Undefined Equations Involving Infinity

I find it interesting that to this day, not all equations involving infinity can be defined. Is this because the answers have yet to be discovered? Or, is there simply no solution? Wikipedia states that the following equations involving infinity are undefined:

+/- infinity multiplied by 0 = undefined

infinity multiplied by negative infinity = undefined

+/- infinity dividided by +/- infinity = undefined

1 to the power of +/- infinity = undefined

Cardinalities

Infinity does not exist as a single number. Rather there are an infinite number of sets that represent infinite ranges of numeric values. Previously, we defined infinity as

1. The size of something infinite.Using the word in the context of sets is sloppy, since different infinite sets aren't necessarily the same size or cardinality as each other.

This definition presents the point that infinity does not refer to one size that can be measured. Rather infinity can have varying sizes. The range from -20 to infinity is a different size than the range of 10 to infinity. However, both are infinite in size. This phenomena can be understood through the concept of cardinality. Cardinality is defined as:

The number of elements in a set. If two setshave the same number of elements (i.e. there is a bijectionbetween them) then they have the same cardinality. Acardinality is thus an isomorphism class in the category of sets.

Cardinality helps to compare and quantify different sets of infinite numbers. This field was pioneered by the great German mathemetician, Georg Cantor. To learn more about Georg Cantor, go to http://en.wikipedia.org/wiki/Georg_Cantor.

Wikipedia (http://en.wikipedia.org/wiki/Infinity#Cardinality_of_the_continuum) talks about Cantor's contribution of cardinalities as a means of explaining infinity as follows:

A different type of "infinity" are the ordinal and cardinal infinities of set theory. Georg Cantor developed a system of transfinite numbers, in which the first transfinite cardinal is aleph-null (), the cardinality of the set of natural numbers. This modern mathematical conception of the quantitative infinite developed in the late nineteenth century from work by Cantor, Gottlob Frege, Richard Dedekind and others, using the idea of collections, or sets. Dedekind's approach was essentially to adopt the idea of one-to-one correspondence as a standard for comparing the size of sets, and to reject the view of Galileo (which derived from Euclid) that the whole cannot be the same size as the part. An infinite set can simply be defined as one having the same size as at least one of its " proper" parts; this notion of infinity is called Dedekind infinite.


References:

infinity. (n.d.). The Free On-line Dictionary of Computing. Retrieved November 19, 2006, from Dictionary.com website: http://dictionary.reference.com/browse/infinity

cardinality. (n.d.). The Free On-line Dictionary of Computing. Retrieved November 19, 2006, from Dictionary.com website: http://dictionary.reference.com/search?q=cardinality

Personal Search for Infinity

My personal interest in infinity began when I was young. I was always a kid that asked why? I would ponder long spans about things that I could not comprehend, such as flight, space, stars and wars.

One concept I could never get my arms around was infinity. When I was told the universe had no boundaries, I became very intrigued. I also became intrigued when my early math teacher told me that any real number added to infinity or subtracted from infinity (besides infinity) would result in infinity. These answers never made sense.

Through the years, I casually explored these concepts. I learned that the universe is indeed finite. Science has proven that the universe is expanding. However, the universe cannot expand unless it is finite in size (http://en.wikipedia.org/wiki/Universe). I also learned that infinity can have a limit and can be expanded or contracted.

For example: The range of "5 to infinity" has a lower limit of "5". However, the range of numbers is still infinite. This infinite range can be expanded by 4 if we reduce the lower limit to "1" resulting in the range of "1 to infinity". Both ranges are infinite, but both have different "cardinalities".

A cardinality is defined by the Free Online Dictionary of Computing as:

The number of elements in a set. If two setshave the same number of elements (i.e. there is a bijection between them) then they have the same cardinality. Acardinality is thus an isomorphism class in the category of sets.aleph 0 is defined as the cardinality of the first infinite ordinal, omega (the number of naturalnumbers).

This perplexed me and began my quest to understand this amazing mathematical concept. I will delve into the concept of cardinality (hence the title of my blog) in my next post.

References:

cardinality. (n.d.). The Free On-line Dictionary of Computing. Retrieved November 17, 2006, from Dictionary.com website: http://dictionary.reference.com/browse/cardinality