I don’t pretend to remember very well what it was like to learn about numbers and especially arabic numerals. I do remember that I was then terribly impressed by authority and perhaps I would believe things about numbers merely because someone told me that something was so. I try to imagine here what it might be like as a skeptical child learns about numerals.

That 1, 2, 3 up thru 9 represent how many things are in a pile seems innocent enough—innocent and useful. It is hard to believe that some cultures don’t even have this much. At first 10 seems a bit mysterious but something has gotta come after 9. 11, 12, 13 ... get back on track sort of; at least getting from 11 to 19 is easy because it is like getting from 1 to 9. 20 is spooky, sort of like 0 follows 9 but not quite. This confusion seems to be the reason that 0 follows 9 on telephones. Perhaps someone tells us now that digits xy describe a pile with x sub-piles each with 10 things, together with a sub-pile of y things. This seems plausible but still 10 seems to belong to family 1 thru 10, instead of the family 10 thru 19. How come the first family only has 9 members while the rest have 10? I think it would be good at this point to introduce the pile with nothing in it at this point and reveal 0 as a numeral.

Pretty soon one can count from 0 to 99. 100 is déjà vu. The pattern is unclear. Should 110 come next, 111? No 101 then 102 come next. When we get to the transition from 109 to 110 a glimmer occurs. Then the idea hits that 1xy is one hundred more than xy. But first you have to swallow that 02 really means 2! Then even the idea that xyz is a pile with x sub piles each with one hundred things, together with yz things. Certainly these ideas are not described with letters x, y and z, but these ideas somehow comes thru.

At this point there seems to be no stopping us. After 999 comes 1000 then 1001. Whee!! The sceptic wonders about the end. There is an end to everything. Perhaps when you have no more room on the paper. But I might run out of paper before you and so that does not seem right. Surely some adult will tell us when to stop.

One limitation that will deter some is the English names of things like 1000. You quickly learn “million” and maybe “billion” but you probably forget the rest. If you think that “billion” is more real than 1000000000 then perhaps this is what tells us where the end of the numbers is. You are probably old enough now to look in an unabridged dictionary where you may find vigintillion with 63 zeros if it is an American dictionary. Maybe this is the end. You gotta believe the dictionary after all and that is all there is!

Still you have become enough of a mathematician to know that you can keep going; you can add one to any number to get another that is bigger. It is all very distressing.


Some years later you take set theory and learn that 4 is the set {0, 1, 2, 3}. Well it isn’t really but we are playing a grand game by re-imagining the mathematical universe, and we want now to believe. We see an axiom that says that there is a set that for every element x in the set, x∪{x} is also in the set. If we believe this axiom we are forced to believe that there are numbers without end. We probably already believed this already but it is a confirmation that reminds us of the conclusion that we suspected as a youth. At least now we can boldly make the claim and accept the axiom with open arms.