To See a World in a Grain of Sand

Gaze up at the night sky and you catch a glimpse of infinity. The stars seem numberless; the immensity of the universe is profoundly impressive, leading us to wonder about the nature of our cosmic home. Is space finite or unlimited in extent? If it is bounded, what lies outside? The notion of infinity is … Continue reading To See a World in a Grain of Sand →

The Sizes of Sets

The sizes of collections of objects may be determined with the help of one or other of two principles. Let us denote the size of a set $latex {A}&fg=000000$ by $latex {\mathfrak{Size}(A)}&fg=000000$. Then AP: Aristotle's Principle. If $latex {A}&fg=000000$ is a proper subset of $latex {B}&fg=000000$, then $latex {\mathfrak{Size}(A) < \mathfrak{Size}(B)}&fg=000000$. CP: Cantor's Principle. $latex … Continue reading The Sizes of Sets →

The Whole is Greater than the Part — Or is it?

Euclid flourished about fifty years after Aristotle and was certainly familiar with Aristotle's Logic. Euclid's organization of the work of earlier geometers was truly innovative. His results depended upon basic assumptions, called axioms and “common notions”. There are in total 23 definitions, five axioms and five common notions in The Elements. The axioms, or postulates, … Continue reading The Whole is Greater than the Part — Or is it? →

Cantor’s Theorem and the Unending Hierarchy of Infinities

In 1891, Georg Cantor published a seminal paper, U"ber eine elementare Frage der Mannigfaltigkeitslehren --- On an elementary question of the theory of manifolds --- in which his ``diagonal argument'' first appeared. He proved a general theorem which showed, in particular, that the set of real numbers is uncountable, that is, it has cardinality greater … Continue reading Cantor’s Theorem and the Unending Hierarchy of Infinities →