  ### Infinities

April 30, 2018

As a middle school math student, I was quite intrigued by the concept of mathematical infinity. Infinity had a sexy symbol, ∞, invented by mathematician, John Wallis, and used in his 1655 book, De sectionibus conicis. Infinity is generated simply in division by zero, a concept that even a child could comprehend.

The ∞ symbol was in the title sequence of the early 1960s television series, Ben Casey, that premiered the year that I started high school. In that title sequence, cast member, Sam Jaffe, drew symbols for man, woman, birth, death, and infinity on a blackboard. Surprisingly for that era, the birth symbol was not the ankh. Symbols for male gender, female gender, a not principally recognized symbol for birth, the Egyptian hieroglyphic symbol for life, ankh, and symbols for death and infinity. These symbols, without the ankh, were in the title sequence for the 1960s television series, Ben Casey. (Created using Inkscape

Some years after middle school, I discovered that there were more than the one infinity I had learned. The principle that some infinities are greater than others was proved by the mathematician, Georg Cantor (1845-1918), in 1874, who published a general method of proof of such, his diagonal argument, in 1891. This proof, outlined in the figure, was used by Kurt Gödel (1906-1978) in his incompleteness theorems, one of which states that the truth or falsity of some mathematical statements cannot be determined; that is, there are limits to mathematical knowledge. Cantor's diagonal argument for the proof that there are more real numbers than natural numbers.

Suppose that you create a table of real numbers, in this case the binary fractions shown, and associate each of these numbers with a natural number sn.

There is an infinite number of natural numbers.

Next, create a new number s by making its n-th digit a different digit from the numbers sn.

This real number is obviously not associated with a natural number, and the idea that the members of the set of real numbers is greater than the members of the set of natural numbers makes one infinity greater than another.

(Wikimedia Commons image)

The technical term for the size of a set is its cardinality, and Cantor named a progression of infinite cardinalities using the Hebrew letter, aleph, א, with (א)0 symbolizing the smallest infinity and successive aleph numbers (א)n symbolizing larger infinities. The cardinality of the natural numbers is the smallest infinity, (א)0. The cardinality of the real numbers is usually considered to be (א)1; however, a Gödel incompleteness theorem makes this one of the undecidable propositions of mathematics.

The earliest example of the idea of infinity is in the works of the Pre-Socratic Greek philosopher, Anaximander (c.610 BC-c.546 BC), a disciple of Thales of Miletus (c.624 BC - c.546 BC). I mentioned Thales in a previous article (The Square Root of Two, June 6, 2016). Anaximander is known, also, for the idea that humans descended from the sea from parasites living in the mouths of fish.

The mathematical properties of infinity were first explicated by Zeno of Elea (c.490 BC - c.430 BC) in his paradoxes, one of which I wrote about in an earlier article (Landslide Sensor, November 15, 2010). Fast forward more than two millennia, and Isaac Newton (1642-1727) was creating his infinitessimal calculus and writing about equations with an infinite number of terms in his 1699 De analysi per aequationes numero terminorum infinitas (On Analysis by Equations with an infinite number of terms). A good summary of Newton's adventures with the infinite can be found in ref. 3.

Aristotle (384-322 BC) considered that the four elements of the world, Earth (Γη), Air (ʾΑηρ), Fire (Πυρ), and Water (ʿΥδωρ), had a tendency to go to their "natural place" in the absence of any holding force. Thus, Earth will tend towards the geocenter, Aritotle's center of the universe as identified as the center of the Earth. Water seeks also to reach the geocenter, but it's impeded by the presence of Earth. Since Air bubbles out of Water, its place is next above Earth and Water, while Fire seeks the highest accessible place just short of the celestial sphere. These four elements were later augmented by a fifth element, the Aether, which didn't have a natural place. The four classical elements, (Earth ΓΗ, Air ʾΑΗΡ, Fire ΠΥΡ, and Water ʿΥΔΩΡ), along with the four qualities of matter (Dry ξηρον, Wet ʿυγρον, Hot θερμον, and Cold ψυχρον).

Earth is both cold and dry, Air is both hot and wet, Fire is both hot and dry, and water is both cold and wet.

The particular properties of a composite material derive from an amalgamation of these element qualities in their proportion.

(Created using Inkscape. Click for larger image.)

Juliano C. S. Neves of the Universidade Estadual de Campinas (UNICAMP) and the Instituto de Matemática, Estatística e Computação Científica (Campinas, SP, Brazil), has recently published an article on arXiv in which he examines how conformal infinities in general relativity are analogous to the idea of natural places in Aristotelian physics. While modern physicists don't consciously try to emulate Aristotle in their theories, it's interesting how the same ideas circulate in human thought in different forms.

As an example, Neves gives the Carter-Penrose diagram for Minkowski spacetime, which combines the three Euclidean dimensions of space with an extra time dimension. A Carter- Penrose diagram is a two-dimensional representation of the causal relations between different points in spacetime where the horizontal axis represents space, the vertical axis represents time, and lines slanted at 45° correspond to light rays. Carter-Penrose diagram for Minkowski spacetime with r = 0 indicating the origin of the coordinate system.

This diagram contains the following infinities: i (past timelike infinity), i+ (future timelike infinity), and i0 (spatial infinity). Also indicated are the lines of past null infinity and future null infinity.

(Created using Inkscape from data in ref. 4.)

Timelike, spacelike, and null geodesics lead to the concepts of timelike infinity, null infinity, and spacelike infinity as places for bodies along these geodesics. As seen in the diagram, bodies on spacelike geodesics have the point i0, spatial infinity, as their origin and end, while bodies traveling on a timelike geodesic have i, the past timelike infinity, as an origin, and i+, the future timelike infinity, as an end to their trajectory. Photons have i+ and i as their natural places, while massive particles have the two lines of future and past null infinity as the boundaries of their natural places.

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