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Blackboard Notes 1 | Mathematical Logic | Psychology & Cognitive Science

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Some notes for class
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  Blackboard Stu ff  : 1  /  30 and 2  /  1 M  /  PHL 344K 1. Two principles of naive set theory:(a)  Extensionality : Twosetsareidenticalifandonlyiftheyhaveexactlythe same members. ã ∀ x ∀  y ( x  =  y  ↔ ∀  z (  z  ∈  x  ↔  z  ∈  y )(b)  Comprehension : Given any property P, or any open sentence S inone free variable, there is a set of all and only things having P, orsatisfying S. ã ∃ x ∀  y (  y  ∈  x  ↔  S (  y ))(For the cognoscenti: this is an axiom schema, and hence speci-fiesaninfinitesetofaxioms(oneforeachopensentenceSinourpreferred logical language).)2. A contradiction in naive set theory: ã  Let S be the open sentence  x    x . Then by Comprehension, ∃ x ∀  y (  y  ∈ x  ↔  y    y . By instantiating  y  to  x , we get the contradiction  ∃ x ( x  ∈ x  ↔  x    x ).3. The standard revised axiomatization of set theory (we didn’t go over thisin class, and won’t really need these details, but here they are, in caseyou’re interested):(a)  Extensionality : twosetsareidenticalifandonlyiftheyhaveexactlythe same elements. (As before.)(b)  Pairing : Given any two sets  x  and  y , there is a set  { x ,  y } .(c)  Union : Given any set  x , there is a set of all members of members of  x .(d)  Restricted Comprehension : Given any set  x  and any open sentence S , there is a set of all members of   x  that satisfy  S .(e)  Infinity : There is an infinite set.(f)  Replacement : Given any set  x  and function  f  , there is a set of theimages of the members of   x  under  f  .(g)  Foundation : Every (non-empty) set contains a member with whichit has an empty intersection.1  (h)  Power Set : Given any set  x , there is a set of all subsets of   x .(i)  Choice : Given any set  x , there is a set that contains exactly onemember of every non-empty member of   x .4. Some properties of binary relations (in class, these were given treating  R asasetoforderedpairs. HereIgivethemtreatingRasabinarypredicate):(a)  Reflexive :  ∀ x Rxx (b)  Irreflexive :  ∀ x ¬ Rxx (c)  Symmetric :  ∀ x ∀  y ( Rxy  →  Ryx )(d)  Asymmetric :  ∀ x ∀  y ( Rxy  → ¬ Ryx )(e)  Antisymmetric :  ∀ x ∀  y (( Rxy ∧ Ryx )  →  x  =  y )(f)  Transitive :  ∀ x ∀  y ∀  z (( Rxy ∧ Ryz )  →  Rxz )(g)  Intransitive :  ∀ x ∀  y ∀  z (( Rxy ∧ Ryz )  → ¬ Rxz )5. Equivalence relations:(a) AnequivalencerelationonAisabinaryrelationonAthatisreflexive,symmetric, and transitive.(b) Given an equivalence relation R on A, the  equivalence classes  gen-erated by R are sets of the form  { b  :  b  ∈  A ∧  Rab } , for each  a  ∈  A .(c) A  partition  on A is a collection of non-empty subsets of A which arepairwise disjoint and collectively exhaustive of A.(d)  Fact1 : If  R isanequivalencerelationonA,thenthesetofequivalenceclasses generated by  R  form a partition of A.(e)  Fact 2 : If   P  is a partition of   A , the relation  being in the same partitionelement as  is an equivalence relation on A.(f)  Fact 3 : If   R  is an equivalence relation, then the equivalence relationgenerated by the partition generated by  R  is  R . And if   P  is a parti-tion on  A , then the partition generated by the equivalence relationgenerated by  P  is  P .2
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