We have rules of arithmetic and algebra that help us manipulate numbers, and we also have gut feelings, or intuitions, about numbers. It is clear, for example, that if we have a collection S of points, and add some more points to get a larger collection T, then the number of points in S is strictly smaller than the number of points in T. We also know that ‘100 per cent of S’ means the same as “all of S”.
These rules and intuitions are based on finite numbers, because that is what we are familiar with. When we start examining infinite numbers, some of those rules and intuitions break down.
For example, let S be the collection of points on a line; note that the line has finite (and positive) length, but it contains infinitely many points, and each point has no length. This already flies in the face of our intuitions – we have a line whose length is greater than 0, and it consists of points which each have length 0. This also means that, if we remove a single point P from S, we will have removed 0 per cent of the length, and the number of points will be reduced by 0 per cent; we are left with 100 per cent of the points (and 100 per cent of the length) of S, even though P has been removed.
We can do this over and over again, removing two points, or 10 points, or a million million points (any finite number in fact), and we will still have removed 0 per cent of the points. That is just how large infinity is, that even a millon million is 0 per cent of infinity.
Now consider a line T that is longer than S. We can easily map every point in S to a point in T, and have infinitely many points in T left over.
But we can also bend S and T into circles, and put the smaller circle inside the larger one, as shown in the figure. Now, for every point A on the outer circle, we draw a line from A to the centre; this passes through a point B on the inner circle. Every point on the outer circle maps to a unique point on the inner circle, and vice versa, so there are exactly as many points on the outer circle as there are on the inner circle.
Sound bites
• From the international scene: A graph G consists of a set of points called vertices, and some pairs of vertices may be joined by an edge. Consider a vertex v; the number of edges that include v is called the degree of v. If v has degree k, then there are k vertices (called neighbours) joined to v by an edge; if the neighbours have degrees d1, d2, ..., dk, then we say that v has vertex-type (d1, d2, ..., dk). Consider a graph with 10 vertices v0, v1, v2, …, v9. What is the smallest possible degree of v0 (for example)? What is the largest possible degree of v0? If v0 has degree 0, can any other vertex have degree 9? This demonstrates that the vertices of G cannot all have different degrees. However, they can all have different vertex-types. Schreyer, Walther and Mel’nikov [Vertex-oblique graphs, Discrete Mathematics 307 (2007)] showed that “almost all” graphs have different vertex-types. They also constructed graphs G such that the vertex-types of G and G’ are all different, where G’ is the complement of G, that is, the graph where two vertices are joined by an edge if, and only if, they are not joined in G.
• From the local scene: Alastair Farrugia [Dually vertex-oblique graphs, Discrete Mathematics 307 (2007)] showed that many of Schreyer’s graphs have a unique vertex-type sequence (the list of vertex-types of the vertices of a graph). Farrugia also constructed graphs G where the vertex-types of G are all different but the same as those of G’. Moreover, he showed that such graphs never have a unique vertex-type sequence.
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Did you know?
• Georg Cantor (1845-1918) was the first mathematician to show that there are different “levels” of infinity. Poincaré referred to Cantor’s ideas as a “grave disease” infecting mathematics, and that “most of the ideas of Cantor should be banished”. Kronecker called Cantor a “scientific charlatan”, “renegade” and “corrupter of youth”. The philosopher Wittgenstein, decades after Cantor’s death, dismissed Cantor’s concepts as “utter nonsense” that is “laughable” and “wrong”. However, David Hilbert wrote of the “paradise” that Cantor unfolded, and from which “we shall not be expelled”.
• Cantor defined ordinal numbers (used when counting) and also cardinal numbers (used to tell how large a collection of objects is). Finite ordinals are the same as cardinals; for example, we count 1, 2, 3, 4 (four different ordinal numbers), and we can have bags with 1, 2, 3 or 4 oranges (four bags of different sizes or “cardinalities”). After the natural numbers 0, 1, 2, ... comes the first infinite ordinal, ω, followed by ω+1, ω+2, ...; however, ω, ω+1, ω+2, ... describe bags of the same size (or “sets” of the same “cardinality”).
• At the other end of the scale are “infinitesimals”, things so small that there is no way to measure them. They were used by Archimedes (287-212 BC), Leibniz (1646-1716), Robinson (1918-1974) and others. Ironically, while Cantor was being attacked for his work on infinity, he himself described infinitesimals as “a cholera bacillus of mathematics”.
For more trivia see: www.um.edu.mt/think