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

A Philosophical Interpretation of Infinity

Infinity, like zero, goes beyond simple numerics and mathematics. It is a philosophical principle as well. Many great minds have pondered the role of infinity in philosophy. Some of my favorite interpretations include:

The Traversal of the Infinite: Discusses the impact of infinity on the philosophies of Kant and Cantor

Philosophy and the Infinite: A link to various interpretations of infinity through the lense of philosophy. It includes some fascinating looks at how the infinite has been defined and applied to the liberal arts through the ages.

Forum for the Philosophy of Infinity: Provides a lot of good discussion about the application of infiity to non-mathematical concepts such as time, existence and life.

To understand infinity, you have to step outside of mathematics and try to determine how it applies to life and the universe we live in. Trying to understand infinity and its potential merely through equations would be like trying to learn basketball by reading a book rather than playing the game.

Cardinalities of Infinity

Welcome to the Cardinalities of Infinity Blog! This blog is dedicated to exploring infinite sums and the underlying mathematics and philosophical interpretation thereof.

Before diving into the more abstract facets of infinity, we need to understand what it is and is not.

Infinity, in its broad application, is defined by www.dictionary.com as:


1. the quality or state of being infinite.
2. something that is infinite.
3. infinite space, time, or quantity.
4. an infinite extent, amount, or number.
5. an indefinitely great amount or number.
6. Mathematics.
a. the assumed limit of a sequence, series, etc., that increases without bound.
b. infinite distance or an infinitely distant part of space.
7. Photography.
a. a distance between a subject and the camera so great that rays of light reflected from the subject may be regarded as parallel.
b. a distance setting of the camera lens beyond which everything is in focus.

A more useful definition for the purpose of this blog is provided by The Free Online Dictionary of Computing:

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 cardinality as each other.

In the next blog, we will discuss some of the basic principles and misnomers surrounding infinity.

References:

infinity. (n.d.). Dictionary.com Unabridged (v 1.0.1). Retrieved November 15, 2006, from Dictionary.com website: http://dictionary.reference.com/browse/infinity

infinity. (n.d.). The Free On-line Dictionary of Computing. Retrieved November 15, 2006, from Dictionary.com website: http://dictionary.reference.com/browse/infinity