𝘼 𝙩𝙝𝙚𝙤𝙧𝙮 𝙤𝙛 𝘾𝙖𝙣𝙩𝙤𝙧'𝙨 𝙉𝙪𝙢𝙗𝙚𝙧 ℵ₀ 𝙞𝙣 𝙩𝙝𝙚 𝙘𝙤𝙢𝙥𝙡𝙚𝙭 𝙨𝙥𝙝𝙚𝙧𝙚 ℂ ▪️

Depuis je crois les années de collège ça me tourmente que l'on ne puisse user de l'infini comme d'un nombre. Je nous ai donné le droit de le faire !

▪️

L'ex-Reine de l'ex-Satan qui était venue des Enfers pour tout faire pour m'empêcher de travailler n'a réussi qu'à me donner l'occasion de prouver que tout handicap est surmontable!

▪️▪️▪️

𝘼 𝙩𝙝𝙚𝙤𝙧𝙮 𝙤𝙛 𝘾𝙖𝙣𝙩𝙤𝙧'𝙨 𝙉𝙪𝙢𝙗𝙚𝙧 ℵ₀ 𝙞𝙣 𝙩𝙝𝙚 𝙘𝙤𝙢𝙥𝙡𝙚𝙭 𝙨𝙥𝙝𝙚𝙧𝙚 ℂ ▪️

🔹❗𝗜𝗻𝗶𝘁𝗶𝗮𝗹 𝗾𝘂𝗲𝘀𝘁𝗶𝗼𝗻𝘀 

1️⃣ Are the limit of an ordered sequence toward positive infinity +∞ and negative infinity - ∞ the same in the set ℕ of integers and in the set ℝ of real numbers? Or different?

2️⃣ What is the specific infinity it tends toward?

⌈ ?

⊨ ⊤| n ∈ ℕ • x ∈ ℝ • 𝑙𝑖𝑚 ₙ→ ͚= 𝑙𝑖𝑚 ₓ → ͚= ?

⌊ 

3️⃣ What about the realm of complex numbers then?

🔹𝗔𝘅𝗶𝗼𝗺𝘀 & 𝗱𝗲𝗳𝗶𝗻𝗶𝘁𝗶𝗼𝗻𝘀

🔹🔹 Function identity 𝑖𝑑

The identify function maps a set member to itself.

⌈ def. 𝑖𝑑

Let Φ be a set •

 𝑖𝑑Φ: Φ→Φ ⋅ χ ∈ Φ ⋅ χ ↦ χ

⌋ 

🔹🔹 Existence of ℵ₀ in ℕ⁺ 

We postulate the existence of the Cantor number ℵ₀ defined as the cardinal of the set of natural number.

⌈ def. ℵ₀ ≝ card(ℕ) ⌋ 

▪️

ℵ₀ must be greater than any integer.

⌈ Axiom [A1]

⊨ ∃ℵ₀ ⋅ ∀ n ∈ ℕ ⋅ n<ℵ₀

⌊ 

▪️

A corollary is that ℵ₀ must be an absorbing element for addition operation.

⌈ Axiom [A2]

⊨ ℵ₀ + 1 = ℵ₀ • 

⌋ 

🔹🔹 Continuity Theorem

Given a fonction 𝑓 over the real numbers 𝑓: ℝ → ℝ and a value 𝑎 ∈ ℝ , if the left limit of 𝑓 towards 𝑎 equals the right limit of 𝑓 towards 𝑎 then 𝑎 belongs to the definition domain of the function 𝑓.

 ⌈ [Continuity Theorem]

⊢ ∀𝑓: 𝑑𝑜𝑚(𝑓) ⊂ ℝ → ℝ ⋅ ∀ x ∈ 𝑑𝑜𝑚(𝑓) ⋅ ∀ a ∈ ℝ ⋅ 𝑙𝑖𝑚ₙ→₋ₐ 𝑓 = 𝑙𝑖𝑚ₙ→₊ₐ 𝑓 ⟹ 𝑎 ∈ 𝑑𝑜𝑚(𝑓)

⌋ ∎ 

🔹𝗧𝗵𝗲𝗼𝗿𝘆'𝘀 𝗧𝗵𝗲𝗼𝗿𝗲𝗺𝘀

🔹🔹

The limit of identity function over the natural numbers 𝑖𝑑ℕ() towards positive infinity is ℵ₀:

⌈ Theorem [T1]

 ⊢ 𝑙𝑖𝑚ₙ→ ͚ 𝑖𝑑ℕ (n) = ℵ₀ 

⌋ (dem¹)

▪️

🔹🔹" black hole" theorems : ℵ₀ is an absorbing element for addition and soustraction operations in ℕ .

▪️

🔹🔹🔹 Theorem [T2]: ℵ₀ is absorbing for addition operation + in ℕ .

⌈ Theorem [T2]

⊢ ∀n > 0 ∈ ℕ ⋅ ℵ₀ + n = ℵ₀ 

⌋ (dem. by recursivity ∵ [A2])

▪️

🔹🔹🔹Theorem [T3] : ℵ₀ is absorbing for substraction operation - in ℕ .

⌈ Theorem [T3]

⊢ ∀n > 0 ∈ ℕ ⋅ ℵ₀ - n = ℵ₀ 

⌋ (dem²)

▪️

🔹🔹🔹 Theorem [T4] : ℵ₀ absorbs itself in addition operation.

⌈ Theorem [T4]

⊢ ℵ₀ + ℵ₀= ℵ₀ 

⌋ ∵ [T2] ∎ 

▪️

🔹🔹🔹 Theorem [T5] : ℵ₀ absorbs itself in substraction operation.

⌈ Theorem [T5]

⊢ ℵ₀ - ℵ₀ = ℵ₀ 

⌋ (dem³)

▪️

🔹🔹🔹 Theorem [T6] : identity through negation.

⌈ Theorem [T6]

⊢ ℵ₀ = -ℵ₀ 

⌋ (dem⁴) 

▪️

🔹🔹🔹Theorem [T7]: negative infinity is ℵ₀.

⌈ Theorem [T7]

⊢ ∀ n ∈ ℕ ⋅ 𝑙𝑖𝑚ₙ→₋ ͚ 𝑖𝑑ℕ(n) = ℵ₀

⌋ (dem⁵)

▪️

🔹🔹 Theorem [T8] : equality of limits at both plus and minus infinities.

⌈ Theorem [T8]

⊢∀ n ∈ ℕ ⋅ 𝑙𝑖𝑚ₙ→₊ ͚ 𝑖𝑑ℕ(n) = 𝑙𝑖𝑚ₙ→₋ ͚ 𝑖𝑑ℕ(n) = ℵ₀ 

⌋ ∴ [T1] ∧ [T7] ∎ 

▪️

🔹🔹Theorem [T9] : equality of limits toward infinity of the identity function over the natural and the real numbers.

Argument:

When one visualise the Riemann's Sphere it's "obvious" the line described by ℕ and the one described by ℝ must converge toward one same "point" ∞, since both curve are superposed.

⌈ Theorem [T9]

⊢ ∀ n ∈ ℕ ⋅ ∀ x ∈ ℝ ⋅ 𝑙𝑖𝑚ₙ→ ͚ 𝑖𝑑ℕ(n)= 𝑙𝑖𝑚ₓ→ ͚ 𝑖𝑑ℝ(x) 

⌋ ∵ ℕ ⊂ ℝ ∎ 

▪️

🔹🔹 Theorem [T10] : The limit of identity function over the real numbers 𝑖𝑑ℝ() towards infinity is ℵ₀.

⌈ Theorem [T10]

⊢ ∀ x ∈ ℝ • 𝑙𝑖𝑚ₓ→ ͚ 𝑖𝑑ℝ(x) = ℵ₀ 

⌋ ∵ [T9] ∧ [T1] ∎

▪️

🔹🔹 Theorem [T11] : the Cantor infinity Aleph Zero ℵ₀ defined as the cardinal of the set of natural integers is a real number.

⌈ 

⊢ ℵ₀ ∈ ℝ 

⌋ ∵ [T10] ∧ [Continuity Theorem]∎ 

▪️

🔹🔹 Theorem [T12] : the Cantor infinity Aleph One ℵ₁ defined as the cardinal of the set of real number is a real number.

⌈ Theorem [12]

⊢ ℵ₁ ∈ ℝ 

⌋ ∵ ℵ₁= 2 ^ ℵ₀ ∧ ℵ₀ ∈ ℝ ∎

▪️

🔹🔹 Theorem [T13] : the complex Aleph Zero ℵ₀ + 𝘪⋅ℵ₀ = 𝘦^𝘪⋅ℵ₀ belongs to the complex plane ℂ .

⌈ Theorem [13]

⊢ 𝘦^ 𝘪⋅ℵ₀ ∈ ℂ 

⌋ ∵ ℵ₀ ∈ ℝ ∎ 

▪️

🔹𝗔𝗽𝗽𝗹𝗶𝗰𝗮𝘁𝗶𝗼𝗻 𝘁𝗼 𝗥𝗶𝗲𝗺𝗮𝗻𝗻'𝘀 𝗦𝗽𝗵𝗲𝗿𝗲

The projective complex line ℙ¹(ℂ) called Riemann's Sphere defined as the set comprised of the complex plane ℂ and the Complex Aleph Zero is actually ℂ.

⌈ ℙ¹(ℂ) = ℂ ⌋ ∎ 

▪️

🔹𝗘𝘅𝘁𝗲𝗻𝘀𝗶𝗼𝗻 𝘁𝗼 𝗮𝗿𝗯𝗶𝘁𝗿𝗮𝗿𝘆 𝗺𝘂𝗹𝘁𝗶-𝗱𝗶𝗺𝗲𝗻𝘀𝗶𝗼𝗻𝗮𝗹 𝘀𝗽𝗮𝗰𝗲𝘀

It comes trivially that this result can be extended to any 𝑛-dimensional space:

⌈ {ℝ₁, ..., ℝₙ} ∋ (ℵ₀¹, ..., ℵ₀ⁿ) ⌋ ∎

🔹𝗗𝗲𝗺𝗼𝗻𝘀𝘁𝗿𝗮𝘁𝗶𝗼𝗻𝘀

🔹🔹dem¹: Demonstration of [T1]

Theorem: the limit of 𝑖𝑑() in ℕ is ℵ₀ •

⌈ dem¹

• ⊤ ?| [T1] 𝑙𝑖𝑚ₙ→ ͚ 𝑖𝑑ℕ (n) = ℵ₀ 

Let define Λₙ as the set of ordered integers from 0 to n.

⌈ n ∈ ℕ • Λₙ ≝ [1...n] ⌋

We have :

 ⌈ theorem [T1.T1]

 ⊨ card(Λₙ) = n 

⌋ (dem. by recursivity)

and :

⌈ axiom [T1.A1]

⊨ ∀ n,β ∈ ℕ • 𝑙𝑖𝑚ₙ→ᵦ 𝑖𝑑ℕ( n ) = β 

⌋ 

▪️

Let define Λ ͚ as the set described by Λₙ when n tends toward ∞ :

⌈ def. Λ ͚ 

 Λ ͚ ≝ [1.. 𝑙𝑖𝑚ₙ→ ͚ 𝑖𝑑ℕ(n)] = [1...∞]

⌋ 

⌈ 

[1...∞] describes the whole set of integers ℕ . Therefore:

⊨ [1...∞] = ℕ

⟹ Λ ͚ = ℕ 

⟹ card(Λ ͚) = card(ℕ) ∵ [T1.T1]

⟹ card(Λ ͚) = ℵ₀ ∵ [def. ℵ₀]

⌋ 

▪️

∴ ⌈ ⊨ 𝑙𝑖𝑚ₙ→ ͚ 𝑖𝑑ℕ(n) = ℵ₀ ∵ [T1.A1]⌋ 

∎⌋ 

▪️

🔹🔹dem²: Demonstration of [T3]

⌈ dem²

• ⊤ ?| [T3] ∀n > 0 ∈ ℕ ⋅ ℵ₀ - n = ℵ₀ 

[T2] ⟺ ∀n > 0 ∈ ℕ ⋅ ℵ₀ + n = ℵ₀

⟺ ℵ₀ + n - n = ℵ₀ - n

⟺ ℵ₀ = ℵ₀ - n

∴ ⊤| [T3] 

∎⌋ 

▪️

🔹🔹dem³ : Demonstration of [T5]

⌈ dem³

• ⊤ ?| [T5] ℵ₀ - ℵ₀ = ℵ₀

∵ [T3] ∴ 𝑙𝑖𝑚ₙ→ ͚ 𝑖𝑑ℕ(ℵ₀ - n) = ℵ₀ =

= ℵ₀ - 𝑙𝑖𝑚ₙ→ ͚ 𝑖𝑑ℕ(n)

= ℵ₀ - ℵ₀

∴ ⊤| [T5]

∎⌋ 

▪️

🔹🔹 dem4: Demonstration of [T6]

⌈ dem⁴

• ⊤ ?| [T6] ℵ₀ = -ℵ₀

[T6] ⟺ ℵ₀ + ℵ₀ = ℵ₀ -ℵ₀

⟺ ℵ₀ = ℵ₀ -ℵ₀ ⟺ ⊤| [T5]

∴ ⊤| [T6]

∎ ⌋ 

▪️

🔹🔹dem⁵: Demonstration of [T7]

⌈ 

• ⊤ ?| [T7] ∀ n ∈ ℕ ⋅ 𝑙𝑖𝑚ₙ→₋ ͚ 𝑖𝑑ℕ(n) = ℵ₀

Remind. [T1]: 𝑙𝑖𝑚ₙ→₊ ͚ 𝑖𝑑ℕ(n) = ℵ₀

∴ 𝑙𝑖𝑚ₙ→₋ ͚ 𝑖𝑑ℕ(n) = - [𝑙𝑖𝑚ₙ→₊ ͚𝑖𝑑ℕ(n)] = -ℵ₀ = ℵ₀ 

∎ ⌋ 

🔹 𝗥𝗲𝗳𝗲𝗿𝗲𝗻𝗰𝗲𝘀

▪️ ℵ numbers - cardinality of well ordered infinite sets : https://en.m.wikipedia.org/wiki/Aleph_number

▪️ Riemann's Sphere : https://en.m.wikipedia.org/wiki/Riemann_sphere

▪️Logic symbols used in this article :

https://en.m.wikipedia.org/wiki/List_of_logic_symbols

🔶◾🔸▪️🔶⬛🔸⚜️🔸⬛🔶▪️🔸◾🔶