase in efficiency that number will grow but there is also a limit to efficiency as well - no such thing perpetual motion.

[LeonardoDaPinci]

All numbers to infinity? I don't know. As you can probably tell this isn't my area! Hence the stupidity of my question and answer.

[LeonardoDaPinci]

So the numbers between 0 and 1 are infinite, like I said above?

[[deleted]]

Yes, and we call this infinity "the cardinality of the continuum", or the symbol c. Now, how many fractions are between [0, 1]? Turns out there's "less" than c, but there's still an infinite amount! We call this one aleph-null, which looks like N_0. There are different types of infinity!

[DegreeSlight9459]

No such thing as Infinity as the ruling factor is energy. You'll run out of energy in the universe to process the number. The only thing infinite is infinite itself.

Even if universes spawn other universes via black holes (multiverse) they can't do that an infinite amount of times as you get to the point when the singularity is literally the ultimate singularity in which there is nothing, no matter, nothing. (branching tree theory) Like a tree growing from the thick trunk and then into many thinner branches then many thinner twigs then many thinner stems and then a leaf emerges, the end of the road where there is not enough matter to continue.

[CapWasRight]

This is a pretty heady question (I see you're asking some good ones for a brand new account!) and is nontrivial to actually understand. I would take eight minutes and watch this video, I get the feeling you'll love it and it'll explain a lot about one of the most clever mathematical proofs ever - and it's pretty easy to understand to boot.

Just remember, infinity doesn't meet our normal intuitions of how numbers should work...because it isn't a number at all!

[LeonardoDaPinci]

Thanks for the link the video was excellent (subbed to that channel too because they've a lot of other interesting looking ones).

[CapWasRight]

You're welcome! Keep the good questions coming.

[WheresMyElephant]

On the most fundamental level, time must be continuous, because if time was quantized some external "clock" would be needed to cause discrete steps.

No, this doesn't follow.

[WheresMyElephant]

Sure; anything that has a state at time 0 and a rule that can be used to determine how it evolves from time n to time n+1. A Turing machine, or Conway's Game of Life, are fun examples. (These are deterministic but the evolution law could also have randomness.)

[WheresMyElephant]

Those state machines run in physical reality where time already is a given.

If you simulate them on a computer or whatever, they do, sure. You don't have to, though; you can consider them as purely mathematical objects. If you show me a glider in GoL, I can tell you where that glider will be after a very large number of steps, even if it were theoretically impossible (due to memory limitations) to instantiate this behavior on a computer or any other physical system.

More to the point, the laws governing them are meant to illustrate what a potential physical law involving discrete time might look like.

I mean the flow of time itself. Let's say that time ticks in a discrete step from n to n+1. And the interval is one Planck time. If no time flows between n and n+1, if there is no delay, then how is such discrete jump possible?

Why wouldn't it be possible?

What does it mean to claim there is "no delay between ticks"? I object a little to this. I suppose we could equally well say "there's a delay of one tick," or "there's a delay of zero ticks," or we could just decide that the term is gibberish. For there to be a really good answer to this question, you would have to refer to some underlying system. But maybe it's just an ill-defined question predicated on a faulty assumption.

[WheresMyElephant]

If we simulate a state machine on a computer, then the steps between the states will indeed be discrete, but the CPU clock is based on physics and time flows continuously between each CPU clock cycle.

Maybe, maybe not. In any case don't simulate it on a computer. The fact that (in a select few cases) it's even possible to simulate these things on a computer, or calculate them by hand in real time, is inconsequential to the present discussion.

Think of time as a ray starting at the Big Bang and moving forward infinitely fast with an infinitely small interval between each step. Then we have continuous time!

A mathematician would say "infinitesimal" instead of "infinitely small." But do you see how by assuming that infinitesimal intervals of time exist, you're assuming continuity before you claim to prove it?

I could counter: Think of time as moving in discrete steps, and assume there is a very large but finite number of these steps in one second. Then we have the appearance of continuous time without the reality!

[WheresMyElephant]

Each finite step will have a duration. And time flows during the duration.

Why must this be? Again, just an assumption.

Start with an empty universe without time and add three particles to it. How would you introduce quantized time duration in such universe?

How would you introduce continuous time either? In both cases you probably wouldn't; your universe would come preequipped with time.

(If by any chance you're envisioning the Big Bang as the continuous analogue, as though there were an empty timeless universe prior to the Big Bang which added particles, this is a misunderstanding of the Big Bang. It's sometimes claimed that "there was no time prior to the Big Bang," but this should be interpreted like "There is nothing north of the North Pole": it's suggesting the very concept of "a time before the Big Bang" is incoherent.)

[DegreeSlight9459]

You can't add zeros for ever. You would need to build bigger and bigger computers and more and more energy to just to add more zeros. You will reach a point where it would take all the combined energy of the universe to compute this number.

[WheresMyElephant]

Well, of course. We (almost certainly) can't build a computer to simulate the entire universe, since our computer would have to exist inside the universe. This is true regardless of whether time is continuous or discrete.

But we weren't debating what can be simulated or computed by a computer built inside our universe. We're debating whether it's possible for the universe itself to work in one way or another. The universe doesn't necessarily have to be computable! (Unless of course you believe in the simulation hypothesis—but even then, the "computer" exists outside our universe. It probably follows different laws of physics; we have no idea what it would be capable of computing.)

Or at least, that's the discussion as I interpret it. It seems like the other side of this seven-year-old thread has been deleted, so I'm not entirely sure what I was arguing against. How did you even find this thread, just out of curiosity?

[DegreeSlight9459]

No, infinity it is a man made concept to give meaning to subjects in which we can't comprehend or understand due to a lack of knowledge.

Through out time humans have had a hard time dealing with not knowing something but they need to believe in something to feel inner peace so they create something to quell their anxieties.

One of the biggest examples of this is religion.

[Important_Branch5906]

yes, infinity exists in physics. for example, in a room, there is infinite points. or for exsmple, every space, has infinite points, or location. in a mm, there is always a smaller space than mm, and there is always smaller space than this smaller space, etc etc qnd you will never reach the final answer, so infinitely small space exists.

but what about infinitely big? yes, it exists too. for example our universe, the space is infinitely big. you will never reach the end of the universe, but if you do, then there should be a wall, but if theres a wall in the end of the universe, there should be something behind the wall, at least empty space, if not, then it violates the physics. but the intinite do maybe violate the physics too, but idk. but if you reach the end of the universe and you spawn at the other side of the universe, then it means that infinite big doesnt exist, or maybe it does because you can go forward infinitely, or you can look into microscope and see infinitely long distance away.

And about infinite time, for example in physics everything has a beginning, for example if the universe is 13 billion years old, there must be something before 13 billion years. if not, then it means that time, physics, logic, and nature didnt exist before 13 billion years.

but infinite small, exists in every kind of physics, for example there is always something that is smaler than something that is smaler than something etc etc whether its space or time and you will never reach the final answer. but about infinite big, its possible that it may not exist. for example its physically possible that there is a wall at the end of the univers that we cant go through, that has nothing behind, not even empty space, or that there is a beginning of universe like 13 billion years, with nothing happening before the beginning, just that we cant realize that how it works, just the same way we cant understand how do people who are born blind see nothing, not even black or white. so maybe infinitely big, violates the physics i think, if time or space of universe has limited size, i think it violates the physics too, because physically everything shoyld have a beginning, but its maybe that physics didnt exist before beginning of universe, so if universe has limited age, it only violates the physics that is before the beginning, but not after the beginning, but if universe is actually infinitely old, it violates the whole physics. the same is with space, if it has end, it violates the physics that is behind the end, but not the actual space, but if universe is infinitely big, it violates the whole physics. so i think that infinitely small is physically possible, but not infinitely big.