"Infinity" as a problematic issue. A list of some of its paradoxes

"Infinity" as a problematic issue

"Infinity" as a problematic issue. A list of some of its paradoxes

"Infinity" as a problematic issue

Infinity as a mathematical concept is highly abstract and at the same time very attractive. Infinity has certain paradoxical properties. I will list some of these paradoxes:

(1) A standard example would be the fact that you can take, say, the number 1 and correlate it with the number 2, take the number 2 and correlate it with the number 4, take the number 3 and correlate it with the number 6. In this way you can gradually, step by step, correlate all integer numbers (1, 2, 3, 4, etc.) with all even numbers (2, 4, 6, 8, etc.). And that's funny, because this would imply at first glance that the number of integers is equal to the number of even integers. That's a paradoxical property of infinity.

(2)To the Greek thinking, infinity as a problematic issue was expressed in this statement. "How will you ever get from Point A to Point B if there are infinitely many things along the way?"

Infinity seems to be more than anything, and for this reason it is something that you will never reach. That's the famous paradox. How will you ever get from Point A to Point B if there are infinitely many things along the way? To get from A to B, you would first have to reach half the way. But to reach half the way, you'd first have to reach half of half the way, and then to get there, you'd have to reach half of the half of the half, and so on. It would appear that you would never be able even to make a start. There are always stages along the way that you haven't reached. If things can be divided into infinitely many stretches along the way, then there is simply no way you will ever cross from one point to another.

(3) It follows from the 2nd that whatever we're doing getting from one place to another, counting things, doing things, we're doing finite things. But how can anything finite encompass within it something that is infinite in character? This leads to a famous study by Cantor on his finite set [0, 1]

(4) Cantor developed the notion of a finite and infinite set, which has infinitely many objects. He proved that over [0, 1] we can define an infinite subset and once you have an infinity, an infinite set, you can create another infinite set that is bigger than the one you started with. There are infinitely many infinities, stretching all the way up.

"Infinity" remains a problematic issue where it was 2,500 years ago and is likely to remain that way.

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