Is there such thing as 'Infinite'?

Answer 1

It's not quite clear what you mean, so let me ramble a while about infinity (and very large numbers which are not infinity) and see if that satisfies what you were trying to ask.

Infinity is an abstract mathematical concept. We don't know whether the universe is infinite or not (we do know the observable universe is definitely not infinite).

Children and other mathematical unsophisticates tend to think of infinity as just "a really, really big number" and come up with phrases like "infinity + 1". This is not really what's going on.

There's a number called Graham's number, obtained by repeating a process 64 times. The basic equation is gn = 3 ^^^... (repeated gn-1 times) 3. g0 is 4, so g1 is "just" 3 ^^^^ 3. g1 is incredibly, unimaginably large; if you're the sort of person who says things like "infinity plus one" it's almost certainly bigger than whatever you think of as "infinity".

Another way of writing 3^^^^3 is to say that it's the end result of a process that starts like this:

f(1) = 3

f(2) = 3^3^3 = 7625597484987

f(3) = 3^3^3^3^ ... ^3 where the number of 3s is f(2)

f(4) is the same, except the number of 3s is f(3)

g1 is f(7625597484987). It's worth noting that f(3) is already much, much larger than the number of atoms in the observable universe, so that may give you some idea of how big g1 is going to be.

Oh, and Graham's number? It's not g1. It's g64. And it's no closer to infinity than 0 is.

If that's not enough for you yet, there are different kinds of infinity. Mathematicians sometimes refer to "countable" infinities and "uncountable" infinities (though "listable" might be a better phrase than "countable"). A countable infinity is one that can be put into one-to-one correspondence with the counting numbers.

The set of positive integers is obviously countably infinite, since it basically is the counting numbers. It's less obvious that the set of all integers is countably infinite, but it's true. It's even less obvious that the set of all rational numbers is countably infinite, but again it can be proven that this is the case.

However, the set of all real numbers is uncountably infinite; you cannot put the set of all real numbers into one-to-one correspondence with the counting numbers. It's a "larger" infinity, sometimes called C.