Infinity

Infinities

In math, infinities come in different sizes. The most basic kind of infinity, and the smallest one, is the infinity of natural numbers. This is called a countable infinity because one can produce a bijection between the set of natural numbers and the set of countable numbers. Actually, the two sets are the same set of naturals, it's just that the natural numbers, most commonly starting with one, are called counting numbers when used for their namesake purpose.

Counting things means labelling each things with a different counting/natural number, starting with the number one and proceeding sequantially. In the usual circumstances, people are interested in the final number, that is the cardinality of the set of the enumerated things, but mathematicians are more interested in finding a bijective function that associates each element from the counting number set and another, also infinite, set that is being enumerated. If such bijective function is found, it is said that the other set is also countable, and moreover, that it has the same size as the set of natural numbers.

The infinity of natural numbers is the smallest infinity. It is a countable infinity, and it is taken as the basic model against which the infinities are "measured".

Georg Cantor showed that the powerset of natural numbers is a larger infinity then natural numbers, and unlike naturals, it is uncountable infinity.

Cantor's theorem is a fundamental theorem which states that the cardinality of any set is strictly smaller then the cardinality of its powerset.

For finite sets, Cantor's theorem can be shown by enumerating the elements of both sets, and it is known that if the cardinality of a finite set, |S| = n, then the cardinality of its powerset, |P(S)| = 2^n.

Much more significant is Cantor's discovery of an argument that is applicable to any set, which showed that the theorem holds for infinite sets, countable or uncountable, as well as finite sets.

ℕ is countably infinite set with cardinality ℵ0 = card(ℕ). A particularly important consequence of Cantor's theorem is that the power set of ℕ is uncountably infinite and has the same size as the set of real numbers, a cardinality that is referred to as the cardinality of the continuum,

𝔠 = card(ℝ) = card(𝒫(ℕ))

The relationship between these cardinal numbers is often expressed symbolically by the equality and inequality:

$\mathfrak{c} = 2^{\aleph _0} > \aleph _0$

There is an infinite number of infinities, and an infinite number of cardinality classes they belong to.

The infinity of irrational numbers is greater than infinity of rational numbers.

It has been said that the rational numbers are like the starts in the night sky but the irrationals are like the blackness.

Countable set of ℵ₀ cardinality

• ℕ : card(ℕ) = ℵ₀

• ℤ : card(ℤ) = ℵ₀

• ℚ : card(ℚ) = ℵ₀

Uncountable sets:

• 𝒫(ℕ): card(𝒫(ℕ)) = card(ℝ) = 𝔠

• ℝ: card(ℝ) = 𝔠

others?

• 𝒫(ℝ)

• ℙ primes

• 𝔸 algebraic numbers

• ℂ = ℝ ∪ 𝕀

• ℍ

By iteratively taking the power set of an infinite set and applying Cantor's theorem, we obtain an endless hierarchy of infinite cardinals, each strictly larger than the one before it. Consequently, the theorem implies that there is no largest cardinal number; or, colloquially, there's no largest infinity.

Refs

https://en.wikipedia.org/wiki/Cantor%27s_theorem https://en.wikipedia.org/wiki/Countable_set https://en.wikipedia.org/wiki/Uncountable_set