A child asked me this question yesterday:
Would it be faster to count to the infinite going one by one or two by two?
And I was split with two answers:
- In both case it will take an infinite time.
- Skipping half of the number should be really faster.
Which brings me this question:
Could an infinite be greater than another one?
YES and NO.
Galileo made the “discovery”, the so-called Galileo’s paradox, that if you start with the succession of natural numbers:
and you map it into the succession of even numbers :
you may map (i.e.associate) every number into its double (today, we call it one-to-one mapping).
So, you have the same “number” of numbers and of even numbers.
Modern set theory (from Cantor on) solved the paradox extending the “counting” process to infinite sets, but proving that the euclidean common notion that “The whole is greater than the part” [see Euclid, The Elements, trans T.L.Heath, Dover, Common notions 5] will not hold for an infinite set.
According to modern set theory, the two above sets can be put in one-to-one correspondence, so they have the same cardinal number, and their “type” of infinity is called denumerable (a set is called denumerable exactly when it can be put in one-to-one correspondence with the set of natural numbers).
But, again by a result of Cantor, not all infinite sets can be put in one-to-one correspondence: there are infinite sets “more infinite” than another. A set of “a larger kind” of infinity is the set of real numbers; it is not denumerable, in the meaning defined above.
As for counting “faster”: of course, if you count two by two, after a while you will be way “ahead” of your friend that is counting by one. The only problem is that you will not “end” before him, because there is no final goal to be reached!