By Paul Howard

This booklet, effects of the Axiom of selection, is a accomplished directory of statements which were proved within the final a hundred years utilizing the axiom of selection. every one end result, also called a kind of the axiom of selection, is assigned a host. half I is a list of the kinds by means of quantity. during this half each one shape is given including an inventory of all statements recognized to be resembling it (equivalent in set idea with no the axiom of choice). partially II the kinds are prepared via subject. partially III we describe the versions of set conception that are used to teach non-implications among varieties. half IV, the notes part, comprises definitions, summaries of vital sub-areas and proofs that aren't on hand somewhere else. half V provides references for the relationships among varieties and half VI is the bibliography. half VII is contained at the floppy disk that is enclosed within the booklet. It includes a desk with shape numbers as row and column headings. The access within the desk in row $n$, column $k$ offers the prestige of the implication ``form $n$ implies shape $k$''. software program for simply extracting info from the desk is usually supplied. good points: entire precis of the entire paintings performed within the final a hundred years on statements which are weaker than the axiom of selection software program supplied supplies entire, handy entry to information regarding relationships among many of the results of the axiom of selection and concerning the types of set thought descriptions of greater than a hundred types utilized in the research of the axiom of selection an intensive bibliography in regards to the software program: Tables 1 and a pair of are available at the PC-compatible software program incorporated with the booklet. additionally, this system maketex.c within the software program package deal will create TeX records containing copies of desk 1 and desk 2 that may then be published. (Tables 1 and a couple of also are to be had on the authors' websites: http://www.math.purdue.edu/$\sim$jer/ or http://www.emunix.emich.edu/$\sim$phoward/.) precise directions for constructing and utilizing the software program are incorporated within the book's advent, and technical aid is out there without delay from the authors.

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**Example text**

FORM 83. E(IJI) (Howard/Yorke [1989]): T-finite is equivalent to finite. Tarski [1924] and note 94. [83 A] E(Ia,II): Every T-finite set is amorphous. Howard/Yorke [1989], notes 57 and 94. FORM 84. E(IIJII) (Howard/Yorke [1989]): (\/x)(x is T-finite if and only if V(x) is Dedekind finite). Howard/Yorke [1989] and note 94. FORM 85. C(oo,Ho): Every family of denumerable sets has a choice function. 13. F O R M 88: C ( o o , 2 ) : A X I O M O F C H O I C E F O R PAIRS 39 [85 A] C(oo, < Ho): Every family of non-empty countable sets has a choice function.

14 AO] Suppose R is a commutative ring, A is a proper ideal in R, and S is a multiplicative semigroup in R not meeting A. Then there is a prime ideal p in R such that ACp and p n S = 0. Rav [1977] and note 80. [14 AP] Suppose to each finite subset F of a set I there corresponds a set &(F) of functions whose domains are subsets of I including F and such that (a) F1 C F2 implies $(F 2 ) C $(F 2 ) and (b) Vi G / , {

G. 8). FORM 21. If S is well ordered, {Ax : x G 5 } and {Bx : x G 5} are families of pairwise disjoint sets, and \AX\ — \BX\ for all x G 5, then | [jxeS Ax\ = \ [jxeS Bx\. G. Moore [1982]. FORM 22. UT(2*°, 2N°, 2*°): If every member of an infinite set of cardinality 2*° has power 2*°, then the union has power 2 N°. Borel [1898] and G. Moore [1982]. FORM 23. (Va)(Z7T(Ka, N a , N a )): For every ordinal a, if A and every member of A has cardinality N a , then | (J A\ = N a . G. Moore [1982]. FORM 24.