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

== <span style="color: #FFFFFF;">Remembering</span> ==
* '''Infinity''' — "The Quality" of "Being Without Bound" or "End": "In Mathematics," "Precisely Defined" by "Cantor's" "Set Theory."
* '''Potential Infinity''' — (Aristotle). "A Process" that "Goes On Forever" but "Is Never" "Actually Completed": "Counting 1, 2, 3…" without "End."
* '''Actual Infinity''' — (Cantor). "A Completed" "Infinite Totality": "The Set" of "All" "Natural Numbers" as "An Actual" "Object."
* '''Countable Infinity''' — (ℵ₀, "Aleph-Null"). "The Smallest" "Infinite Cardinal": "The Size" of "The Natural Numbers" (1, 2, 3, ...). "A Set" is "Countable" if "Its Elements" can be "Listed" (Put in "1-1 Correspondence" with "Natural Numbers").
* '''Uncountable Infinity''' — "A Set" "Too Large" to be "Listed": "The Real Numbers" are "Uncountably Infinite" — "Proven" by "Cantor's" "Diagonal Argument."
* '''Cantor's Diagonal Argument''' — "A Proof" that "The Real Numbers" between "0" and "1" "Cannot Be Listed" — "There Are More Reals" than "Natural Numbers."
* '''The Continuum Hypothesis''' — "Cantor's" "Conjecture": "There Is No" "Infinite Set" "Strictly Between" "The Natural Numbers" (ℵ₀) and "The Real Numbers" (ℵ₁). "Proved" to be **"Independent" of "ZFC"** by "Gödel" (1940) and "Cohen" (1963).
* '''Transfinite Ordinals''' — "Cantor's" "Extension" of "Ordinary Numbers" beyond "The Finite": "ω, ω+1, ω·2, ω², ωω, ε₀, ..."
* '''The Absolute Infinite''' — (Cantor). "The Totality" of "All" "Ordinals" — "Cannot Be" "A Set" (Russell's Paradox) — "God" in "Cantor's" "Theology."
* '''Large Cardinal Axioms''' — "Additional" "Axioms" for "Set Theory" (Measurable Cardinals, Woodin Cardinals) "Beyond" "ZFC" — "Generating" "A Rich" "Hierarchy" of "Consistency Strength."
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