Infinity is a hard concept to understand, it has a lot of miseries hidden there, and it requires a lot of imagination and a sort of ‘math devotion’. Infinity is something the human mind cannot properly compare with anything real, and it transforms your way of thinking after you encounter it. Let us start with a definition everyone knows:

Infinity(symbol: ) is an abstract concept describing somethingwithout any limit.

This is something that everyone knows and it is also a concept used in other areas not only math or science. But if you first start to think more about it and you look at the symbol “” it is a little confusing. If you look at the symbol do you really think at infinity? Be serious with me? ( I wait for you answer in the comment box bellow) Because it doesn’t represent infinity to me, it really does not, but I cannot think of another way of representing it neither.

The first time I heard about infinity was in my calculus course, when we were talking about the existence of limits and other properties of limits. I did not gave it that much sense or importance, it was something interesting but nothing that big strange at that point. The first more important discovery of infinity was in my Sets and Algebraic Structures course. It was there when I did Cantor’s set and other types of infinities. I will quotes from my post Impossibility about this :

I am pretty sure that you know know that there are an infinity of natural numbers, but how surprising (shocking??) is the fact that there are as many even numbers as natural numbers? Those infinities are kind of ‘the same’. Kind of hard to properly understand, even if you have the proof just under your eyes. And if one tries to think about integers, they are as many as natural numbers, and rational numbers the same. The proofs are kind of hard, but they show us something that for a long period of time we consider to be impossible. And the most elegant proof concerning this was the one regarding the real numbers. There is another infinity, a different one , a ‘bigger’ one. The proof that shows that in the set (0,1) there are more numbers than in the set of natural numbers was absolutely fascinating.

Thinking about all these infinities, what do you think about The Universe? What kind of infinity is it? Is it a true infinity or we just don’t know that many things about it? **The deep questions of maths can leave you feeling very small in a vast universe.**

But now The Infinity is another well known symbol and everyone uses it. Also you can find it in a lot of interesting areas from art to jewelry; even some well know business have taken and used the symbol or the concept. We love the symbol, we try to explain it in many ways, we try to represent it in many way, we try to understand it, but can we? Do we understand it’s hidden mysteries?

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A very nice post Ioana! To answer your question about the ‘infinity symbol’ (you asked us for our opinions, so here’s mine 😉 ) Kurt Vonnegut put it so well: “Everything is nothing, with a twist.” Not only does this express the visual truth of ‘twisting’ a circle/loop into a lemniscate, representing the ‘zero’ symbol and ‘infinity’ symbol… but on a ‘second level’ of significance, ‘infinity’ (as you said in its ‘primitive’ definition of being boundless), to the human mind is a state of indistinguishability, and is therefore also indistinguishable from ‘nothing’ (shameless plug: I talk of this under the ‘Liar’s Paradox’ section of my latest post here: http://taomath.org/2015/06/adjacent-existents-a-theory/). But it does make one think: “Why ‘with a twist’?” – given what I just said, wouldn’t just “Everything is nothing” suffice? That’s where we realize that ‘infinity’ – even as a ‘boundlessly bounded’ (or infinitely-distinguished) thing itself also becomes indistinguishable from ‘nothing’ in that the human mind can no longer ‘make out’ distinct things at this ‘scale of focus’… Further to the ‘representativeness’ of the lemniscate as ‘infinity’, the circle, if you imagine it as the ‘edge’ or contour of a Pringles chip (the topological ‘saddle’ shape), and give it a quarter-turn to look at it so that the ‘front downwards’ part of the contour ‘crosses’ the ‘back downards’ part, it looks kind of like a lemniscate…. and, to no surprise to anyone, a circle is continuous, or ‘non-terminating’…

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I will say that I totally looove your explanation… Never thought of this before and it is quite impressive for me. Thank you very much for sharing your thoughts ^_^

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