Changes

The simplest place to see this seeming paradox is in the fact that all integers are finite, but that there are an infinite number of them.

The simplest place to see this seeming paradox is in the fact that all integers are finite, but that there are an infinite number of them.

*Why are all integers finite?  Because, if infinity were an integer, what would <math>\infty+1\,</math> be?

*Why are all integers finite?  Because, if infinity were an integer, what would <math>\infty+1\,</math> be?

*Why is the ''set'' of all integers an infinite set?  Because they go on forever.  There can't be a "biggest" integer, because we can always add one to any integer.  This kind of thinking is often schoolchildren's first introduction to [[logical reasoning]].   

*Why is the ''set'' of all integers an infinite set?  Because they go on forever.  If the set were finite, there would be a biggest integer.  But that can't be, because we can always add one to any integer.  This kind of thinking is often schoolchildren's first introduction to [[logical reasoning]].   

While infinity is not a number, it appears in many other contexts.  As we have seen, the size of the set of integers is infinite.  We say that the [[cardinality]] of the integers (this set is often denoted <math>\mathbb{Z}</math>) is infinite.  The rational numbers (<math>\mathbb{Q}</math>) and the real numbers (<math>\mathbb{R}</math>) also have infinite cardinality.  An interesting result from set theory (see below) is that <math>\mathbb{Z}</math> and <math>\mathbb{Q}</math> have the same cardinality, while <math>\mathbb{R}</math> has larger cardinality than the other two.

While infinity is not a number, it appears in many other contexts.  As we have seen, the size of the set of integers is infinite.  We say that the [[cardinality]] of the integers (this set is often denoted <math>\mathbb{Z}</math>) is infinite.  The rational numbers (<math>\mathbb{Q}</math>) and the real numbers (<math>\mathbb{R}</math>) also have infinite cardinality.  An interesting result from set theory (see below) is that <math>\mathbb{Z}</math> and <math>\mathbb{Q}</math> have the same cardinality, while <math>\mathbb{R}</math> has larger cardinality than the other two.

*<math>\lim_{x\to 0}\frac{1}{x} = \infty\ \ </math>Yes.  This is what was presumably meant by the incorrect statement above.

*<math>\lim_{x\to 0}\frac{1}{x} = \infty\ \ </math>Yes.  This is what was presumably meant by the incorrect statement above.

*<math>\int_0^1\frac{1}{x} = \infty\ \ </math>Infinity has a special meaning for integrals.

*<math>\int_0^1\frac{1}{x} = \infty\ \ </math>Infinity has a special meaning for integrals.

*<math>|\mathbb{Z}| = \infty\ \ </math>The cardinality of the integers is infinite.

*<math>\|\mathbb{Z}\| = \infty\ \ </math>The cardinality of the integers is infinite.

*<math>\mu(\mathbb{R}) = \infty\ \ </math>The (Lebesgue) measure of the reals is infinite.

*<math>\mu(\mathbb{R}) = \infty\ \ </math>The (Lebesgue) measure of the reals is infinite.