cardinality of the set of all functions from natural numbers to real numbers

The set of all possible functions from the set of natural numbers to the set of real numbers is denoted by ℝℕ, where ℕ is the set of natural numbers and ℝ is the set of real numbers. Each function in this set can be thought of as an infinite sequence of real numbers, such as (0, 1, 2, 3, …), (π, e, √2, √3, …), or (1, -1, 1, -1, …).

The cardinality of the set of all possible functions from the set of natural numbers to the set of real numbers is uncountable, meaning that it is larger than the cardinality of any countable set, such as the natural numbers or the rational numbers. One way to show this is by using a technique called Cantor’s diagonal argument1, which proves that there is no way to list all the elements of ℝℕ without missing some.

Another way to show that the set of all possible functions from the set of natural numbers to the set of real numbers has uncountable cardinality is by using some facts about cardinal arithmetic2, which is a way of performing operations on cardinal numbers. A cardinal number is a symbol that represents the size of a set. For example, ℵ₀ (aleph-null) is the cardinal number that represents the size of any countable set3. The cardinal number that represents the size of any uncountable set is usually denoted by ℵ₁ (aleph-one) or higher.

One fact about cardinal arithmetic is that if κ and λ are cardinals, none of them zero, and at least one of them is infinite, then κ + λ = κλ = max{κ, λ}. This means that adding or multiplying two infinite cardinals gives the larger one. Another fact is that if κ and λ are cardinals and λ is infinite, then κλ = 2κ. This means that raising an infinite cardinal to any power gives 2κ.

Using these facts, we can calculate the cardinality of ℝℕ as follows:

|ℝℕ| = |ℝ|ℕ = (2ℵ₀)ℕ = 2ℵ₀ℵ₀ = 22ℵ₀ = 2|ℝ|

This means that the cardinality of ℝℕ is equal to the cardinality of the power set of ℝ, which is the set of all subsets of ℝ. This cardinality is also denoted by ℵ₂ (aleph-two) or higher.

Therefore, the answer to your question is that the cardinality of the set of all possible functions from the set of natural numbers to the set of real numbers is uncountable and equal to 2|ℝ|.