# Deformation of structures on manifolds defined by by Spencer D.C.

By Spencer D.C.

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Extra resources for Deformation of structures on manifolds defined by transitive, continuous pseudogroups (Ann. of Math., 1962-1965)

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I n every base notation there are two ways to express an integral fraction by an aleph-null string of digits. 2500000 . . 2499999. . Although it is not necessary for the validity of the diagonal proof in decimal notation, it is customary to avoid ambiguity by specifying that each integral fraction be listed only in the form that terminates with an endless sequence of nines, then the diagonal number is constructed by changing each digit on the diagonal to a different digit other than nine or zero.

FIGURE 16 Straight-line configuration FIGURE 17 How to place thc pennies 6. The sixth configuration of 10 pennies having five straight lines of four coins each is shown in Figure 16. 7. To put penny C between two touchilig pennies A ancl B without touching A or moving B, place a fingertip firmly on R and then slide C against B. Be sure, however, to let go of C before it strikes B. The impact will propel A away from B, so that C can be placed between the two previously touching coins. 8. Three pennies can be placed with two heads on one side of a line and two tails on the other as shown in Figure 17.

The set of all subsets of an aleph-null set is a set with the cardinal number 2 raised to the power of aleph-null. This proof shows that such a set cannot be matched one to one with the counting integers. It is a higher aleph, an "uncountable" infinity. Cantor's famous diagonal proof, in the form just given, conceals a startling bonus. It proves that the set of real numbers (the rationals plus the irrationals) is also uncountable. Consider a line segment, its ends numbered 0 and 1. Every rational fraction from 0 to 1 corresponds to a point on this line.