The most naive approach to infinity
is treating it like a number
and using it in calculations like any other number. This lies at the heart of many a false mathematical proof
and is, generally speaking, dead wrong
However, it is also not quite right to say that infinity is not a number, because it is, it's just a transfinite one, and those behave differently than your normal everyday numbers. But there are certain operations that can be performed on them, and there is more than one.
To get to the point, there are two commonly encountered infinities:
- countable infinity, i.e. the number of all natural numbers. This is, strangely enough, the same as the number of all integers and even the same as the number of all fractions, because it is possible to find bijective functions that map between these sets. So even though it intuitively seems like there are twice as many integers as natural numbers, and infinitely as many fractions, the numbers are really the same, and you see why infinities have to be treated specially.
- uncountable infinity is, however, different: there are actually more real numbers than there are fractions or integers, because it is impossible to create a bijective mapping between them - the diagonal argument proves this
There are other, even larger cardinalities
, but you are unlikely to ever encounter them unless you choose to do so. However, the two above (and the difference between them), come up quite frequently in many fields of mathematics