Re: Infinity 101

2006-5-1 06:30:00

Larry quoting me:

"3. Infinity or infinite. I am using this term as it is used in mathematics. Whereas endless does not include the end point, infinity does. The infinite includes the beginning, middle and end. In math there is no such thing as infinity plus one because infinity always includes the plus one. Infinity includes the endless end."

Larry writes:

"One mathematical definition of infinity is the value of 1/x as x approaches 0. Mathematical infinity does not include an end point - that is the whole point. Per Cantor some infinities (in mathematics) are actually greater than others (which seems rather odd, but in fact subject to mathematical proof)."

JJ:

I suppose a number of mathematicians are playing around with different ideas and definitions, but when I took math in High School and college there was only one consistent definition in all the classes I took with several different teachers. Basically it is the one I have put forth which agrees with this online mathematical dictionary which gives it as: "A reference to a quantity larger than any specific integer."
(Reference: http://www.mathpropress.com/glossary/glossary.html#I)

In other words infinity is larger than any number that can be written.

Since any number can be reduced to writing then infinity does not exist -- only as a theoretical idea.

Wikipedia gives a similar definition: "Infinity is the state of being greater than any finite (real) number however large."

They also gave a reference to a train of thought that thinks as I do that infinity does not exist. It is found at:

http://en.wikipedia.org/wiki/Mathematical_constructivism

Here are some interesting comments from another math page at:

http://www.math.toronto.edu/mathnet/answers/infnotnumber.html

No "infinity" concept exists in the context of any number system, if by number system one means a collection of concepts that have operations like addition and multiplication the way familiar numbers do, operations which obey the usual properties of arithmetic.

One way to see this is to think, what would infinity minus 1 be? It couldn't be a finite number, since no finite number plus 1 equals infinity. So it must be infinite, and this would mean Infinity minus one equals infinity.

From this one can immediately see that the rules of arithmetic must be violated, since if they held one could subtract infinity from both sides to conclude that -1 = 0, which isn't true.

Therefore, there is no number system which possesses the usual rules of arithmetic and in which infinity exists. In other words, infinity does not exist, if by "exist" one means in the context of a number system.