Cantor and Infinity

After a brief respite for evaluation, I posted a new essay with more details on Georg Cantor’s definition of Infinity. You can read it here at www.jacob2012.com/essays/cantor. This essay discusses the impact of Cantor’s work in mathematics and infinity, specifically the differences between finite numbers, a countable infinity, and uncountable infinity.

We seldom, if ever, come upon an uncountably infinity in our daily lives. We may think that Bill Gates and Warren Buffet have an infinite amount of money – and $56 billion may seem like an infinite amount of money that would take an entire lifetime to spend, but it is still finite. An example of a countable infinity are the natural or counting numbers (0, 1, 2, 3, 4, 5, …….). Even the set of all fractions – the Rational numbers – are a countable infinity since they can be mapped directly onto the natural numbers.

An example of an uncountable infinity are the Real Numbers. You can think of the Reals as all of the numbers between 0 and 1 – also denoted as the closed set [0,1]. If every person on Earth spent their entire lifetime counting the Real Numbers, we would never, ever count them all!

For a more in-depth discussion, go here http://pirate.shu.edu/projects/reals/infinity/uncntble.html and infinity http://www.mattababy.org/~belmonte/Publications/Books/CSaW/5_infinity.html

If you really concentrate on infinity and you get a headache or stomach-ache, then you’re on the right track in your attempt to feel and/or intuit infinity. Keep up the good work!

Questions for Cantor and Hubble

Let us consider only the discoveries of Cantor and Hubble for today. Both men significantly contributed to expanding the boundaries of Mankind’s understanding of the Universe. Cantor defined infinity as a mathematical entity in the 1890s. Hubble proved through astronomical observation that the Universe was considerably larger than the galactic womb of the Milky Way. The full essays are at www.jacob2012.com/essays/hubble and www.jacob2012.com/essays/cantor.

Cantor provided the definitions of countable and uncountable infinities. Cantor’s work on infinity was only a mathematical exercise before Hubble. Why? Before Hubble published his findings regarding the true size of the Universe in 1923, the heavens consisted of the Sun, Moon, the planets from Mercury to Saturn, and the Milky Way Galaxy which included all of the fixed stars and fuzzy nebulae. But Hubble’s work destroyed the ancient notion of the heavens by pushing the boundary of the known Universe out to the edge of Infinity. Since then, other astronomers have shown that the Universe is approximately 14 billions light years across.

Is space infinite? Scientists tell us that the Universe is expanding and that space-time has expanded along with the Universe. But is the Universe embedded in an infinite space? If space is not infinite, then prove there is nothing beyond the edge of the Universe. Ah, now that’s a question for only the heartiest of explorers to ponder.

Discussion of 5 Major Discoveries

Yesterday I posted a new essay on my companion website www.jacob2012.com/essays/5discoveries regarding 5 important things that were discovered over the last 120 years. Here they are again:
1) Cantor’s Definition of Infinity in the 1890s
2) Plank’s Discovery of Quantum in 1900
3) Einstein’s Theory of Relativity in 1905
4) Hubble’s Discovery of Galaxies in 1923
5) Aspect’s Experiments Demonstrating that Reality is Non-local in 1982

(This list is not meant to be definitive. They’re the important discoveries that fit within the scope of the point I am making. You may have other significant intellectual improvements besides the ones listed. Please feel free to send me a comment with your important discoveries and why you consider them significant.)

I list these 6 discoveries since they have led to a deeper understanding of Reality. Each has had a major impact and expanded the frontiers of our understanding of the really big, really small, and everything in between.

For example, let’s look at #1 Cantor’s Definition of Infinity. Mathematics is an ancient subject. Gauss considered mathematics “the Queen of the Sciences” (1). There have always been at least 2 main branches of math – one for daily use and one strictly theoretical (called Pure Mathematics). Daily math grew in complexity in relation to how business transactions grew more complicated. Pure math has witnessed some titans: Euclid, Pythagoras, and Archimedes in ancient times to Newton, Leibniz, Descartes, Gauss, Euler, and others too numerous to mention (plus I don’t wish for this to become a lesson in math history).

The subject of infinity was always beyond description and comprehension. Maybe infinity didn’t even exist. Have you ever seen one? With all humor aside, Georg Cantor devoted his life to providing rigorous definitions to the concepts of sets, transfinite numbers, cardinal numbers, and the distinction between countable and uncountable infinity. By 1900, his work established the study of infinity as a branch of mathematics.

From a metaphysical viewpoint, Cantor’s work is significant because it finally allowed mankind to discuss infinity with reliable terminology. Cantor provided definitions of the new terms he used to describe infinite quantities and mathematical proofs of the concepts. It should be noted that Cantor was a deeply religious Lutheran (2). I will be using Cantor’s work in future discussion in the hope of updating our definition and understanding of the only Infinite Being, God.


References
(1) The Quotation Page (2007) Retrieved from http://www.quotationspage.com/quote/30184.html
(2) Hedman, B. (1993) Cantor’s Concept of Infinity: Implications of Infinity for Contingence. Retrieved from http://www.asa3.org/asa/PSCF/1993/PSCF3-93Hedman.html