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Set concept has skilled a fast improvement in recent times, with significant advances in forcing, internal types, huge cardinals and descriptive set idea. the current publication covers every one of those parts, giving the reader an figuring out of the tips concerned. it may be used for introductory scholars and is wide and deep sufficient to deliver the reader close to the bounds of present learn. scholars and researchers within the box will locate the ebook useful either as a learn fabric and as a computer reference.

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3) The union of a countable family of countable sets is countable. 3) let A, be a countable set for each n E N. For each n, let us choose an enumeraA,: tion (ank: k E N) of A,. That gives us a projection of N x N onto u:=o Thus u;=,, is countable. A, (n* k ) H a n k 1. 40 AXlOMATlC SET THEORY In a similar fashion, one can prove a more general statement. 1. [US1I IS\ . s u p { I x I : x E S } . Proof: Let K = IS I and 3, = sup{ I X I : X E S}. We have S = { X u : a < K} and for each a < K, we choose an enumeration X, = { a a a :p < I,},where I, I 1.

It can be verified that if a and a’ are two different sequences, then ii # a’. 13. Every continued fraction is irrational. [Assume that ii = p / q . Then q = p . a. 1 +p . 14. If c is an irrational number, 0 < c < 1, then there is a continued fraction such that c = ii. [Construct the sequence (ao, a l , .. ] - Thus if we assign to each a E A’”the continued fraction F ( a ) = a + 1, where a + 1 = ( a , + 1 : n E N), we have a one-to-one correspondence F between the Bake space and the set of all irrational numbers in the unit interval [0, 11.

For every set X , I W ) l f 1x1 Proof: Let f be a function from X into P ( X ) . The set Y = {x E x :x # f ( x ) } 3. 23 CARDINAL NUMBERS is not in the range off: If z E X were such thatf(z) = Y, then z E Y if and only if z @ Y, a contradiction. Thus f is not a function of X onto P(X). Hence IP(X)I f 1x1. The ordering of cardinal numbers is defined as follows: 1x1 IPI if there exists a one-to-one mapping of X into Y. 1x1 < I Y I * 1x1 I I Y I ~ I X fI I Y I The relation I is clearly transitive.