By A. Grzegorczyk
Recent years have noticeable the looks of many English-language hand books of common sense and various monographs on topical discoveries within the foundations of arithmetic. those guides at the foundations of arithmetic as an entire are really tough for the novices or refer the reader to different handbooks and numerous piecemeal contribu tions and likewise occasionally to principally conceived "mathematical fol klore" of unpublished effects. As exact from those, the current booklet is as effortless as attainable systematic exposition of the now classical leads to the principles of arithmetic. as a result the ebook will be invaluable specially for these readers who are looking to have all of the proofs conducted in complete and all of the recommendations defined intimately. during this experience the publication is self-contained. The reader's skill to bet isn't assumed, and the author's ambition was once to minimize using such phrases as obvious and visible in proofs to a minimal. for the reason that the ebook, it really is believed, might be beneficial in instructing or studying the root of arithmetic in these occasions within which the scholar can't confer with a parallel lecture at the topic. this can be additionally the explanation that i don't insert within the publication the final effects and the main modem and stylish methods to the topic, which doesn't improve the basic wisdom in founda tions yet can discourage the newbie by means of their summary shape. A. G.
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Additional info for An Outline of Mathematical Logic: Fundamental Results and Notions Explained with all Details
The operations determined by formulae (65) and (66) do not, as can easily be verified, lead outside the set iY, so that if a, b E iY, then (a+=b) E iY and (ax b) EiY. It can also easily be seen that these operations are associative and commutative, that multiplication is distributive with respect to addition, and that the other laws of the arithmetic of rational numbers hold. The system (iY, +=, x) is thus the mathematical domain called the field of rational numbers. EXERCISES 1. Define the addition of integers using equivalence relations of the form xe(a+b) == ...
U E [Y]R, by (67), and hence u E a. We have thus proved that, for any element u, if u E a, then U E b, and conversely, if U E b, then u Ea. 51 FOUNDATIONS OF MATHEMATICS Hence, u e a if and only if u e b (the rule of logic (23) has been used here). The conclusion just arrived at is formulated in symbols 1\ u(uea == ueb). By the axiom of extensionality this formula implies a = b. We have thus proved formula (70). The following formulae are easy to prove: (71) (72) x E [X]R for any x e Z, xRy == [X]R = [Y]R for any x, y e Z, and will be used below.
Where the right hand member of this equivalence contains only the constants V, /\, V, .... , ==, =, E, and variables. 2. Reformulate definition (40) without using the pair symbol side of the equivalence symbol. Use the result of exercise 1. 1\ . <>on the right 3. How many different subsets are there of a set containing n elements? ) Check it for 2, 3, 4 and give a general proof. 4. How many different ordered pairs are there in an n-element set? 5. How many different functions are there defined on an n-element set and taking on values in a k-element set?
An Outline of Mathematical Logic: Fundamental Results and Notions Explained with all Details by A. Grzegorczyk