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S~r Figure I. Boolean operations. (7) 28 1. Origins of Formal Structure This subset is sometimes called the extension of the property H, to emphasize the notion that differently formulated properties may have the same extension-and that Mathematics has to do with extensions rather than with meanings. This in turn involves the "extensionality" axiom for sets-that a set is completely determined just by specifying its elements. This means that the equality of two subsets of X is described by the statement S = T ¢:;> (For all x in X, XES ¢:;> x E 1), (8) while the inclusion of one subset S in another is described by SeT ¢:;> (For all x in X, XES ~ x E 1); (9) here the arrow ~ stands for "implies".

By the recursion diagram (5), with X replaced by N " there is such a homomorphism. Since N' with s' also satisfies the Peano postulates, the same recursion diagram produces a unique homomorphism g: N' ~N in the opposite direction, as in the bottom row of the commutative diagram below: Now compare the composite function g-j: N ---N with the identity function I: N ---N. They both make the diagram 0 -N ~N I"g·! II IiII,g.!

The labeled vertices are all different; that is, fk fm implies k m; one says that the function f is injective (an injection, or one-one into). With these labels, each motion T: X ..... X of the square sends each vertex to a vertex, so determines a permutation # T: Y ..... Y of the set Y of vertices. Thus # T does to k what T does to fk; in other words, = = (2) for k = 1,2,3, or 4. This equation can be written in terms of composites of functions as (3) or displayed in a diagram of the corresponding functions as #r • y------ f~ y ~f x - - - - - -• x .