Infinity
From Conservapedia
Infinity is a designation for that condition which goes on without end. It is not an actual number that can be quantified, but a value which is approached as a limit. It can not be empirically tested as one can never have a sample size of infinity, but logically we know it exists. Infinity has important theoretical applications in mathematics.
An example where infinity can be seen as a limit would be in any attempt to divide by zero. The result is an undefined value, but it can be seen with the function Fx = 1 / x that as x approaches the value of zero, that the resulting answer approaches infinity.
Georg Cantor's diagonal argument is an elegant proof demonstrating that the infinity of real numbers is greater than the infinity of countable integers. 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.
To formalize countably and uncountably infinite, we need the set theory concept of cardinality. Using set theory it can be shown that there are infinitely many distinct infinite cardinalities.
Infinity is written using the symbol ∞
However, in some non-standard model of Peano Arithmetic, ∞ is treated as an actual number.
In Zermelo-Fraenkel set theory, one must assert that an infinite set exists. This is known as the Axiom of Infinity. Constructive mathematicians and practicians of elementary techniques deny this axiom believing that the world and human thought are necessarily finite, the realm of the infinite being the realm solely of God.
