Changes

Another place where infinity is considered acceptable is in [[Floating-point]] arithmetic in computers.  The IEEE-754 standard (which all modern computers support) allows the values <math>\infty\,</math> and <math>- \infty\,</math>. But their treatment is very different from other numbers, and computer floating-point numbers don't faithfully model the real numbers in any case.

Another place where infinity is considered acceptable is in [[Floating-point]] arithmetic in computers.  The IEEE-754 standard (which all modern computers support) allows the values <math>\infty\,</math> and <math>- \infty\,</math>. But their treatment is very different from other numbers, and computer floating-point numbers don't faithfully model the real numbers in any case.

Also, in some non-standard models of [[Peano Arithmetic]], <math>\infty\,</math> is treated as an actual number. But, once again, this is not standard mathematics.

Also, in some non-standard models of Peano Arithmetic, <math>\infty\,</math> is treated as an actual number. But, once again, this is not standard mathematics.

[[Georg Cantor]]'s [[diagonalization|diagonal argument]] is an elegant proof demonstrating that the (infinite) cardinality of [[real number]]s is greater than the (infinite) cardinality of [[countable]] [[integer]]s.  The essence of the argument is that in any proposed list of all real numbers, a new real number not in the list can be constructed by taking the digits in a [[diagonal]] through the list and changing them to construct a new real number that differs from the nth entry at the nth position right of the decimal point.   

[[Georg Cantor]]'s [[diagonalization|diagonal argument]] is an elegant proof demonstrating that the (infinite) cardinality of [[real number]]s is greater than the (infinite) cardinality of [[countable]] [[integer]]s.  The essence of the argument is that in any proposed list of all real numbers, a new real number not in the list can be constructed by taking the digits in a [[diagonal]] through the list and changing them to construct a new real number that differs from the nth entry at the nth position right of the decimal point.