We will give a simple philosophical "proof" of the negation of Cantor's continuum hypothesis (CH). (A formal proof for or against CH from the axioms of ZFC is impossible; see Cohen [1].) We will assume the axioms of ZFC together with intuitively clear axioms which are based on some intuition of Stuart Davidson and an old theorem of Sierpinski and are justified by the symmetry in a thought experiment throwing darts at the real number line. We will in fact show (...) why there must be an infinity of cardinalities between the integers and the reals. We will also show why Martin's Axiom must be false, and we will prove the extension of Fubini's Theorem for Lebesgue measure where joint measurability is not assumed. Following the philosophy--if you reject CH you are only two steps away from rejecting the axiom of choice (AC)--we will point out along the way some extensions of our intuition which contradict AC. (shrink)

Banach introduced the following two-person, perfect information, infinite game on the real numbers and asked the question: For which sets $A \subseteq \mathbf{R}$ is the game determined????? Rules: The two players alternate moves starting with player I. Each move a n is legal iff it is a real number and $0 , and for $n > 1, a_n . The first player to make an illegal move loses. Otherwise all moves are legal and I wins iff ∑ a n exists (...) and ∑ a n ∈ A. We will look at this game and some variations of it, called Banach games. In each case we attempt to find the relationship between Banach determinancy and the determinancy of other well-known and much-studied games. (shrink)

This isnotabout the symbolic manipulation of functions so popular these days. Rather it is about the more abstract, but infinitely less practical, problem of the primitive. Simply stated:Given a derivativef: ℝ → ℝ, how can we recover its primitive?The roots of this problem go back to the beginnings of calculus and it is even sometimes called “Newton's problem”. Historically, it has played a major role in the development of the theory of the integral. For example, it was Lebesgue's primary motivation (...) behind his theory of measure and integration. Indeed, the Lebesgue integral solves the primitive problem for the important special case whenfis bounded. Yet, as Lebesgue noted with apparent regret, there are very simple derivatives = 0,F=x2sinforx≠ 0) which cannot be inverted using his integral.The general problem of the primitive was finally solved in 1912 by A. Denjoy. But his integration process was more complicated than that of Lebesgue. Denjoy's basic idea was to first calculate the definite integralf dxover as many intervals as possible, using Lebesgue integration. Then, he showed that by using these results, the definite integral could be found over even more intervals, either by using the standard improper integral technique of Cauchy, or an extension technique developed by Lebesgue. (shrink)