Hardegree, Set Theory ; Chapter 5: Cardinal Numbers page 1 of14 14 5 Cardinal Numbers 1. Introduction. 2 2. Equipollence. 3 3. Finite and Infinite Sets. 4 4. Denumerable Sets.

 

Hardegree, Set Theory ; Chapter 5: Cardinal Numbers page 1 of14 14 5 Cardinal Numbers 1. Introduction. 2 2. Equipollence. 3 3. Finite and Infinite Sets. 4 4. Denumerable Sets.

 

arXiv:1204. 6228v1 math. CT 27 Apr 2012 Abstract. 6 ,wherewecon forsettheory. settheory. Shelah suchasShelah. Weincludeasmall dictionary in. culttobe morefamiliar. Thus,ifamanbred. shouldtake.

 

1-0 Specifyingsets informal 1-4 Speci s paradox EscapingRussell 1-5 A symbol. recursively formulasarede. Inmoredetail ::: Variables a;b;:::;x;y;::: Connectives:!, : plus , Quantier:8 plus 9 Brackets.

 

• 20 Questions total PAH: Set Theory Quiz Overview • 1-4: Know the difference between elements and subsets and know the symbols for both! • 5-8:.

 

Hardegree, Set Theory , Chapter 3: Functions page 1 of17 17 3 FUNCTIONS 1. Introduction. 2 2. Functions as Relations. 2 3. Functions, Arguments, Values. 3 4. Injections, Surjections, Bijections,.

 

see6-26-0B1Set Theory: Outline notes6 ZF9 TheAxiom ofReplacemen t, ZF8 ThePrimitiv ZF9 Regularity 6-1 Ev erynon-empt ysetisw ell-founded thatis,con tainsanelemen tminimalw. r. t. 2 :8a a6 ! 9x x2a x a : TypoinGoldrei 1st edn :a6 omitted.

 

04set-ax Zermelo ZF1 ality 8x x2a x2b a b ZF2 EmptySetAxiom Thereisaset : x2a ZF3 b ,whosemembers 8x x2c x a _ x b denotedfag singletonset. ZF4 x2b 9y x2y y2a Notation:Sfa;b g isdenoteda b. ZF5 a x2b x a shorthand:a b standsfor8x x2a!x2b. ZF6 SeparationAx

 

Basic Set Theory Gary Hardegree Department of Philosophy University of Massachusetts Amherst, MA 01003 1. Introduction. 2 2. Membership. 2 3. Extensionality. 2 4. The Empty Set. 3 5. Simple Sets;.

 

Hardegree, Set Theory , Chapter 3: Functions page 1 of17 17 3 FUNCTIONS 1. Introduction. 2 2. Functions as Relations. 2 3. Functions, Arguments, Values. 3 4. Injections, Surjections, Bijections,.

 

Math111Problems AliNesin 1. Dene a x y x isasubsetofy , b x y. 2. Showthat ;. 3. ; isequaltop2. 4. ;. 5. Let A beaset. Showthatfa2A:a 2ag 2A. 6. ShowthatifX ;thenX ;. 7. Weknowthatif X X isaset. WhathappensifX ; 8. Consider8y 8z z 2x z 2y x y. Forwhat x s

 

Hardegree, Set Theory , Chapter 2: Relations page 1 of35 35 2 Relations 1. Ordered - Pairs. 2 2. Reducing Ordered - Pairs to Unordered - Pairs. 3 3. The Cartesian Product.

 

Hardegree, Set Theory , Chapter 2: Relations page 1 of35 35 2 Relations 1. Ordered - Pairs. 2 2. Reducing Ordered - Pairs to Unordered - Pairs. 3 3. The Cartesian Product.

 

Hardegree, The Natural Numbers page 1 of36 36 4 The Natural Numbers 1. Numerals and Numbers. 2 2. The Axiom of Infinity. 5 3. Mathematical Induction. 8 4. Examples of Mathematical Induction.

 

The Axiom of THEOREM: The follo wing statemen tsareequivalent. AC cf Choice function formof AC :Givenanon-empt yindexsetIanda ysets, thereexists. t. 8i2I f i 2Ai: AC set Po w ersetformof.

 

9-0 Introductionto WO and AC Arithmeticin ! ,set-wise 9-1. QuestionForm;n2 ! ,howdothesetsm n,m nandm n ,asde thesetsmandn Answer 1 m n m f0g n f1g. 2 m n m n. 3 mn f functionsg:n!mg. Proofs :Exercise Do 1 , 2 , 3. SeeGoldrei cardXand card Y respectively.

 

Silv er s Theorem Theorem1. 1. Let be a 0. ,2. Then2. Let beasingular 0. Denotecf by andletf iji gbeastrictly. ,2 i i. f is a stationary subsetof f ,f i 2 i. f ;Dom g. 2 ; Dom f Dom g ,f 6 g. f. tually disjoin t. Lemma1. 2. LetF Mbeeventual.

 

languagef2; ;U;Sg,2isa binary relationsymbol, isthe unary relationsymbols. The ts also called atoms willbe denedbelow. Itis a. It is denotedbyZFA. Itsaxiomsare. 1 8x U x _S x 8x: U x S x. 2 8x8y.

 

Gettingamodelof ZF Fnd is calledZFA ZermeloFrankel with atoms. ;U;Sg,2isa binary relationsymbol, isthe unary relationsymbols. The ts also called atoms willbe denedbelow.

 

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Basic Set Theory Gary Hardegree Department of Philosophy University of Massachusetts Amherst, MA 01003 1. Introduction. 2 2. Membership. 2 3. Extensionality. 2 4. The Empty Set. 3 5. Simple Sets;.

 

languagef2; ;U;Sg,2isa binary relationsymbol, isthe unary relationsymbols. The ts also called atoms willbe denedbelow. Itis a. It is denotedbyZFA. Itsaxiomsare. 1 8x U x _S x 8x: U x S x. 2 8x8y.

 

Hardegree, Set Theory , Chapter 3: Functions page 1 of17 17 3 FUNCTIONS 1. Introduction. 2 2. Functions as Relations. 2 3. Functions, Arguments, Values. 3 4. Injections, Surjections, Bijections,.

 

Basic Set Theory Gary Hardegree Department of Philosophy University of Massachusetts Amherst, MA 01003 1. Introduction. 2 2. Membership. 2 3. Extensionality. 2 4. The Empty Set. 3 5. Simple Sets;.

 

AmendedMarch5, 2006 15-0B1Set Theory: Lecture 15 Ordinals, again andordinals Hartogs Theorem Equivalentsof A C , again CC implies WO ZFC In ZFC. From w ell-o rderedsetstoo.

 

10. Prof. Dr. PeterKoepke,Dr. PhilippSchlicht. 12. 2012 Problem35. Suppose isanin nitecardinal. ,there isasequence Xnjn2! suchthat Sn2!Xnandotp Xn n inordinal exponentiation. Problem36.

 

11. Prof. Dr. PeterKoepke,Dr. PhilippSchlicht. 12. 2012 Problem39. a !. Showthat X isclosedin ! ifandonlyif X !. b Let!!Card,cof ! 1. Let Ci! i ! andlet cof !. Problem40. Suppose! 1 isregularandf:! !. Showthattheset ! f iscubin !. ! ofthisform.

 

12. Prof. Dr. PeterKoepke,Dr. PhilippSchlicht. 01. 2013 Problem43. a LetF Q A A. Thencard F. b Let! cof 2Card. Let2 forall! 2 Card. Then2. Problem44. SupposeS ! 1 isstationary.