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: [ax­io­mof­choicedefi­ni­tion­in­tu­tive 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:

  1. 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. Definitions

  2. 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! 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.-

  1. 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.

  2. 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

  3. 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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