Recall Euclid's proof that there is no limit to the list of prime numbers. One way to show this is that, by assuming that some number $latex {p}&fg=000000$ is the largest prime, we arrive at a contradiction. The idea is simple yet powerful. A Non-constructive Proof Suppose $latex {p}&fg=000000$ is prime and there are no … Continue reading The Axiom of Choice: Shoes & Socks and Non-constructive Proofs
Tag: Set Theory
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
Limits of Sequences, Limits of Sets
In undergraduate mathematics, we are confronted at an early stage with "Epsilon-Delta" definitions. For example, given a function $latex {f(x)}&fg=000000$ of a real variable, we may ask what is the value of the function for a particular value $latex {x=a}&fg=000000$. Maybe this is an easy question or maybe it is not. The epsilon-delta concept can … Continue reading Limits of Sequences, Limits of Sets
Sets that are Elements of Themselves: Verboten
Can a set be an element of itself? A simple example will provide an answer to this question. Let us define a set to be small if it has less than 100 elements. There are clearly an enormous number of small sets. For example, The set of continents. The set of Platonic solids. The set … Continue reading Sets that are Elements of Themselves: Verboten
The Size of Sets and the Length of Sets
Cardinals and Ordinals The cardinal number of a set is an indicator of the size of the set. It depends only on the elements of the set. Sets with the same cardinal number --- or cardinality --- are said to be equinumerate or (with unfortunate terminology) to be the same size. For finite sets there … Continue reading The Size of Sets and the Length 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
The Size of Things
In Euclidean geometry, all lengths, areas and volumes are relative. Once a unit of length is chosen, all other lengths are given in terms of this unit. Classical geometry could determine the lengths of straight lines, the areas of polygons and the volumes of simple solids. However, the lengths of curved lines, areas bounded by … Continue reading The Size of Things
Aleph, Beth, Continuum
Georg Cantor developed a remarkable theory of infinite sets. He was the first person to show that not all infinite sets are created equal. The number of elements in a set is indicated by its cardinality. Two sets with the same cardinal number are ``the same size''. For two finite sets, if there is a … Continue reading Aleph, Beth, Continuum
The Empty Set is Nothing to Worry About
Today's article is about nothing: nothing at all, as encapsulated in the number zero and the empty set. It took humanity millennia to move beyond the counting numbers. Zero emerged in several civilizations, first as a place-holder to denote a space or gap between digits, and later as a true number, which could be manipulated … Continue reading The Empty Set is Nothing to Worry About
Stan Ulam, a mathematician who figured how to initiate fusion
Stanislaw Ulam, born in Poland in 1909, was a key member of the remarkable Lvov School of Mathematics, which flourished in that city between the two world wars. Ulam studied mathematics at the Lvov Polytechnic Institute, getting his PhD in 1933. His original research was in abstract mathematics, but he later became interested in a … Continue reading Stan Ulam, a mathematician who figured how to initiate fusion
The Birth of Functional Analysis
Stefan Banach (1892–1945) was amongst the most influential mathematicians of the twentieth century and the greatest that Poland has produced. Born in Krakow, he studied in Lvov, graduating in 1914 just before the outbreak of World War I. He returned to Krakow where, by chance, he met another mathematician, Hugo Steinhaus who was already well-known. … Continue reading The Birth of Functional Analysis
Do you remember Venn?
Do you recall coming across those diagrams with overlapping circles that were popularised in the 'sixties', in conjunction with the “New Maths”. They were originally introduced around 1880 by John Venn, and now bear his name. John Venn Venn was a logician and philosopher, born in Hull, Yorkshire in 1834. He studied at Cambridge University, … Continue reading Do you remember Venn?
Degrees of Infinity
Many of us recall the sense of wonder we felt upon learning that there is no biggest number; for some of us, that wonder has never quite gone away. It is obvious that, given any counting number, one can be added to it to give a larger number. But the implication that there is no … Continue reading Degrees of Infinity
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