# Is it faster to count to the infinite going one by one or two by two? [closed]

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?

• It almost like comparing $\infty$ and $\frac{\infty}{2}$ both of which are infinity. You can never compare infinity with another infinity as you do not exactly know what it is.
– lsp
Commented Feb 3, 2014 at 10:17
• so to finalize this question,simple talk to child that there is no different,because infinity is not number Commented Feb 3, 2014 at 10:26
• You then could ask a follow-up question: what is faster, counting one-by-one, or, counting all the odd numbers first, and after that counting all the even (i.e. $\omega$ versus $\omega+\omega$)? Commented Feb 3, 2014 at 11:38
• After each counting step your distance to the goal $\infty$ equals $\infty$ itself and stays constant, so in both cases your speed is zero :) Commented Feb 3, 2014 at 15:30
• I recommend that any kid who has this question, should read The Phantom Tollbooth. Then again, I recommend that any kid at all read The Phantom Tollbooth, but in any case, it has a great chapter on exactly this concept. (Though sadly, it doesn't introduce the concept of different cardinalities. It totally should have.) Commented Feb 3, 2014 at 23:23

YES and NO.

$1, 2, 3, ...$

and you map it into the succession of even numbers :

$2, 4, 6, ...$

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!

• Another word for "denumerable" is "countable". You might encounter the terms "countably infinite" and "uncountably infinite" in your travels. Commented Feb 3, 2014 at 16:32