Universe: Infinity

Universe: Infinity

Published: February 14, 2008, 12:00 am
Updated: April 21, 2009, 8:44 pm
Lead Author: Sten Odenwald
Topics: Cosmology The Universe

Introduction

Philosophers have debated the issue of "Infinity" for as long as they have debated whether the Void exists. One of the problems in understanding infinity is that there are actually many different kinds of infinity known to mathematicians.

Cantor's Continuum

In 1873 Georg Cantor (1845 - 1918) thought about the question of how many points exist between 0 and 1 on the number line, although in the language of set theory this question is 'What is the cardinality of the set of points between 0 and 1? His answer was that there existed an uncountable infinity of points in such a set. In other words, there were more points between 0 and 1 than there were numbers in the set [1,2,3,4,.....]. Since the cardinality of the set [0,1,2,3...] is represented by the so-called first transfinite number, Aleph-null, the cardinality of the set of points between 0 and 1 must be a new kind of inconceivably vaster infinity called "Aleph-one." Because of the way in which transfinite numbers are multiplied together, the cardinality of an infinite 1-D line is also Aleph-one. Cantor then went on to ask what the cardinality of a 2-D infinite plane, a 3-D infinite volume and an N-dimensional manifold would be. Although he could not prove it, Cantor believed that the cardinalities of these sets would be the new transfinite numbers Aleph-2, Aleph-3 and so on.

Cantor's Continuum Problem was eventually solved this century when it was demonstrated that all N-dimensional manifolds have the same cardinality, namely, Aleph-one. The point of this is that there are countable infinities, and uncountable infinities, and mathematical manifolds are members of the second class no matter what their dimensionality. Against the uncountable infinity of points in a mid-sized 4-D manifold, our physical spacetime is a shining dwarf. Spacetime may approach the complexity and smoothness of a true mathematical manifold. This would only arise, however, if spacetime were constituted from a large enough network of virtual gravitons. I am not aware, however, of any proof that the number of virtual gravitons available to construct spacetime are as numerous as the uncountable infinity of points in a true mathematical manifold. This is especially true if there is a natural limit to the divisibility of spacetime set by the Planck scale, and the finite age of our universe.

Volume of the Visible Universe

Consider the total volume of our visible universe at the present time. In terms of Planck units, if we were to divide a sphere 15 billion lightyears in radius into boxes measuring 10-33 centimeters on a side, our visible universe would contain 10186 of these cells by the present time. In terms of a 4-dimensional volume, the age of the universe is 15 billion years, and a Planck unit of time is 10-43 seconds so the universe is 5 x 1060 Planck times old. The total 4-D volume is then about 10247 Planck Units. This is a big, in fact horrific, number but certainly not infinite and incomprehensible. It is also nowhere close to the incomprehensible magnitude of Aleph-one.

We can also ask whether a physical, spatial infinity really exists, or whether infinity in the physical world is just loose talk for saying that the rest of the universe beyond our local horizon is 10, 100 or 10 billion times bigger than our observable corner of it? Some cosmologists feel that a truly infinite universe, where the Cosmological Principle reigns supreme, is not a very appealing place to live. If the universe is truly infinite, then every event that can occur with even a miniscule but finite probability is destined to repeat itself an infinite number of times. As cosmologist Edward Harrison writes,

"What can be the point of [all this] when once is often more than enough? If eternity is silliness then the infinity of space is shear madness." 

Endless repetitions of events and configurations of matter, from DNA to planets and civilizations, occur in a truly endless universe. Somewhere "out there" beyond our observable universe lies the infinite stretches of a vaster cosmos containing an uncountable infinity (there's Aleph-one again) of events where at this very moment in Cosmic Time, countless copies of you and I exist.

Inflationary Cosmology

Since 1980, a new window onto this issue has opened up with the development by Alan Guth and Andrei Linde of Inflationary Cosmology. This elaboration of Big Bang cosmology proposes that our universe emerged from a tiny patch of a pre-existing spacetime that originally measured one trillion trillionth of a centimeter in size. Through the mechanism of inflation, it expanded by such an amount that, today, the actual size of this patch may be billions of times larger than our observable horizon. Inside this vastly inflated domain, the universe looks very much the way we see it here, but outside this patch in the remaining parent spacetime, things could be very queer indeed. According to Russian Cosmologist Andre Linde,

"After inflation, the universe became divided into many [expanding] domains with different kinds of [physical laws] and even with different dimensionality of its spacetime."

The conception of what we mean by infinity now becomes the idea of what we mean by it as something that can be realized in the physical world. In Gott's Bubble Cosmology, each bubble can be an infinite universe; one of many that inhabit the pre-existing de Sitter spacetime which is itself infinite. In Linde's meta-cosmology, infinity is also populated by domains in which the manifold property of dimensionality is not consistent so that the meta-spacetime is not a true mathematical manifold.

Infinite Universes

The idea that our universe is only one of an infinite number was first advocated by philosopher Gottfried Leibniz (1646 - 1716) in a 1714 book "The Monadology."

"There is an infinite number of possible universes, and as only one of them can be actual, there must be sufficient reason...which leads [God] to decide upon one rather than another."

To this is often tacked-on the statement rendered by Voltair's character Dr. Panglos in "Candide" who states that "This is the best of all possible worlds." Einstein himself often wondered openly whether God had any real choice in creating the universe the way he did, considering how well all of the natural laws fit so logically and harmoniously together.

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Preview Image

"MyCn18: An Hourglass Nebula" - Astronomers have recently used the Hubble Space Telescope (HST) to make a series of images of planetary nebulae, including the central star of this hourglass-shaped planetary nebula. Here, delicate rings of colorful glowing gas (nitrogen-red, hydrogen-green, and oxygen-blue) outline the tenuous walls of the "hourglass."   (Source: NASA's Astronomy Picture of the Day, January 18, 1996. NASA, R. Sahai and J. Trauger (JPL), WFPC2 Science Team, NASA.)

Citation

Odenwald, Sten, Ph.D. (Contributing Author); Bernard Haisch (Topic Editor). 2009. "Universe: Infinity." In: Encyclopedia of the Cosmos. Eds. Bernard Haisch and Joakim F. Lindblom (Redwood City, CA: Digital Universe Foundation). [First published February 14, 2008].
<http://www.cosmosportal.org/articles/view/138902/>

 

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