Review: Willard Van Orman Quine, Set Theory and Its Logic

tendency, Thiele prefers everywhere to deal with relations instead of (uniform) functions. He defines Abbildungen as sets of ordered pairs, and when /is an Abbildung and m is a subset of the domain of/, he uses the notation f(m) for the set of elements y of the codomain satisfying the condition (Ex)(x em&<.x, y~> «/). Hence in the case of a uniform Abbildung (mapping)/, what usually is called the value of/for x is the sum set of/({*}) ({*} being the unit set with the element *)• Thiele avoids the use of descriptions. As to equality, he defines a = b = Df (*)(x € a <-* x « b) & (x)(a t x <-* b e x), so that the axiom of extensionality gets the form (x)(x ta*-*xtb)-*(atc-+btc). The other axioms are almost all in the usual form. Only the schema of the replacement axiom is strengthened to (x)(x « m -*• (Ey)A(x, y)) -*• {Ez)(x)(x em-> (EyKy tz & A(x, v))). This schema applies to arbitrary relations A(a, b), not only to relations that are uniform with respect to b. This strengthening is advantageous for some purposes. —The Fundierungsaxiom is not included. —Thiele is sparing, when possible, with applications of the axiom of replacement. In many cases this can be done by making use of the power-set axiom. In treating transfinite recursion he formulates and proves only the specialized recursion theorem relating to the case where the Abbildungen occurring are into a set b given from beginning. Concerning the general case he only remarks that one can, for any particular application, prove the existence of such a set b, applying the axiom of replacement if necessary. The formulation of the specialized recursion theorem could be simplified a bit, since a dispensable complication here as in several other passages is caused by Thiele's representing any well-ordered set by an ordered pair <m, r> where m is the set and r is the well-ordering. This in fact can be dispensed with, as indeed m is determined by r, being the domain of r. No use is made of the possibility of defining ordinal numbers directly as sets with certain distinguishing properties—an approach by which it quickly follows that the ordinal numbers are well-ordered by the e-relation, so that the theory of well-ordering is replaced for the most part by the theory of ordinal numbers, which in many respects is more handy. On the other hand, in several cases Thiele dispenses with the application of the theory of well-ordering by using Zorn's lemma. In particular, as he points out, the comparability of the power of any two sets results from this lemma without any reference to well-ordering. Errata. P. 179, in the footnote, line 3, instead of "eine Abbildung" read "eine eindeutige Abbildung"; p. 189, lines 12-13, the ordinal number 0 is, it seems, omitted; p. 190, line 3 from bottom, instead of p put (p; p. 191, line 15, in the subscript of II the a should be a0; p. 193, line 8, instead of "Ordinalzahl" put "Limeszahl"; p. 194, line 20, at the end of the formula replace ax by ax; also in this formula (lines 19-20) an omission occurs similar to that mentioned on page 189; in fact, in the last implication to its first member (its premiss) should be added "AX # m". PAUL BERNAYS