Metaphysics of Infinities

All are welcome! Just take a look at a few links if you want some ideas. What are infinities (or what is infinity?) Is the concept of infinity useful? Does anything infinite exist? How do we obtain knowledge about infinities? What kinds of infinities are there? And if interest permits: is it possible to "discover" a measure over an infinite space of possible worlds that might explain observations in our universe?

Links of interest (provide any of your own as a reply!):

General info on Infinity
https://en.wikipedia.org/wiki/Infinity

Actual Infinity
https://en.wikipedia.org/wiki/Actual_infinity

• Distinguishing between the idea of the potentially infinite and the actually infinite.

Zeno's paradoxes
https://en.wikipedia.org/wiki/Zeno%27s_paradoxes

• Some important paradoxes about motion and time.

Standard Calculus
https://en.wikipedia.org/wiki/Calculus

• An important area involving the application of the concepts of infinity.

Ross-Littlewood Paradox
https://en.wikipedia.org/wiki/Ross%E2%80%93Littlewood_paradox

• A paradox demonstrating the counterintuitiveness of infinity in terms of the indeterminate form "infinity minus infinity."

Hilbert's paradox of the Grand Hotel
https://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel

• If an infinite number of buses show up, each with an infinite number of people in them, and you have an infinite number of rooms listed 1, 2, 3, ... etc., then how can you accommodate everyone with a room? And related issues.

Cardinality of the continuum
https://en.wikipedia.org/wiki/Cardinality_of_the_continuum

• The set of natural numbers {1, 2, 3, ...} is countably infinite. This link discusses the cardinality of the continuum, which is uncountably infinite (in other words, a larger kind of infinity.)

Banach-Tarski Paradox
https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox

• The Ross-Littlewood paradox involves countable infinity, whereas the Banach-Tarski paradox shows an even stranger result involving the cardinality of the continuum. A sphere is partitioned into five components which are then translated and rotated to produce two exact copies of the original sphere.

Cardinal Numbers
https://en.wikipedia.org/wiki/Cardinal_number

• A cardinal number measures the size of a collection (or set).

Ordinal Numbers
https://en.wikipedia.org/wiki/Ordinal_number
0 = {}, 1 = {{}} = {0}, 2 = { {}, {{}} } = {0, 1}, 3 = { {}, {{}}, { {}, {{}} } } = {0, 1, 2}, n = {0, 1, 2,3, ..., n - 1}, w = {0, 1, 2, 3, ..., n, ...}, w + 1 = {0, 1, 2, 3, ..., w}, and it keeps going. Each ordinal number consists of all the ordinal numbers preceding it. There are countable and uncountable ordinals. Infinite ordinals can be used to measure the strength of a mathematical theory.

Large Countable Ordinals
https://en.wikipedia.org/wiki/Large_countable_ordinal

• Infinite ordinal numbers that are countable.

Large Cardinals
https://en.wikipedia.org/wiki/Large_cardinal

• Large cardinal axioms can be added to a mathematical theory to allow for infinities of various unimaginable sizes.

Absolute Infinite
https://en.wikipedia.org/wiki/Absolute_Infinite

• Cantor's concept of a kind of infinity which transcends in size all of the infinities describable in a given set theory.

Measure Theory
https://en.wikipedia.org/wiki/Measure_(mathematics)

• The theory of assigning numbers to subsets of a set which intuitively capture how large each subset is relative to every other subset. If the universe is infinite, or if there are an infinite number of universes, then for the sake of science one has to define a measure over the the space of possible worlds; for example, what proportion of all worlds in the universe/multiverse are habitable? If there are infinitely many habitable worlds, and infinitely many uninhabitable worlds, then how do we determine what proportion of worlds are habitable?