The History of Mathematics: A Brief Course

542 19. LOGIC AND SET THEORY x^2 = ÷ ÷(1 — ÷) = 0, that is, no object can have a property and simultaneously not have that prope ...

LOGIC 543 proudly proclaimed that "Not only... will 'no knowledge of mathematics beyond the simple rules of Arithmetic' be req ...

544 19. LOGIC AND SET THEORY against the principles of induction and cause in the preceding century. Venn's reduction risked exp ...

SET THEORY 545 His object was to show that F(x) must be a linear function, so that G(x) — F(x) — Ax — Â — \a.QX^2 would be a q ...

546 19. LOGIC AND SET THEORY we now call it) might be easily describable, Cantor needed to discuss the general case. He needed t ...

SET THEORY 547 the Borel sets, which is the smallest class that contains all closed subsets and also contains the complement o ...

548 19. LOGIC AND SET THEORY Beyond that point, everything was human-made and therefore had to be finite. If you spoke of a numb ...

SET THEORY 549 set A, one could conceive of another set whose members were the subsets of A. This set is nowadays denoted 2 A ...

550 19. LOGIC AND SET THEORY it is consistent to assume that the real numbers can be expressed as a countable union of countable ...

SET THEORY 551 when referring to an object that was defined effectively, through a finite number of uses of well-defined opera ...

552 19. LOGIC AND SET THEORY The most influential figure in mathematical logic during the twentieth century was Godel. The probl ...

PHILOSOPHIES OF MATHEMATICS 553 numbers on this one relation and nine axioms, together with a symbolic logic that he had devel ...

554 19. LOGIC AND SET THEORY if we apply Cantor's argument to this mapping, we are led to consider S — [E : Å £ E}. By definitio ...

PHILOSOPHIES OF MATHEMATICS 555 approach (which can be reversed, once the analysis is finished, to synthesize a given well-for ...

556 19. LOGIC AND SET THEORY mathematicians, since such proofs seldom give insight into the structures being studied. For exampl ...

QUESTIONS AND PROBLEMS 557 that C £ C. For an object ÷ to belong to C, it no longer suffices that ÷ £ ÷; it must also be true th ...

558 19. LOGIC AND SET THEORY FIGURE 1. The Brouwer fixed-point theorem. Why should you conclude the latter rather than the forme ...

QUESTIONS AND PROBLEMS 559 one of the regular class meetings the following week. One of the brighter students then reasons as fo ...

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Literature Note: CSHPM/SCHPM = Canadian Society for the History and Philosophy of Mathematics/Societe Canadienne d'Histoire et P ...