Is anything infinite in the physical world? Although the concept of infinity has a mathematical basis, we have yet to perform an experiment that yields an infinite result. Even in maths, the idea that something could have no limit is paradoxical. For example, there is no largest counting number nor is there a biggest odd or even number. Although it seems like there should be half as many odd or even numbers as there are numbers in total, if both sets are infinite, in some sense we have the same amount of each.
Contradictions are more palatable in the realm of the abstract than in the real world bound by physical laws. But once we have irrationals there is a continuity to physical space that lets us 'move', or at least prove that we did.
If we were simply drawing a line with a pencil on a paper we couldn't get from A to B with only the rationals, but once we have the irrationals we can draw the line continuously. I'm also implying that this sort of knowledge of rationals, irrationals, and infinities is what lets the infinite series converge and hence allows us to 'arrive' at a particular destination - or number thanks!
Close, but no cigar. It's true to say that there exist many irrational numbers which are seperated by an infinite number of rational numbers. But not that there are an infinite number of rational numbers between any two irrational numbers , since there are irrational number between which zero rational numbers exist, between which one exists, between which two exist, etc..
There actually are an infinite number of rationals and irrationals between any two rationals or irrationals. You claimed that sometimes two irrationals have no rationals between them.
That's clearly false. For example, if you picked these two irrationals:. Then you can find a rational between them like this. Just truncate the larger number after the first digit where it differs from the smaller number:. That is guaranteed to be a rational that's between the two irrationals. If you prefer rationals written as fractions, then just write it over a power of ten:. And that fraction is guaranteed to be between the two irrationals. So you can never have zero rationals between them.
In fact, by truncating later and later, you can generate an infinite number of rationals, all of which lie in between the two given irrationals. And clearly, the above procedure works, no matter which two irrationals you choose, as long as they're different and positive.
And it works with slight modifications for negatives, too. Therefore the amount of time between each point as well as the distance gets smaller. The amount of time is therefore finite, even if the amount of jumps are infinite.
Zeno's paradox is an interesting mathematical construct, but does it really exist in the real world? Do you really need infinity to describe motion? Only in an infinite universe. A finite universe doesn't require infinity to describe motion. Think of it like pixels in a video game. It doesn't make sense to say that there are an infinite amount of pixels between two pixels, because you know there are a finite amount of pixels on your screen x for example.
But yet you see motion in your video game. Motion is an illusion. There are only instantaneous state changes, that occur so rapidly and at such small scale our brains interpret them as the abstract concept of motion.
Our brains are also finite and therefore can only process a finite amount of information in a given period of time. However we want to draw conclusions and make predictions even though we only ever observe a tiny bit of the information thrown at us. We can use mathematics to make assumptions about the information we never directly observed.
This proves helpful in most situations, but could also lead us to incorrect conclusions. Could the real universe be the same way? There are a finite amount of atoms between two points on a sheet of paper. Are there a finite amount of particles within an atom? An electron? A quark? No one can say for sure I have wondered about this.
I like your pixel analogy, I think I will use that when I try to describe this issue to people. One of the things that im curious about this is that, doesn't Zeno's paradox prove that we must live in a finite universe? I hope Im thinking of the right paradox here. Is this the one to do with finishing the race? If we do live in the pixel universe then interesting questions regarding neutonian motion crop up.
Particularly his laws concerning the conservation of energy regarding momentum. In a pixel world im using pixel here to denote 'the smallest possible unit of space' surely in order for objects to move at different speeds some will need to effectively 'stop' for a short period of time, and then after a certain really really small amount of time has passed, 'teleport' to the next pixel along?
Zeno's paradox says that motion is not possible, hence the paradox. It was solved by using infinite summations, i. However if space is not smooth, which an integral requires, but quantified then it is just an abstract mathematical model to the real problem.
The math would then not correspond to reality. Reality would them be more like a video game in where particles vanish in one position and appear in another. The video game description is similar to the condensate describe in quantum field theory in where particles can vanish into the condensate or be extracted from the condensate and give the illusion of a particle moving through space. But the video game analogy fails, because the real world is not made of pixels- it is continuous.
That's the whole point of the dichotomy- for any given interval, you can always halve the interval. There is no atomic unit at which point further division is impossible at least, conceptually impossible- certainly there are intervals that are not practically or physically divisible.
So, the dichotomy doesn't require the assumption that space is infinite i. And what, exactly, is paradoxical here depends on the construction of Zeno's argument- on the classic Aristotelian interpretation, we end up with an infinite number of sub-intervals, each taking some positive non-zero duration to traverse. If we have to do an infinite number of tasks, each with some non-zero duration, then it would take us an infinite duration- we would never arrive.
Or so the argument goes. Obviously, contemporary maths to the rescue here. But one other construction of the paradox is to conclude that there is no FIRST interval, such that the traversal of which sees us begin our journey, and no LAST interval, the traversal of which sees us arrive at our destination. Though this is more of a counter-intuitive result rather than any contradiction, it can't be dealt with quite as easily as the traditional version of the paradox. You are just assuming the real world is continuous.
There is absolutely no evidence for that. Current physics would have us believe that the Planck distance is the smallest unit of distance, i. Is it? We don't know. But there is certainly no evidence that the universe is continuous. Infinity could be defined as something that resides outside of space and where time doesn't flow and never did.
If space and time have a beginning then they could only come from something that doesn't have a beginning. The concept of beginning is tied to time, if time didn't exist, then whatever it came from, is infinity. The universe could be infinite but it isn't infinity.
Infinity is everything that is. It's difficult for us to understand because we are inside of the universe. Our thoughts are limited by space and time, we can't think outside of them because we can't imagine any existence outside of them. For us, nothing can exist outside of space and time. Yet, if we see that time had a start, something before it must not have had one.
The same goes for space, if it had a start then it must have been that it didn't exist before, this is hard to express: there must be a place where space and time do not exist at all, a place that isn't a place where time doesn't flow?
Some fields of science advance the universe came from infinite potential or possibilities. They seem to support the concept of infinity. Let's say that one possibility the one we exist in for example out of all the infinite possibilities is taken out to put the universe into it. Now we can see that whatever was taken out of infinity is still part of it. The 1 that was taken out can only exist inside of infinity.
Physics looks at this from the inside out, inside the 1 looking out at infinity. Maybe that's why infinities come up everywhere and no one can figure out the mass of the universe.
These infinities may not be flaws or errors, they could be that infinity is showing through there, only we don't account for it because we are looking from the inside. The paradox is how could the universe not dissolve back into infinity. Since it is part of infinity there is no particular identification for that specific potential.
Imagine the universe is a drop of water in the ocean. It has its own identity, well, because it exists and there is something in it that notices the existence.
How is it possible for it to maintain it's identity or existence inside the ocean? In this scenario, the universe exists inside infinity. You could argue the degree of existence it has. One way it could exist without disappearing in infinity is if it's shielded from 'realizing' it is inside and part of it.
Another example with water, if you want to make a hole in it, you have to keep it spinning. As soon as you stop the hole is gone. Maybe infinity has the same kind of property, the universe has to keep spinning so its own space would continue to exist inside of infinity. Maybe, the reason we can't put our finger on the mass of the universe or explain why so many infinities are showing up all the time is because the universe has a finite mass that is growing inside of infinity.
Here is a link to an essay I wrote about the same. It may have some additional info for those interested. If I have 1 slice of pizza.. If I have you see 2. They are actually 2. Its 2 in reality. Its 2 on paper and in reality. I cant chop up my slice and claim that slice was infinite. It was just 1.
So what we have here is human being playing games with paper and denying reality. There are no infinities. There is infinite stupidity as Einstein said. This is bias overruling logic to justify an end. Every example used fails and anyone trained in logic sees that in all of 3 seconds. The principle describes an infinite number of infinitesimals or an infinite series tending to an infinitesimal for Xeno's paradox , not whole numbers.
It is because of this that infinites can produce finite values. Divide your pie by three exactly accurately please. You can not. You will need an infinite amount of time and accuracy to perform this task.
I am surprised you didn't mention continuous, uncountable infinity vs. For example, temperature, while we know that due to quantum mechanics, it is in fact digital, Aristotle would probably have thought that it was analog, smooth.
These are measured as real numbers. You can count to 10, 57, or 4. You can even count to -9, 0, etc, by counting like this: 0,1,-1,2,-2,3, It will be the th item in the list. But, try and count the irrational or real numbers. Basically, I can do this no matter what crazy scheme you count with.
Thank you, thank you, thank you. I do not exclude Cantor!!! There are no infinities larger than any other, because no matter what 'sophist Comparisons and distinctions of quantity or magnitudes cannot be made of 'things' concepts or otherwise that are NOT quantities or magnitudes!
Every type of 'infinity', by definition, goes on forever, it never concludes. Demonstrating that a counting system can be set up for some and not for others says nothing about the magnitude, or relative magnitude of either. Each goes on forever regardless of whether we can conceive one-for-one relationships between or among them.
So how does it "act like a number"? Additional comment: I don't care what kind of infinity or infinite set you may define, I can count them one-for-one as you produce another number from it.
So what's so deep about showing gaps in any proposed ordered counting system? Infinity is not a number. Any infinity can be counted by the natural numbers forever. It's smoke and mirrors to say any infinite set is larger than another assuming, without accepting, that the term 'infinite set' actually has any meaning.
Can a "set" be an unbounded amount? Possibly only in fantasy, which is what most discussion of infinities amounts to.
The set of all the natural numbers exists, and therefore, the size of the set also exists. That size happens to be a non-finite number called Aleph-Null.
Your argument is wrong. And it does behave like a number because you can do basic arithmetic with it. In regards to time, surely 'infinity' is practically impossible. Since the concept of something neverending means that the number is always unattainable. It continues, infinitely, throughout time, but never can it BE infinite.
At no point in time is something ever 'of infinite value'. But presently, it continues. And it continues. And it continues, always. This is often described as potential infinity and it means a system has the potential to be infinite if infinity were to exist. This is in contrast to actual infinity - which is the belief of modern mathematics, and is a system which exists infinitely all at once. So the biggest number depends on what exactly you're asking. If you're asking what the biggest observed number is - it is exactly that.
If you are asking what potentially is the biggest number, what is beyond the bounds of the universe, unlike mathematicians that believe in infinity - I can only tell you how to get there, not what it actually is.
In classical mathematics we learn that there is not just one infinity, in fact there are several , and some are bigger than others! The main two infinities are the so called Natural Numbers - which means every integer number. And the Real Numbers - which means every decimal number, including every decimal expansion, including those which expand infinitely.
Here are a few examples of ones with finite length expansions. The natural numbers and the real numbers have some different interesting properties. For example I can always write down a natural number. It might be really long and take page and pages but it should be possible to write it down. For this reason I can also always communicate a natural number to someone.
The natural numbers allow us to count objects by pairing each one with something else. For example if we had a pile of rocks we could put aside one for each natural number, and when we were done this would tell us how many rocks we had. For this reason sometimes they are called the countable numbers. But there is no way to pair the natural numbers with the real numbers. There are simply more real numbers than there are natural numbers.
Any time we find a pairing that might work, it is possible to create a new real number which was not paired. This is called Cantor's Diagonal argument and is why the real numbers are considered a larger infinity.
Consider the real numbers which have an infinite decimal expansion but where the expansion follows no pattern. It is impossible to talk about a specific one of these numbers.
They are completely impossible to express. Okay so a few we can talk about such as pi and the square root of 2, but most of them just follow no pattern, have no properties, and go on infinitely. This makes them impossible to communicate. In fact nothing can be done with these numbers. Because they can't be expressed they can't even be used in mathematics. The vast majority of real numbers are like this.
The only ones that aren't are those that correspond to the natural numbers. The set of real numbers is somehow padded out with all these indescribable and fundamentally useless elements - and so many of them that in proportion it appears the set is completely full of these indescribable numbers.
The question is, do these indescribable numbers, which have no interaction individually with the rest of the mathematical universe, and only exist due to the axiom of infinity, really exist? One worry about dropping the concept of infinity from mathematics is that many theories of mathematics rely on infinity in one form or other. Calculus is often the first topic that comes to mind. If we remove infinity from mathematics wont we lose a lot of it's usefulness? Will we be unable to integrate, differentiate, calculate areas and boundaries?
Will we be able to use Turing Machines in computer science? Could we still calculate the area of a circle? As it turns out almost all useful forms of infinity in mathematics can be modelled perfectly well by potential infinities. A great example of this is limits , which in a finite mathematical world are like function with a promise - given some infinite resource - they would eventually calculate some value.
In a finite mathematical world a limit never actually calculates the value it expresses - it can only be used to find approximations of it - but this doesn't matter. Thanks to the greatness of algebra we can still find rules and ways to manipulate these objects in symbolic form without computing anything infinitely. Sums with infinite terms don't actually have to be resolved. Things like integration can be performed with just a few switches and modifications of symbols.
If anything Calculus is a shining example of finitist mathematics! Unless you actually believe that when performing symbolic integration you are calculating an infinite number of additions in your head We can also consider the value of pi.
In a infinite mathematical world actual pi - the full decimal expansion of pi - is said to exist. But is this really necessary? Everywhere we use the symbol pi it is equally adequate to instead talk about pi defined in limit form - a finite form. What is the usefulness of pi existing in full decimal expansion form? So what can't be done if we remove infinity from the axioms?
Well we can't talk about things such as the Real Numbers that follow no pattern infinitely - we can't describle a turing machine which prints to the tape without following any pattern and goes on forever.
We can't use actual infinities. But we also can't talk about sets of these things - and this is the main issue for many mathematicians. We can't talk about sets of things where each thing can't be described individually. Now this might be an issue for some, but it certainly isn't a problem for useful mathematics. Some would say that this subset exactly describes useful mathematics - because finite mathematics is the only expressible mathematics.
If infinity makes the mathematical world act so weirdly, why was it chosen as an axiom in the first place?
Well, the concept of infinity had always sat in a troubled place in mathematics until Georg Cantor developed the mathematics required to use and understand it.
Much of his work was considered incredible, beautiful and profound - it gave clarity to many of the questions of infinity. Cantor's theories were undeniable beautiful and vast. They described stacks of infinities of infinities - each one larger and more transcendent than the last - interacting and wrapping around each other, twisting and curling into infinite areas like fractals, or unwrapping into a vastness unimaginable.
Cantor created a kind of mathematics seemingly so fundamental, so deep, that many felt certain it touched on the real roots of mathematics - that it must be ground for development of the axioms of mathematics. And this is what happened. David Hilbert and many more forged forward to build mathematics on axioms based around set theory. The mathematicians, enthused by Cantor's world, built the axioms in the spirit of Cantor's original work.
There was no question that infinite sets existed. As David Hilbert was famous for saying - "No one shall expel us from the Paradise that Cantor has created". There was less potential for opposition. The science of the day didn't have quantum theory. Most scientists probably believed that space was a continuum - that a space could be divided into infinitesimally smaller sections forever. When Cantor was writing the atom hadn't been discovered.
The cosmology we have today didn't exist either. Time and space may as well have been infinite to scientists of the day - there was no reason to believe otherwise. And of course there were no computers. When Cantor was writing the world looked a lot more infinite and the arguments of the Constructivists seemed to fall flat. Constructivists appeared conservative - old fashioned - and resentful that so many of the mathematical community had followed Cantor.
But science and beauty weren't the only reasons for infinity. There was a final, surprising reason - God. Cantor was a devout Lutheran and believe that the theories he'd discovered had been communicated to him by God.
He equated actual infinity directly with the concept of God. Well - they have many similarities - both are transcendental concepts that defy imagination and according to many must exist. Both go beyond imagination and seem to exist in some kind of beautiful unimaginable realm.
Even for the other mathematicians who may not have been religious the presence of a kind of "God" in mathematics was desirable. Actual Infinity was the mathematical concept that lifted mathematics above the plane of everyday life - brought it above humanity and imagination - and made it somehow a deeper study, not of nature, but of something grander.
This idea, they felt, no matter what the paradoxes and oddities, was worth fighting for. I remember learning about infinity at school. It was a pretty hard concept to grasp. Each child seemed to have their own way to explain it.
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