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'''Infinity''' is an amount that is impossible to diminish and impossible to exceed.  Infinity is similar to zero in that neither can be diminished, but infinity is different from zero in that zero can be exceeded.<ref>Merrian-Webster defines infinity as an "unlimited extent of time, space, or quantity ... an indefinitely great number or amount." [http://www.merriam-webster.com/dictionary/infinity Merriam-Webster dictionary] Put another way, infinity is something which "goes on without end."</ref>  An example is the [[Multiplication of the loaves]] and the master's estate in the [[Prodigal Son]], considered perhaps the most insightful story in all of written history.  Also, one's soul is of infinite value to him and cannot be exceeded in value by anything else, as explained by Matthew 16:26.

'''Infinity''' is an amount that is impossible to diminish or exceed.   

 

 

Infinity has surprising similarities to [[zero]].  Neither can be diminished, and the multiplication or division of either cannot change it.  Yet in other ways they are opposites.

 

An example is the [[Multiplication of the loaves]] and the master's estate in the [[Prodigal Son]], considered perhaps the most insightful story in all of written history.  Also, one's soul is of infinite value to him and cannot be exceeded in value by anything else, as explained by Matthew 16:26.

[[Jesus]] taught the existence of infinity in multiple ways.  Indeed, the [[Gospels]] and [[Epistle to the Hebrews]] are like a book of [[logic]] about infinity.  This was contrary to the views of the greatest Greek mathematician and philosopher, [[Pythagoras]] and [[Aristotle]], who both incorrectly rejected that infinity could exist.<ref>See [[Math and the Bible]].</ref>  The [[Old Testament]] has only indirect references to infinity.

[[Jesus]] taught the existence of infinity in multiple ways.  Indeed, the [[Gospels]] and [[Epistle to the Hebrews]] are like a book of [[logic]] about infinity.  This was contrary to the views of the greatest Greek mathematician and philosopher, [[Pythagoras]] and [[Aristotle]], who both incorrectly rejected that infinity could exist.<ref>See [[Math and the Bible]].</ref>  The [[Old Testament]] has only indirect references to infinity.

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. 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]].   

*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]].<ref>Merrian-Webster defines infinity as an "unlimited extent of time, space, or quantity ... an indefinitely great number or amount." [http://www.merriam-webster.com/dictionary/infinity Merriam-Webster dictionary] Put another way, infinity is something which "goes on without end."</ref>

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.