Using cardinals to order infinities
Infinity is a concept (not a number) which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol ∞.
The ancient Indians and the Greeks did not define infinity mathematically, and approached it as a philosophical concept. By the 17th century, European mathematicians started using infinite numbers and infinite expressions systematically until it was formally used in calculus, where infinity was used as an unbounded limit. Similarly, an infinitesimal is a quantity that is closer to zero than any standard real number, but not zero. So in calculus you can approach infinity as a limit but never reach it (for example when trying to add all natural numbers), or approach zero as a limit but never reach it, or approach any number such as 1 by using the notation 0.999..., where the decimals repeat infinitely but the number never reaches 1.
It is a computing challenge to calculate the decimal representation of π to as many digits as possible, but its digits are infinite. Similarly, there is an infinity of prime numbers, but there is an ongoing search for the next largest prime.
Transfinity is different from infinity, in that it can reach really large numbers but never infinity, which is not a number but a concept. The multiverse has a transfinite number of universes (due to causality) but a megaverse and beyond are infinite.
Infinity is fuzzy in that the set of odd or even numbers is clearly smaller than the set of natural numbers. So how is it possible to have larger infinities? In set theory, this is resolved by introducing a system of transfinite numbers where the first transfinite cardinal is aleph-null (ℵ0), the cardinality (count of elements) of the set of natural numbers. So the sequence of all positive odd integers followed by all positive even integers {1,3,5...2,4,6...} is an ordering of the set (with cardinality ℵ0 ) of positive integers.
Transfinity was coined by mathematician Georg Cantor in 1895. He also came up with the Absolute Infinite (symbol omega or Ω) which he defined in set theory to be an amount that is larger than any finite or transfinite quantity. An example is that the set of all real numbers (uncountable infinity) is much larger than the set of natural numbers (countable infinity), in that you can get an infinity of decimal points just between 0 and 1, and so on. This means that any subset of an infinite set has a smaller cardinality than said infinite set, meaning that there's an infinite number of infinities.
Cantor also linked the Absolute Infinite with God who resides in (or is) the omniverse, an eternal abode. An eternity is an infinite amount of time.
A poetic view of infinity is the start of Auguries of Innocence by William Blake, which goes:
- "To see a World in a Grain of Sand, And a Heaven in a Wild Flower, Hold Infinity in the palm of your hand, And Eternity in an hour."
Could this have a deeper meaning, such as suggest there is some truth to the holographic principle?
Infinity breaks the traditional laws of physics and introduces new laws in many ways, including:
- Tachyons increase in speed as their energy decreases, and would require infinite energy to slow down to the speed of light; conversely instantaneous travel would require an infinitesimally small amount of energy, or ZPE.
- At the center of a black hole is a gravitational singularity, a region where the spacetime curvature becomes infinite. The singularity is a single point (0d) with zero volume and infinite density.
- The megaverse has infinite dimensions, but is a subset of the omniverse. The hyperverse contains dimensions from 12 to infinity, of which the omniverse is a dimensional subset. A Hilbert space can be infinite-dimensional.
- Gravity can result in an infinitesimally small amount of gravitons that leak into lower spacetime dimensions.
- Planck units can be infinitesimally small or transfinite, but never infinite.
See also
A final thought
The Infinite monkey theorem states that any sequence of events which has a non-zero probability of happening will almost certainly eventually occur, given unlimited time. The Many-worlds interpretation states that every possible outcome of a quantum event exists in its own universe. Does this imply that a megaverse which contains infinite universe can contain an infinity of fictional, imagined universes... or is that a larger set of infinity? Would this be contained in the infinite-dimensional hyperverse, which includes all thought dimensions?