# – Infinity?

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– Infinity?

In: Mathematics

There are two main “types” of infinity (that have a lot to do with each other, but can be looked at out of context, I think), I’ll try to put them here:

1. “Development over time”: Let’s say you have a drink. Every five minutes you drink half of what’s remaining of your drink. Over time, the amount of fluid in your glass gets smaller and smaller, but you would never finish your drink. However, we know that, if we want you to be arbitrarily close to finishing your drink, we just have to wait some time, we can conclude that, **as time goes on**, the drink is finished (if we care to wait infinitely long).Another example would be: You have a big, BIG and strechable elastic red rubber band with a blue point on it, and a snail on that band. The snail moves with 5cm/h and every hour you strech the band by 1m. No matter how long the gummi band is, the snail will eventually reach the blue point (if it’s after 1m in the unstreched state it takes around 20k years, but snails live that long, right?).The important thing here is the development over time. That’s why you would often hear “as time goes to infinity”, that just means that we want to study the development over time.
2. “Big-ness of a set”: You want to know how big a set is. If you have finitely many elements in it, you can just count all of them and hurray. However, you can easily grasp that the natural numbers {0, 1, 2, …} are infinitely many (if they were finite, there would be a biggest integer k, but k+1 is also an integer that is bigger than k).If you now take any set and if you are able to “arrange” everything in the set in a way that you can count them (take {a, b, …, z, aa, ab, … , zz, aaa, …}) and are able to say: “a is my first element”, “b is my second element”, “aa is my 27th element”, etc. then you can say: “I can count this set”. This implies the question: “Are there sets that you can NOT count?”, to which the answer is yes, there are, for example the real numbers (without proof here, look up Cantor’s diagonal argument).Order theory in mathematics imply that you may have more, bigger “infinities”, however going there would go a bit too far.
Edit: To add, there is a mathematically sharp definition of infinity (Dedekind), however to understand it you would have to know a bit about functions, and even proving that you can compare any two sets as to which one is bigger is quite a bit of work (a good page of math)

Mathematically, it’s the sum of all positive numbers if you’re talking about positive countable infinty, known else as א0. There are many types of infinity and it’s truly facanating how hard is it to the human mind observe infinity.

To understand more about infinity I would suggest watch some videos about the topic.

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What’s the highest number you can think of? Now add 1 to it. You will never get a number you can’t add 1 to and make a larger number. Therefore, there are infinite numbers, there is no largest number.

Infinity isn’t a number, it’s a concept. Take the fraction 1/3. In decimal it’s written as .3333333333… In theory it could go on for ever. The only limit to the number of 3’s you can write is the number of atoms in the universe.

Infinity is a property that sets can have, of being infinite. It is not a number, it is not an object, though sometimes it’s referred to that way for the sake of brevity or intuitions.

A set is finite if you can count the number of objects in it and eventually stop. A set is infinite if you can keep counting forever and ever and never stop. A path is finite in length if you can travel along it and eventually reach the end. A path is infinite if you can travel along it without repeating and keep going forever and ever.

And so on. Note that this property does not uniquely specify one particular thing. You can have two sets that are both infinite but are different from one another, just like you can have to apples that are both red but are different from one another.

In some contexts, it’s possible to create an object that behaves as if it’s bigger than every other number, and people will sometimes call it “infinity”, but it’s not actually a real number, it’s an object in whatever context they created it.