Editing Model Theory (section)

== <span style="color: #FFFFFF;">Understanding</span> ==
Model Theory is about the '''Flexibility of Language'''.

'''1. Language vs. World''':
Imagine the statement: "Everyone has a father."
* In the '''Model''' of "Family Tree," this statement might be true.
* In the '''Model''' of "Single-celled Organisms," this statement is false.
Model theory studies which "Worlds" (Models) make which "Sentences" true.

'''2. The Power of Löwenheim–Skolem''':
This is a shocking finding. It says that no matter how hard you try to describe the "Whole Numbers" (1, 2, 3...), you can never write enough rules to prevent "Giant" models from existing. There are "Models" of arithmetic that contain "Numbers" larger than every whole number, yet they still follow all the rules of math perfectly. These are called '''Non-Standard Models'''.

'''3. Categoricity''':
A "Categorical" theory is a perfect description. It only allows for one type of world.
* Basic arithmetic is '''Not Categorical''' (as shown above).
* Some advanced theories (like "Dense Linear Orders") are '''$\aleph_0$-categorical''', meaning they have only one countable model.

'''Non-Standard Analysis''': By creating a model of math that includes "Infinite" and "Infinitesimal" (infinitely small) numbers, mathematicians can solve calculus problems much more simply than with standard limits. This shows that "The Truth" depends on which model you choose to live in.
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