The relation between prime numbers and irrational numbers are discussed using prime line and pre-irrationality. A rational number is the quotient of 2 whole numbers i and j, coordinates of a points (j, i) in the plane of 2 dimensional natural numbers shown in Figure 1. Each points (j, i) represents a rational number whose value is i/j that equals the slope of the straight line connecting the point (j,i) to the origin (0,0).

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Discussion about Cantor’s first proof using the next-interval-function, potential and actual infinity. Cantor’s first proof of the uncountability of real numbers is the first rigorous demonstration of the notion of uncountability. Countable sets can be put into a list indexed with natural numbers. If a set cannot be listed, then, it has more members than the set of natural numbers and is uncountable. Cantor’s first proof is a proof by contradiction. First, he supposes that all real numbers are listed in any order by the list X=(x1, x2, x3 …). Then, a real number out of this list is found by using a series of intervals, contradicting that X lists all real numbers.

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This article discusses the uncountability of the power set of ℕ proven by using the out-indexes subset contradiction. Cantor’s theorem proves that the power set of ℕ is uncountable. This is a proof by contradiction. Suppose that the power set of ℕ is countable. This allows us to put all subsets of ℕ in a list. The contradiction will come from the indexes.

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PDF On the uncountability of the power set of ℕ
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or Word https://www.academia.edu/21601620/On_the_uncountability_of_the_power_set_of_N