Editing The Philosophy of Infinity and Cantor's Paradise (section)

== <span style="color: #FFFFFF;">Understanding</span> ==
The philosophy of infinity is understood through '''Cardinality''' and '''Independence'''.

'''1. The "Same Size" Paradox (Countability)''':
"There are as many even numbers as all numbers."
* (See Article 226). **"Galileo"** "Noticed": "Every" "Natural Number" "Can Be Paired" with "An Even Number" (1↔2, 2↔4, 3↔6, ...).
* "So" "The Evens" and "The Naturals" have "The Same" **"Cardinality."**
* "A Part" can be "As Large As" "The Whole" — "Only" "For" "Infinite Sets."
* "Infinity" **"Defies" "Common Sense."**

'''2. The "Diagonal" Proof (Uncountability)''':
"Real numbers are a bigger infinity than natural numbers."
* (See Article 229). **"Cantor"** "Proved" that "You Cannot" **"List"** "All" "Real Numbers" between "0" and "1."
* "Method": "Assume" "You Have" "A Complete List." "Construct" "A New Number" by "Changing" "The n-th Digit" of "The n-th Number." "This New Number" "Differs" from "Every Entry" in "Your List."
* "Contradiction": "Your List" was "Incomplete."
* **"Uncountably Many"** "Reals Exist."

'''3. The "Hierarchy" of Infinities (Cardinal Arithmetic)''':
"There are infinitely many sizes of infinity."
* (See Article 226). "Not Only" are "Reals" "Bigger" than "Naturals" — "The Set" of "All Subsets" (Power Set) of "Any Set" is **"Strictly Larger."**
* **"2^ℵ₀ > ℵ₀"** (Power Set of Naturals has more elements than Naturals).
* "This Generates" "An Endless" **"Hierarchy"** of "Infinities": ℵ₀ < ℵ₁ < ℵ₂ < ..."
* "Infinity" has **"Infinite Variety."**

'''Cantor's Diagonal Argument (1891)'''': "The Most" "Elegant" "Proof" in "All" of "Mathematics." "On One Page," **"Cantor"** "Proved" that "The Real Numbers" are "Uncountably Infinite" — "A Bigger" "Infinity" than "The Natural Numbers." "Poincaré" "Called" it **"A Disease."** "Hilbert" "Called" it **"Cantor's Paradise."** "It Is" "Now" "Standard" "Mathematics."
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