# Axiom of Choice: Guide

## Learning the Axiom of Choice

What do you know about set theory, mathematical logic and axiom systems?
a: Almost nothing
wants: [axiom*of*choice*definition*intutive axiom of choice finite sets banach tarski paradox axiom of choice function existence axiom of choice how can something be neither true nor false
-wants: Axiom of Choice: Definition (Formal)
b: Well, I’ve used them for other maths, but haven’t studied them directly.
wants: axiom of choice definition intutive axiom of choice finite sets axiom of choice function existence axiom of choice how can something be neither true nor false
-wants: Axiom of Choice: Definition (Formal) banach tarski paradox
c: I have a good grasp of it, but would like some explanation anyway.
wants: Axiom of Choice: Definition (Formal) axiom of choice finite sets axiom of choice function existence axiom of choice how can something be neither true nor false
-wants: Axiom of Choice Definition (Intuitive) banach tarski paradox
d: I have a good grasp of it and don’t need to hear more.

wants: Axiom of Choice: Definition (Formal)
-wants: Axiom of Choice Definition (Intuitive) axiom of choice finite sets axiom of choice function existence banach tarski paradoxaxiom of choice how can something be neither true nor false
]

This axiom has a rich and interesting history. How much do you want to learn about? a: Give me all of the juicy history side-facts! wants: [6c7 Axiom of Choice: History and Controversy b: Eh, give me a short intro. wants: Axiom of Choice: Introduction -wants: Axiom of Choice: History and Controversy c: Just stick to the mathematics, please. -wants: Axiom of Choice: Introduction Axiom of Choice: History and Controversy ]

This axiom has a rich and interesting history. How much do you want to learn about? a: Give me all of the juicy history side-facts! wants: [6c7 Axiom of Choice: History and Controversy b: Eh, give me a short intro. wants: Axiom of Choice: Introduction -wants: Axiom of Choice: History and Controversy c: Just stick to the mathematics, please. -wants: Axiom of Choice: Introduction Axiom of Choice: History and Controversy ]

How heavy should the maths be that we show you? a: Lay it on me! Give me the formal logical notation. wants: [6c8 -wants: Axiom of Choice Definition (Intuitive) b: Just give me an intuitive description with words. No mathematical notation. -wants: Axiom of Choice: Definition (Formal) wants: Axiom of Choice Definition (Intuitive) ]

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Plan for this guide:

Axiom of Choice: Guide

Conditional paragraphs for concepts being described on later pages.

Questions on what the main page should look like

1 Introduction 2 Getting the Heavy Maths out the Way: Definitions 3 Axiom Unnecessary for Finite Collections of Sets 4 Controversy: Mathematicians Divided! Counter-Intuitive Results, and The History of the Axiom of Choice 5 So, What is this Choice Thing Good for Anyways? 6 Physicists Hate Them! Find out How Banach and Tarski Make Infinity Dollars with this One Simple Trick! 7 How Something Can Exist Without Actually Existing: The Zermelo Fraenkel Axioms and the Existence of a Choice Function 8 How Something can be Neither True nor False: 9 A Rose by Any Other Name: Alternative Characterizations of AC 10 Zorn’s Lemma? I hardly Know her! 11 Getting Your Ducks in a Row, or, Rather, Getting Your Real Numbers in a Row: The Well-Ordering Principle 12 AC On a Budget: Weaker Versions of the Axiom 13 And In Related News: The Continuum Hypothesis 14 Axiom of Choice Considered Harmful: Constructive Mathematics and the Potential Pitfalls of AC 15 Choosing Not to Choose: Set-Theoretic Axioms Which Contradict Choice 16 I Want to Play a Game: Counterintuitive Strategies Using AC

-Guide Questions-

I’d also like the corresponding pages to show or hide some information based on what is chosen here.

Choose one of the pregenerated paths, or customize your own. a. Comprehensive path: Learn more than you wanted to know! Add all pages b. Substantial path: All the most important stuff Add 1,2,3,4,5,6,7,8,10,14 c. Compact path: Only the very important stuff Add 2, 3, 5, 7, 10 d. First-time path: Learn the important basics without getting too bogged down. Add 1. 2 (intuitive), 2 (definition), 3, 4, 5, 6, 7, 8, 10 e. Overview path: Just get a taste of the axiom Add 1, 2 (intuitive), 2 (definition), 3, 4, 5, 7 f. Custom Path

If the user chooses to customize a path:

What do you know about set theory, mathematical logic and axioms? a. Almost nothing Add Definitions (Intuitive) Add 3. Finite Sets Add 6. Banach-Tarski Add 7. ZF Axioms and Existence of a Choice Function Add 8. How can something be neither True nor False b. Well, I’ve used them for other maths, but haven’t studied them directly. Add 2. Definitions

Add 3. Finite Sets Add 7. ZF Axioms and Existence of a Choice Function Add 8. How can something be neither True nor False c. I have a good grasp of it, but would like some explanation anyway. Add 2. Definitions Add 3. Finite Sets Add 7. ZF Axioms and Existence of a Choice Function Add 8. How can something be neither True nor False d. I have a good grasp of it and don’t need to hear more.

Add 2. DefinitionsThis axiom has a rich and interesting history. How much do you want to learn about? a. Give me all of the juicy history side-facts! Adds 1. Introduction Adds 4. Controversy (History) b. Eh, give me a short intro. Adds 1. Intro c. Just stick to the mathematics, please.

-Questions 3-5 are revealed if, and only if, the answer to Question 1 is b,c, or d.-

How much detail would you like to read about the mathematics related to the axiom? a. I’d like to know a lot of the detail. Adds 5. What is this Choice Thing Good For Adds 9. Alternative Characterizations Adds 10. Zorn’s Lemma Adds 12. Weaker Versions of the Axiom Adds 13. Continuum Hypthesis b. I only want the most important extra details. Adds 5. What is this Choice thing good for Adds 10. Zorn’s Lemma c. I only want the absolute essentials.

How much would you like to know about constructive mathematics and its relation to the axiom of choice? a. I don’t care to read about it right now. b. I don’t know what that is, please tell me about it. Adds 14. Constructive Mathematics and Pitfalls of AC Potentially adds a link to an intro on constructivsm. c. I know what it is, but I would like to hear more about how it relates to Axiom of Choice. Adds 14. Constructive Mathematics and Pitfalls of AC

How much do you care about the paradoxes the axiom implies? a. I don’t care about them, I just want to know about the axiom itself. b. Give me a very basic overview. Adds 6. Banach-Tarski Paradox d. Just tell me some interesting ones. Adds 6. Banach-Tarksi Paradox Adds 16. Counterintuitive strategies using AC c. Tell me the whole story! Adds 6. Banach-Tarski Paradox Adds 11. Well-Ordering Principle Adds 15. Axioms which contradict choice Adds 16. Counterintuitive strategies using AC

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Parents:

- Axiom of Choice
The most controversial axiom of the 20th century.