(Remarks: 23) is limited in an obvious way by the set of cards. 24): Note analogy and lack of analogy between the limited supply of cards in 23) & of words in our memory in 24). Observe that the limitation in 26) on the one hand lies in the tool (the abacus of 20 beads) & its usage in our game, on the other hand (in a totally different way) in the fact that in the actual practice of playing the game no more than 20 objects are ever to be counted. In 27) that latter kind of limitation was absent, but the large bead rather stressed the limitation of our means. Is 28) a limited or an unlimited game? The practice we have described gives the limit 40. We are inclined to say this game “has it in it” to be continued indefinitely, but remember that we could also have construed the preceding games as beginnings of a system. In 29) the systematic aspect of the numerals used is even more conspicuous than in 28). One might say that there was no limitation imposed by the tools of this game, if it were not for the remark that the numerals up to 20 are learnt by heart. This suggests the idea that the child is not taught to “understand” the system which we see in the decimal notation. Of the tribe in 30) we should certainly say that they are trained to construct numerals indefinitely, that the arithmetic of their language is not a finite one, that their series of numbers has no end. (It is just in such a case when numerals are constructed “indefinitely” that we say that people have the infinite series of numbers.) 31) might shew you what a vast variety of cases can be imagined in which we should be inclined to say that the arithmetic of the tribe deals with a finite series of numbers, even in spite of the fact that the way in which the children are trained in the use of numerals suggests no upper limit. In 32) the terms “closed” & “open” (which could by a slight variation of the example be replaced by “limited” and “unlimited”) are introduced into the language of the tribe itself. Introduced in that simple and clearly circumscribed game, there is of course nothing mysterious about the use of the word “open”. But this word corresponds to our “infinite”, & the games we play with the latter differ from 31) only by being vastly more complicated. In other words, our use of the word “infinite” is just as straight forward as that of “open” in 31 || 32?), and our idea that its meaning is “transcendent” rests on a misunderstanding.)