Topics Mathematics

Question HERE IS QUITE A CONUNDRUM: Can we meaningfully speak of the "infinity-th" and "infinity+1-th" term of the sequence of natural numbers? If not, then what do we in fact mean by "all" (as distinct from "any" or "each") when applied to an "infinite" set? Given that a real number constructed via the diagonal construction on a F I N I T E set, of n reals, can always be added to the list at position n+1 to give a list of n+1 reals, why couldn't a real number constructed via the diagonal construction simply be included in the "infinite" list of reals at "position" "infinity+1" ??? (Which is to say that, in the "infinite" case, no real could be constructed outside the infinite list of reals at all!) Also, in the case of the natural numbers, if a number m, is defined as the sum from 1 to n of the first n natural numbers, then m is a natural number that is not in the list of the first n natural numbers. If you make this construction on the "entire" set of "all" natural numbers, then by construction, there is always a natural number m that is not counted in the set, but this surely does not imply that the set of natural numbers is uncountable. (Or does it?) Why then is this argument considered valid in the case of reals? is it not equally fallacious in both cases?

Accepted:
August 21, 2008

Comments

No, mathematicians haven't defined any meaning for expressions like "infinity-th" or "infinity+1-th". (The fact that they're so awkward to write should be something of a giveaway!). It's important to appreciate that infinity is not a number. Don't be misled by the fact that we can say things like "there are infinitely many natural numbers", which seems to have the same form as a sentence like "there are three coins in the fountain". The number sequence doesn't go: 0, 1, 2, 3,... 1,000,000, 1,000,001, 1,000,002, 1,000,003,... infinity, infinity+1, infinity+2, infinity+3.... Rather, infinity is a property of certain sets, such as that of the natural numbers. The infinity of that set consists in the fact that, for any member you might care to consider, there will be another member, which is larger than it but which is nevertheless still finite. And we can easily refer to that set as a whole, and we can even quantify universally over its members. We can say, for instance, that they all have successors, or that this set contains them all. (Though I'm not sure just how distinct this is from saying that each of them has a successor, or that each of them is a member of the set).

Given that infinity+1 is not a number, there is no such thing as an infinity+1-th position in a hypothetical enumeration of the real numbers, where you could add an extra one from the diagonalisation of the sequence. But that's no problem, as there's a very simple alternative: just shift all the members up one place, and stick this extra one at the start. But then you'll have a new sequence, and it too will be susceptible to diagonalisation. The diagonally-constructed number you get for this new sequence won't be the same as the one you got the first time (it will differ from that one in at least its first digit): but there will still be one, and it's this that shows that the real numbers are uncountable. The uncountability of a set means that any list, with a first member, a second member, a third member and so forth, will fail to contain all of them. Try to construct such a list of the real numbers, and you'll invariably find that you've missed one out, namely the one you get from diagonalisation.

But the natural numbers are countable. You can place them in a list where the first member is zero, the second is one, the third is two, the fourth is three, etc. (It's tidier with the positive integers!). Now, if you stop after only the first n natural numbers, then, yes, the number m that you've defined will indeed be absent from your list. But this fact doesn't carry over to the infinite case. If you take the whole sequence of natural numbers and add them all together, they won't give you a natural number as a result, whereas the diagonalisation of a purported enumeration of the real numbers will give you a further real number -- that's the difference between the two cases. Mathematicians are comfortable with the notion of sums to infinity of geometrical progressions, but only when these yield finite results. Thus, for instance, the sequence 0.5, 0.25, 0.125, 0.0625... issues a sum to infinity of 1. But there is no sum to infinity defined for the arithmetical progression of 0, 1, 2, 3,.... And, if there should be some temptation to say that such a sum does indeed exist, and it's infinity, just remember again that infinity is not a number, and still less is it a natural number. Consequently, it is not the case there "there is always a natural number m that is not counted in the set", and the set of natural numbers can comfortably remain countable after all.