Is there any simple way to understand it? If our universe was a free fall model would the planet fall for an eternity? And if it isn’t infinite what lies outside the boundary is there any research to study around it
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Infinity is a place holder, its a way to describe a potential value beyond our ability to calculate. Or something that has an expanding value which appears to never have an end, or an end we dont yet know.
Infinity is a mathematical concept of a number so large it cannot be quantified. It goes on forever. This is separate than what you ask in the description. Which is to ask if the universe is infinite, and how that could be.
Assuming the universe is infinite, the earth will not “fall” anywhere. All matter in the universe is constantly moving away from all other matter. Only gravity keeps things together. Eventually, all matter will spread out and cool off. The universe will “burn out” and die a cold quiet death.
There are other asymptotes that might be easier to explain. Looking at a graph of 1/x as x gets near 0. The value becomes very large, but it’s hard to put a specific value on it. We can use ∞ for that.
I’ll try to give an answer that will help you understand the concept. I don’t think a 5 year old would get this though.
There are only so many ways that you can organize protons, neutrons, and electrons. That is the building block of all matter. Basically you have 3 legos that are really tiny and can only fit together in a certain way. Everything is built out of those 3 pieces. Because the number of ways to build matter are finite, there essentially must be an infinite number of Milky Way galaxies with every atom in the same spot as our galaxy if the universe is infinite (and the laws of physics don’t change in infinite ways or something silly).
And if the universe is actually infinite, we can estimate the average distance between our galaxy and the next identical galaxy. That number is something like 10 to the power of 10 to the power of 26 light years. Something absurd and too big to really fathom. But because the universe is infinite, there would still be infinite identical Milky Way galaxies with infinite me’s typing this to infinite you’s.
In addition to those identical Milky Way galaxies, there will be more infinite numbers of galaxies with 1 atom difference. And more with a million atoms difference. There are also infinite numbers of them, but there are MORE of them even though both have infinite numbers. It is a more common infinite to find milky ways with some differences than ours.
But there are still infinite numbers of our exact galaxy, or even our exact known universe. That is, if the universe is infinite and the laws of physics are the same. Which we don’t know.
Infinity is anything bigger than the biggest thing anyone can observe or perceive.
As soon as you observe it to be a value, infinity gets further away, at an infinite distance away each time.
The thing is there isn’t just one notion of infinity, there’s lots of different infinities. Some are “numbers” in a sense, and others aren’t.
Probably the first time you’ll be introduced to infinity is in the context of a *limit* when you first encounter calculus. This is the classic ∞, and in this context infinity sort of represents a direction: saying something “tends to infinity” is shorthand for saying it’s eventually larger than any natural number. This is probably also the notion of infinity you think of when you try to apply infinite quantities to physics, such as when you consider infinite density, or an infinite universe.
This is also the infinity people talk about when they say “infinity isn’t a number, it’s a concept”. They’re right, in a sense, but it’s misleading. There are plenty of infinite numbers (which I’ll get into below) it just turns out that you can’t formalize this specific notion of infinity as something resembling a “number”. What’s important for the moment is that this infinity represents a sort of “upper limit” to the natural numbers.
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The next place mathematics student will encounter infinity is in the formal treatment of infinite sets. We know that the natural numbers, rational numbers, real numbers, complex numbers, etc. are infinite sets, but is there a way to formalize a notion of “size” for them? Well, what mathematicians did was introduce the idea of *cardinality*, which is a number attached to a set that measures its size. For finite sets, this is just the number of elements it has.
The cardinality set of natural numbers is the smallest infinite cardinal, and we say that two sets have the same cardinality if you can pair up their elements so that every element in the first set gets paired with a unique element in the second set. Counterintuitively in this sense the set of natural numbers and the set of even natural numbers are the same size, as you can pair them up like: (1,2), (2,4), (3,6),… and you can check for yourself no element from either set gets left unpaired. From this, you can show that the natural numbers, the integers, and even the rational numbers are all the same size. The set of real numbers, on the other hand, is a larger set, and is associated with a larger cardinal. The set of possible functions from the real numbers to the real numbers is a larger cardinal still.
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Now, where does our previous notion of infinity, (∞) fit in with all this? Well let’s briefly talk about *ordinals*.
You see, in set theory, we *define* the natural number zero to be the empty set, {}. Then, we can recursively define what “1”, “2”, “3”,… are as follows
– 1 = {0} = { {} }
– 2 = {0,1} = { {}, { {} } }
– 3 = {0,1,2} =…
and so on. This gives you a way to recursively construct the natural numbers with just the empty set as a fundamental building block. It’s then pretty easy to define arithmetic operations on these so they work just the way we expect.
Now ordinals are numbers that have representations like this, as sets of smaller ordinals. Now notice, for example, that the cardinality of the ordinal 3 is also 3. In fact, every natural number is an ordinal, i.e a set like above with a cardinality equal to its ordinal value.
So… are there ordinals with *infinite* cardinalities? Of course! In fact, every finite or infinite cardinal can be associated with an ordinal. We call the smallest infinite ordinal ω, and it’s cardinality is that of the set of natural numbers.
However, just like the finite ordinals don’t remotely cover every finite real number, you can constrict the *surreal* numbers that contain these infinite ordinals, but also things like log(ω), ω^π, and √ω. The surreal numbers are *huge*. (too large to be a set, in fact)
Which brings it back around: just as ∞ is the *upper limit* of the real numbers, it can be thought of as the *lower limit* of the infinite surreal numbers. It represents a (nonexistent) quantity larger than every real number but smaller than every infinite hyperreal. In this sense, you can think of the familiar concept ∞ as the *gap* between finite and infinite.
Infinity, everything, all, are the opposites of zero, nothing, none.
Aaaand since it was bonked by the anti-minimalist-bot, I’ll repost it.
A mathematical concept. Just like a perfect circle, it doesn’t exist.
Imagine a few ducks in a row. Congrats, you’ve got your ducks in a row. Now imagine more. Now imagine that row of ducks just never ending. Ever. That’s infinity. It’s impossible to have that many ducks, sure, but a lot of thing in math are impossible. “3” is possible, you can have 3 of something. You can multiply something. Bacteria multiplies. But a perfect mathematical circle is impossible for anything in reality. It’s an idea. Lots of things get close. But nothing can exactly fit.
>If our universe was a free fall model
hehe, wut? Like a universal “down” gravity?
>If our universe was a free fall model would the planet fall for an eternity?
Sure. I mean, essentially it already is. If there was some cosmic constant “downward” force. …We wouldn’t notice. Planet Earth is mostly in free-fall, with the sun pulling it around in an orbit.
>And if it isn’t infinite what lies outside the boundary is there any research to study around it
We’re pretty sure the universe stretches infinitely in every direction, and has done so even a nano-second after the big bang. There is no boundary.
Yeah, infinity just means that something doesn’t have an upper limit, it just keeps going forever. So yes, if the earth were to have some weird downward acceleration applied to it, it would just keep going forever. Empty space is infinite. It might be helpful to think of infinity not as an amount of something but rather as a lack of a limit, it’s unlimited.
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