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## Does infinity exist?

I have been reading a lot about mathematics and astrophysics over my summer holidays. The question I have in my head right now is 'Does infinity exist?' or is it just a term to express what we cannot compute or understand yet?

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## David Irvine

To make a perfect circle you can draw parallel lines and keeping the distance between them, move them through a centre position all the degree's and parts of a degree. for the circle to be perfect, the number of parallel lines has to be infinite!

Problem,

There is no perfect circle, just a notion.

There is no such thing as parallel lines, just a notion

Now if it were possible we would have an infinity.

If we then transpose the circle to a sphere we have infinite infinities !

In the circle example we assume 2d - i.e. lines have no width If we move up a dimension to 3d then infinity becomes infinity to the power of infinity, now add more dimensions and it gets weird. This I think is a problem with some advanced physics, we use these notions and then take them way to far out of context. Maybe an example of proper chaos theory where our initial assumptions are very poor, and immediately we move to N-Dimension thinking like string theory.

Huge subject that I find fascinating, great question, I do think there is an answer, but I believe we do not have the language (spoken, written and mathematical) to provide that answer just yet. We can live in the excitement that there are answers and there are so many we can all help find them, if we just think enough and accept current failings readily.

Great time to be alive!

David

## Goran Malic

Think of it this way: a circle is a specific set of points contained in a plane, that is, it is a subset of the plane. A sphere is a specific subset of space. How many points are there in a plane? And how many in space? The answer is the same, both have an infinite number of points, so neither the circle nor the sphere can have more than an infinite number of points, i.e. they have exactly an infinite number of points, not infinity to the power of infinity.

Our intuition falls apart when dealing with infinity. A clear example is this one: consider the set of all positive integers 1, 2, 3, 4, 5 and so on. Now, cross out all the odd numbers in this set. You'll get the set consisting of all even integers 2, 4, 6, 8 and so on. The question is: which one is larger, the set of all positive integers or the set of both positive and even integers?

The usual answer is that the set of all positive integers is larger since we obtained the other set by crossing out odd numbers, but actually, they are of the same size, both are infinite. To clarify why this is so: well, we just rearrange the counting pattern of the even integers, we consider 2 as the first even integer, 4 as the second even integer, 6 as the third even integer and so on. In such a way we have established a correspondence in which 2 corresponds to 1, 4 corresponds to 2, 6 corresponds to 3, 8 corresponds to 4, 10 corresponds to 5 and so on, that is, to every even integer we can assign a corresponding integer. From this we conclude that there are as many even integers as there are all of the integers.