Editing Set Theory (section)

== <span style="color: #FFFFFF;">Understanding</span> ==
Set theory is the study of '''Membership''' and '''Infinity'''.

'''1. The Basics''':
Sets are defined by their contents, not their order. $\{1, 2, 3\}$ is the same as $\{3, 2, 1\}$.
* '''Universal Set ($U$)''': The "Context" set that contains everything we are talking about.
* '''Complement ($A'$)''': Everything in the universe that is ''not'' in set A.

'''2. The Paradoxes of Early Set Theory''':
Early set theory (Naive Set Theory) was broken by '''Russell's Paradox''': "Does the set of all sets that do not contain themselves contain itself?"
* If yes, it shouldn't.
* If no, it should.
To fix this, mathematicians created '''ZFC (Zermelo-Fraenkel set theory with Choice)''', a strict set of rules about how sets can be built to avoid these logical "loops."

'''3. The Sizes of Infinity (Cantor's Discovery)''':
Georg Cantor proved that not all infinities are the same size.
* The set of all whole numbers $\{1, 2, 3...\}$ is infinite.
* The set of all decimal numbers between 0 and 1 is ''also'' infinite, but it is '''Larger'''.
He proved this using the "Diagonal Argument," showing that you can never make a complete list of decimals—there will always be one missing.

'''The Continuum Hypothesis''': This is one of the most famous unsolved problems. It asks if there is an "In-between" infinity between the counting numbers and the decimals. Surprisingly, mathematicians proved that this question '''Cannot be answered''' using our current rules of math (ZFC).
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