So I got curious today about the concept of infinity.
According to any evidence, basically, the space is an infinite entity, and that the galaxy continues to expand into nowhere, demonstrating that there is so much more to be explored.
Yet, such a human concept of Internet which is measured by more than terabytes now is also infinite, yet is existing in our laptops. So the so called cyber-space and the immeasurable infinity of space and time and the conflict about the competition of the superiority creates a natural conundrum of what is bigger?
It is hard to see what is bigger because either are expanding, and the second one is the product of a human mind which is already endless. Just an interesting question which I could not find an answer to so I decided to call up you guys (Armor Squad XD)
Space is not necessarily infinite. As far as I know, the concept of infinity has no place in physics. According to what I thought was the current state of knowledge on the universe, unlike, say, a bubble in water, there simply is no space outside of the universe. The 'bubble' of space-time in which we are is not expanding in empty space, but kind of making the space?
About internet, I would say that as large as the data handled by it may be, it cannot be physically endless because it relies on hardware.
Well, the original question is "What is bigger?" So space, definitely.
Like HahiHa pointed out, the internet relies on hardware. Specifically, it's size is determined by the amount of data. The amount of data possible is in turn determined by the available Bytes (and kB, MB, GB and the rest) of free space on a global level. As such, definitely not infinite
The universe is not infinite either. But a single galaxy is enough to be "bigger" in a sense than the internet.
Internet is like this two (^ up) said phisically on our hardwers and since they are pike many more things part of universe univers is kinda bigger.
But interasting question.
According to any evidence, basically, the space is an infinite entity, and that the galaxy continues to expand into nowhere, [...]
No. One or the other; not both. An infinite universe cannot increase in size, because -- being infinite -- it has no size.
Yet, such a human concept of Internet which is measured by more than terabytes now is also infinite, yet is existing in our laptops.
No.
>1e12 ≠ ∞
It is hard to see what is bigger because either are expanding, and the second one is the product of a human mind which is already endless.
No, actually, it isn't. Bytes are not units of size: They are dimensionless values. There are also stringent limitations on the capacity of the human mind, as any neuroscientist can tell you. Furthermore, all of the internet's physical storage is housed on Earth; a tiny mote somewhere in a small region of space.
According to what I thought was the current state of knowledge on the universe, unlike, say, a bubble in water, there simply is no space outside of the universe. The 'bubble' of space-time in which we are is not expanding in empty space, but kind of making the space?
That's the idea; that or stretching the edges out of highly compacted space, or expanding the three-dimensional "surface are" of an enclosed topography.
An infinite universe cannot increase in size, because -- being infinite -- it has no size.
There are degrees of infinity though. For example, the amount of numbers between 1 and 2 (infinity) is half as much as the amount of numbers between 1 and 3 (also infinity).
There are degrees of infinity though. For example, the amount of numbers between 1 and 2 (infinity) is half as much as the amount of numbers between 1 and 3 (also infinity).
No. It is not half. They are both infinite. After that, it doesn't matter.
Basically infinity can be defined as the end of a field or a total, a point where that field or total's properties no longer exist. For a tiny (νm) electrically charged object, an infinite distance could even be 1 metre because that's about where the electric field "stops" in a way. Of course it is all depending on a lot of things, so this may be a bad example but basically that's how it works
So, even though the numbers between 1 and 2 are infinite and the numbers between 1 and 3 are also infinite, you cannot talk about halfs or anything else for that matter, because infinity is the point where the total of all numbers ends (something we cannot really visualize, only imagine).
@Doombreed will you at least concede that there are more numbers between 1 and 3 than there are between 1 and 2? Because the set of all numbers from 1 to 3 includes the set of numbers from 1 to 2, with more tacked on at the end.
will you at least concede that there are more numbers between 1 and 3 than there are between 1 and 2? Because the set of all numbers from 1 to 3 includes the set of numbers from 1 to 2, with more tacked on at the end.
That's what common sense says, but what I am trying to explain is that common sense doesn't work with an imaginary boundary, like infinity.
How would you know that the numbers between 1 and 2 are half the ones between 1 and 3? Do you know how much infinity is? And since infinity is the assumed "end" of the numbers, how can we even talk about twice that amount?
In conclusion: they are both infinite. What happens after that, how much they are, and if one can be twice the other don't matter. These questions have no physical meaning/answer
I looked for and found a TED-Ed video on the infinity topic. It starts with precisely the issue addressed by @GhostOfNinja, but it gets ugly fast. I'm not quite sure what conclusion to draw, or whether there is one; I'll have to watch it again. Anyway, here, look for yourselves:
There are quite a few problems with that video. I'll start with Doombreed's latest post, that there are as many even numbers as there are numbers. This mathematical concept that there are as many even numbers as there are numbers is, according to the video, supported by Georg Cantor's set theory which defines infinite (5.b) as "any set that can be put into a one to one correspondence with a subset that is not the given set". However, Cantor also used set theory to prove that there are infinities of different sizes. By that logic the infinity of even numbers should also be a smaller infinity than the total infinity of whole numbers. In fact, it should also be smaller than the infinity of odd numbers. Because while both positive and negative integers may not have an end they do have a beginning at an odd number. 0 being indivisible is neither odd nor even making 1/-1 the beginning of all whole numbers stretching out to infinity from there. Given the fact that odd numbers, even numbers, and whole numbers in general can not truly be put in a one to one correspondence with each other none of them are infinite by set theory's definition of the word.
Next, there's the matter of listing infinities. Even at its most limited and mundane (excluding set theory in which a limited set can seemingly be called infinite), infinite is defined as being immeasurably large. Infinite is more accurately defined as being without limits or bounds. As such, the very definition of the word makes it impossible for any list to truly contain an infinity. Likewise, because an infinity is too unlimited to be measured there is no accurate way to compare sets of infinities and determine which is bigger or even to place them in sets for set theory to define them as infinities.
Finally, the video says that time proved Cantor correct and even displayed an animated version of his wikipedia page as proof. The concept may have been proven important for theoretical mathematicians, but it is not a proven concept. I've already shown that even a limited understanding of the theory produces significant logical errors that need to be considered before relying on set theory. And even the wikipedia page has a section on the unresolved paradoxes of set theory. One of which was even discovered by Cantor himself.
All of which is extremely off topic. Regarding the original topic, since cyber-space is not a physical space with physical measurements it's hard to directly compare with the universe. There are a couple of ways to conceptually match the two though. Ask yourself if the computing power of the internet is capable of rendering a simulation of a universe equal to our own in size, complexity, detail, and resolution. If the answer is no (which it is) then the internet is smaller. You could also consider whether the internet contains all the knowledge of the universe. For the movement of every particle is pages upon pages of calculations that would be needed to define all details of that particles movement, and the universe is filled with such behavior on both a micro and macro scale. If the internet doesn't possess all of the knowledge about how every aspect of the universe is behaving then obviously the universe must be larger, because that information exists within it as an action.
However, Cantor also used set theory to prove that there are infinities of different sizes. By that logic the infinity of even numbers should also be a smaller infinity than the total infinity of whole numbers.
I don't follow this move here. What Cantor is doing is comparing infinite sets to determine their relative size. Suppose we have two sets - A and B. If we can take each element from A and match it to one and only one element in B, then we have what's called a one-to-one correspondence. That's all that is needed for two infinite sets to be the same size.
Because while both positive and negative integers may not have an end they do have a beginning at an odd number. 0 being indivisible is neither odd nor even making 1/-1 the beginning of all whole numbers stretching out to infinity from there.
That's typically how we think of numbers, but what we're talking about here is sets. While sets do have an ordering, that order can be entirely arbitrary. But remember the main point here - that there are as many even numbers as there are numbers. 0 is actually an even number (it can be a source of confusion, though). But think about what it is for a number to be even. It's a multiple of 2 (i.e. 2 x 0) and it's bordered on both sides by odd number (i.e. 1 and -1). So 0 (just like 2, 4, 6, and so on) will be in both the set of all numbers and in the set of even numbers. But so long as there is a one-to-one correspondence (0 matches to itself) then we have two infinite sets of the same size.
You could also consider whether the internet contains all the knowledge of the universe.
I like this move a lot. It clearly shows that the Internet is much smaller than the universe. And we can even measure both by using the same sort of yardstick - that is, information. We can ask how much information is contained in each. We already know that the Internet doesn't (and couldn't) contain an infinite amount of information. But we can use Cantor's method here to see which set is larger.
So we take each piece of information (a bit, if you like) and match to a particular piece of information in the universe. We could pick the location of a particular gluon or molecule of hydrogen and quickly see that the information contained in the universe is much, much larger.
But there's an even simpler way to go about it. Remember, there are 2 sets here: set I is all the information on the Internet. Set U is all the information in the universe. Since the information on the net is contained within the universe, we know that each element in I will be in U. Everything that's left over will be information that is in the set U but not in the set I. So, again, we get the result that the universe is much, much larger than the Internet.
I'm not gonna lie, I kinda prefer the new topic ^^
So do I.
I don't follow this move here. What Cantor is doing is comparing infinite sets to determine their relative size. Suppose we have two sets - A and B. If we can take each element from A and match it to one and only one element in B, then we have what's called a one-to-one correspondence. That's all that is needed for two infinite sets to be the same size.
Now, I can keep doing that for infinity because all numbers increase to a true infinity, but the even number will always have to be double the whole number. If there were some theoretical end to whole numbers beyond human measurement you'd run out of even numbers long before you reached that end counting with whole numbers. Because of this the two cannot be said to truly be equal if there are indeed infinities of varying relative sizes. What Cantor did was limit his thinking to a finite field that was meant to represent an infinite set and if the two finite set portions could be matched one to one he declared it equal in size. If we accept that there can be infinities of differing relative sizes than n≠2n, even if n=∞.
That's typically how we think of numbers, but what we're talking about here is sets. While sets do have an ordering, that order can be entirely arbitrary.
While the ordering can be arbitrary, if you're looking at all whole numbers as a set then from -∞ to +∞ there is still the distinct central point at which -1 transfers to 0 which transfers to 1. This series of two odd numbers with no even numbers in between creates a difference between the relative sizes of the infinities for odd and even numbers, if we're taking the video's description of set theory as accurate.
0 is actually an even number (it can be a source of confusion, though). But think about what it is for a number to be even. It's a multiple of 2 (i.e. 2 x 0) and it's bordered on both sides by odd number (i.e. 1 and -1). So 0 (just like 2, 4, 6, and so on) will be in both the set of all numbers and in the set of even numbers.
The requirement of being an even number is being divisible by 2 with no remainder. 2/2=1 4/2=2 6/2=3 0 is indivisible (you can't get part of nothing, you just get nothing) and therefore would not fit this description. 0 being considered even is a matter of simplicity for other fields of mathematics from basic arithmetic to graph theory. But take a moment to really look at the number Zero. 0 is neither positive nor negative. Every other number in existence has a positive and negative integer. 0 is not divisible nor is any other number divisible by 0. 0 times anything will only equal 0 (which is only proof of 0 being even if you define it as such before the multiplication) leaving 0 on both sides of an equation that would appear unbalanced if we didn't simply accept the fact that anything times 0 equals 0. 0 itself is a visual representation of nothingness, the antithesis of infinity.
All of that is irrelevant to the main point though, that there is a defined moment within the infinite set of whole numbers at which more odds exist than evens. 0 does not have a positive and negative integer while all other numbers do. That means that going on into infinity, for every positive odd number there is a negative odd number. But if you count 0 as an even number, then you have a single even number that does not have two integers.
Doombreed will you at least concede that there are more numbers between 1 and 3 than there are between 1 and 2?
Well, there aren't. The very notion of "more than an infinite number" is flawed. One set can encompass another set with smaller boundaries, but this does not change the fact that the contents of both sets are infinitely divisible.
[...] supported by Georg Cantor's set theory which defines infinite (5.b) as "any set that can be put into a one to one correspondence with a subset that is not the given set".
That is not how 'infinite' was defined. That's one definition of an infinite set.
As such, the very definition of the word makes it impossible for any list to truly contain an infinity. Likewise, because an infinity is too unlimited to be measured there is no accurate way to compare sets of infinities and determine which is bigger or even to place them in sets for set theory to define them as infinities.
Are you suggesting that there is a limit to the number of decimal places that a value between two integers can have? People have got up to 13 trillion for pi with no end in sight.
Finally, the video says that time proved Cantor correct and even displayed an animated version of his wikipedia page as proof.
No. It says that his ideas are accepted by all research mathematicians. Accepted ≠ Proven
I've already shown that even a limited understanding of the theory produces significant logical errors that need to be considered before relying on set theory.
Much like a limited understanding of Cantonese results in severe grammatical errors. Hitting a logical disjunction when trying to use the theory does not necessitate that the theory itself is illogical.
Now, I can keep doing that for infinity because all numbers increase to a true infinity, but the even number will always have to be double the whole number. If there were some theoretical end to whole numbers beyond human measurement you'd run out of even numbers long before you reached that end counting with whole numbers.
You have two premises here: 1 Both sets continue without end. 2 The even set would reach the end first.
These two clearly don't go together. There is no end, and therefore no way to run out.
If we accept that there can be infinities of differing relative sizes than n≠2n, even if n=∞.
It was never about n being equal to 2n. It's about n1 being equal to n2.
While the ordering can be arbitrary, if you're looking at all whole numbers as a set then from -∞ to +∞ there is still the distinct central point at which -1 transfers to 0 which transfers to 1. This series of two odd numbers with no even numbers in between creates a difference between the relative sizes of the infinities for odd and even numbers, if we're taking the video's description of set theory as accurate.
Well, no, it doesn't. Regardless of the value of the greatest odd number in both directions, the next consecutive number is even. We may as well say that the even numbers are now ahead by one, because at this point everything except 0 will be matched.
The requirement of being an even number is being divisible by 2 with no remainder.
That really isn't the requirement, but:
2∙0 = 0
0 = 0/2
Therefore, 0 is divisible by two with no remainder.
0 is neither positive nor negative. Every other number in existence has a positive and negative integer.
-0 = 0. This does not mean that -0 isn't a real value.
0 itself is a visual representation of nothingness, the antithesis of infinity.
A completely endless nothingness, whose lack of content is without bounds.
