Thinking about infinity

[robvalue]

Yeah, an infinity is countable if each element can be mapped to a unique natural number.

So obviously the natural numbers are countable. Basically, it means you can put them in a straight line and count them.

The integers are also countable. You can just alternate between positive and negative to count them all.

Rational numbers (fractions) are countable too. (A/B) <=> (2^A) * (3^A) gives unique natural numbers due to unique factorisation into primes.

But real numbers are not countable. They are a "bigger infinity". It's impossible to form such a map between these and the natural numbers. I can show you why if you like.

As for whether a finite object being represented by an infinite number of subsections is "real", I'm not really sure how to answer it. A line is a continuum anyway, and consists of an infinite number of points along the way. So you move through infinitely many iterations to move a finite distance. This is just a different way of "counting" the distance, rather than adding it continuously. So yes, I'd say it's a logically possible way of viewing an object. Obviously it doesn't become split up into all this bits just because it's possible; but it still is the sum of all those bits. If you removed any, you'd no longer have the whole length.