By A. Grzegorczyk

Recent years have visible the looks of many English-language hand books of good judgment and diverse monographs on topical discoveries within the foundations of arithmetic. those courses at the foundations of arithmetic as a complete are particularly tough for the rookies or refer the reader to different handbooks and diverse piecemeal contribu tions and likewise occasionally to principally conceived "mathematical fol klore" of unpublished effects. As targeted from those, the current publication is as effortless as attainable systematic exposition of the now classical ends up in the rules of arithmetic. therefore the ebook can be worthy specially for these readers who are looking to have the entire proofs performed in complete and all of the recommendations defined intimately. during this feel the ebook is self-contained. The reader's skill to bet isn't assumed, and the author's ambition used to be to minimize using such phrases as glaring and visible in proofs to a minimal. reason why the e-book, it really is believed, should be priceless in instructing or studying the root of arithmetic in these events within which the coed can't confer with a parallel lecture at the topic. this can be additionally the explanation that i don't insert within the e-book the final effects and the main modem and classy ways to the topic, which doesn't increase the fundamental wisdom in founda tions yet can discourage the newbie through their summary shape. A. G.

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**Sample text**

When comparing Definitions (45) and (46) of cardinal number with Definitions (67) and (68) we see that in fact X = [X]eqnum and CN = Abseqnum, so that cardinal numbers are abstraction classes of the relation eqnum in agreement with the above definitions. If we assume that a relation R is an equivalence relation, then, by (67) and (68), (69) a E AbsR == a E 2 z 1\ V Y E Z(z /\ (z € a == zRy», which also may be considered a definition of the set of abstraction classes of the relation R. On comparing Definition (64) of rational numbers with Definition (69), we can see that "fII" = Abs sfs for Z = dJt, so that the rational numbers are abstraction classes of the relation sfs between fractions.

The set of right-hand fingers is equinumerous with the set of left-hand fingers because we can put both hands together so that each finger of one hand touches one and only one finger of the other hand. This establishes a one-to-one correspondence between the elements of both sets. In general we shall say that two sets X and Yare equinumerous, X eqnum Y, if there is a one-to-one mapping f of the set X onto the set Y, that is, such that the whole set Y is exactly the image of the set X. Each element of the set X is then associated by the mapping f with exactly one element of the set Y, and conversely, each element of the set Y is associated by the mapping f with exactly one element of 39 FOUNDATIONS OF MATHEMATICS the set X.

Instead of a property, reference is made to the set of all elements having that property. Thus, the number 5 is the set of all sets containing five elements, or the set of all sets containing as many elements as a right hand has fingers. The number 2 is the set of all those sets which have as many elements as a normal man has ears, etc. Thus, in set theory the concept of equinumerosity is defined independently of the concept of number. By making this reasoning more precise we arrive at the strict concept of a natural number as a finite cardinal number.