By Iain T. Adamson
Offers a proper description of set concept according to the Von Neumann-Bernays-Godel axiomatic strategy utilizing the concept that of periods. Covers the root of the speculation, kin, ordinals, cardinals, and the axiom of selection. Paper. DLC: Set concept.
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Additional resources for A Set Theory Workbook
Ifwe denote the elements by the numbers 1,2,3,4,5,6, 7, 8, then the following are the combinations of2 (or, more simply, pairs) that can be generated from them 12 13 I 4 2 3 2 4 25 34 35 3 6 4 5 4 6 4 7 1 5 I 6 2 6 3 7 4 8 2 7 3 8 1 7 2 8 5 6 5 7 5 8 6 7 68 78 1 8 We can see very well that the number of these pairs is (from right to left) 1+2+3+4+5+6+7 On the other hand we might have argued that any element can be paired with the remaining 7, so the 8 elements would yield 8 x 7 pairs; but we have counted every pair twice, first when we were pairing the first term and then when we were pairing the second.
Susie said this instead of 1+2+3+4+5+6+7 She said 4 instead of 1, that is 3 more. But she also said 4 instead of 7, that is 3 less, and these equal out. In the same way it is true that 4 is 2 more that 2, but it is 2 less than 6 and so these too equal out. ' I had to give Eve her due; I could never have explained it so well myself. These little unprejudiced researchers make some extraordinary observations. 'It's like an exercise book,' exclaimed Mary, another little pupil. ' Pure interest was guiding these little researchers.
So the numbers in the Pascal Triangle occur as coefficients of descending powers of lOin the expression of the powers of 10 + 1. The second term in 10 + 1 is 1, and every power of! is still 1 (since 1 X 1 = 1), and so it does not appear that the powers of the second term come into the expression at all. But we can smuggle them in in the following way: 1IS = 1331 = 1000 + 300 + 30 + 1 = 1 X 103 + 3 X 102 X 1 + 3 X 10 X 12 + 1 xis We see that while the powers of the first term decrease, the powers of the second term increase.