However it is the insight of Dedekind’s analysis that we obtain the powerful principle of induction, If we take away the requirement that the sequences be shrinking, does by taking an intersection over (what one can show is a proper) class of all inductive sets. xP⋂jPIAj↔(@jPI)(xPAj). Why do we need this? Example1.28 (i) Let I=N.Each Ai=R.Then∏iPIAiis the same asNRthe set of infinite sequences of Definition1.1We letP(x)denote the class{y∣yĎx}. Definition1.16 (Kuratowski) Let x,y be sets.Theordered pair setof x and y is the set We Definition1.19We defineorderedk-tupleby induction:⟨x,x⟩has been defined; if⟨x 1 ,x 2 ,⋯,xk⟩has But the above definition by recursion makes matters correct. HenceTC(x)is the smallest transitive set t satisfying xĎt. Axiom of the Empty SetThere is a set with no members. forming sets if we are to be free of contradictions.There followed a period of intense discussion as to Then(N,<)is a wellordering, but(Z,<)is not. We conclude that⋃xĎx. IfXis the set of We shall use the standard notationP for the Q.E.D. We have a number of basic laws that⋃and⋂satisfy: @i(iPI→AiĚC)→⋂iPIAiĚC. fundamental to the notion of ordinal number. all points in the plane, andYthe set of all circles, the ‘pis the centre of the circleS’ determines a relation Exercise1.6 Show that for any setX: a)⋃P(X)=Xb)XĎP(⋃X); when do we have=here? So suppose⟨x,y⟩=⟨u,v⟩.Case 1 x=y.Then⟨x,y⟩=⟨x,x⟩={{x},{x,x}}= Conversely: if⋃xĎxthen for anyyPxby definition of⋃, (ii)(x,y,zPX∧xăy∧yăz)→(xăz). To understand this consider an analogous situation in which we consider human beings in the place of sets and elements, and x ∈ A means x is an ancestor of A. the same for{∅,{∅},{∅,{∅}}}. Implicit in this is the idea that wecancollect together all the subsets of a given set. Hence, by Extensionality{u}={x}, and so, again using This leads to: Definition1.20(i) Let A,B be sets. ⋃x=d fx;⋃x=d f⋃x,⋃x=d f⋃(⋃x), ,... ,⋃n+x=d f⋃(⋃nx)... as in the next definition.]. Which of the defining conditions above do we need? Recall Def.1.31. Both Zermelo’s and von Neumann’s bers , , ,.. .. A set is determined by its elements Remark 1. specification defined asetof objects. sets that the Axiom of Subsets will always consistently yield sets. elementhoodrelation:xPAwill be read as “ the setxis an element of the collectionA”. including formal number theory: such theories cannot prove their own consistency. of Inf. andmaximalelements are defined in the corresponding way. However we shall want to see how we can specify relations underScannot be finite:∅,S(∅),S(S(∅))are all distinct (although we have not proved this yet). does not imply that least elements will always exist (think of the total ordering(Z,≤)). However: Lemma1.17(Uniqueness theorem for ordered pairs) ● In generalAˆB≠BˆAand further, theˆoperation is not associative. (v) A partial ordering can also be considered a relation: R={⟨x,y⟩∣xĺy}. (iv) Show that Trans(x)←→Trans(P(x)). case⋂jP∅Aj=V. If we have a set and an object, it is possible that we do not know whether this object belongs to the set or not, because of our lack of information or knowledge. This latter system has the advantage that “n” has exactlyn yĎ⋃x, henceyĎxand thus Trans(x). set theory and forcing lecture notes jean-louiskrivine translated by: christian rosendal typeset by: jessica schirle Antisymmetric (x,yPX∧xRy∧yRx)→x=y) perhaps recognizing the importance of this concept and generalizing it.The theory of wellorderings is correctly rule out all obviously inconsistent ways of forming sets. ● IfX=P(A)for some setAand we tookxRy⇔xĎyforx,yPXthenRis reflexive, antisym- minimalelement if@yPX(¬yăx). (In general we designate any collection of They are not guaran-teed to be comprehensive of the material covered in the course. Axiom of Infinity:There exists an inductive set: represented by sets. betweenXandY. If this equals⟨u,v⟩then we must haveu=v(why? Type of relation Defining condition. So there is no setzequal to (i) xPX→¬xăx ; ).The moral of Russell’s argument (which he took) is that we must restrict our ways of His papers on the subject appeared between 1874 to 1897. Introduction to Logic and Set Theory-2013-2014 General Course Notes December 2, 2013 These notes were prepared as an aid to the student. been defined, then⟨x,⋯,xk,xk+⟩=d f⟨⟨x 1 ,⋯,xk⟩,xk+1⟩. (ii)@i(iPI→AiĎC)→⋃iPIAiĎC. Once the dust eventually cleared, the following axiom scheme was seen to suppose for a contradiction it was a set; apply the axiom of union.]. (ii)!is the class of natural numbers. ● Thus⟨x,x,x⟩=⟨⟨x,x⟩,x⟩,⟨x,x,x,x⟩=⟨⟨⟨x,x,⟩x⟩,x⟩etc.Note that once we have objects as aclassand we reserve the termsetfor a class that we know, or posit, or define, as a set.The These notes were prepared using notes from the course taught by Uri Avraham, Assaf Hasson, and of course, Matti Rubin. X=Y. structure on∏iPIGito turn it into a group. We shall be more interested in relations between elements of a single set, thatis when Then comes!+,!+,.. .. (iii) LetXbe a class of transitive sets. In words: for any non-empty set Z there is another set,⋂Z, which consists precisely of the members of of Extensionality). IfX≠∅, then⋂Xis transitive. then we should havezPz! foundations of arithmetic (which tried to derive the laws of arithmetic from purely logical assumptions) that plagued the early theory and led many to doubt that the theorycould be made coherent. result in set theory: it was his discovery of the uncountability of the realnumbers, which he noted on {{∅}},{∅,{{∅}}}are not. We have talked about relationsRthat may hold between objects, and even used the notationĺif we members, and is the set of all its predecessors in the usual ordering. a class is not, or cannot be, a set, then we call it aproper class. (It will turn out have setsX,Yand a relationRthat holds between some of the elements ofXand ofY. =d f{}={∅}=∪{}=S(), (below) that for any setxthere is a smallestyĚxwith Trans(y).) Exercise1.3 We define therankof a setx(‘(x)’) to be the least such thatxĎV. x=u. Trans(x)by⋃xĎx. If so thenzRz. how to “correctly” define sets. Intuitively though we can see that any inductive set which has to be closed We say that elements has one member, and the other two. It is natural to adopt some kind of notation )Hence one of these (ii)P(∅)={∅};P({∅})={∅,{∅}};P({a,b})={∅,{a},{b},{a,b}}. these. Exercise1.15 Show thatAˆ(B∪C)=(AˆB)∪(AˆC). damental and, as it turned out, fatal error to his programme. Hence{x,y}= Axiom of Power SetFor any set xP(x)is a set, the power set of x. So{x}={u}and It is important to notice that this axiom is a non trivial assertion about belonging. This course will start with the basic primitive concept ofset, but will also make use along the way This required him to investigate the notion of such infinite systems. dering. ă(orĺ). {u,v}Ð→x=u∧y=vfails. (1862-1943). Suppose the setZin the above definition were empty: then we should have that for anytwhatsoever Set theory Q.E.D. AˆAˆ⋯ˆAk+=d f(AˆAˆ⋯ˆAk)ˆAk+ numbers. reflexivity to hold, then we useĺ, so that thenxĺxis allowed to hold. ematical discourse.How can we do this? Appeal to the Ax. =d f{, }={∅,{∅}}=∪{}=S(), So of these last two sets, if they are the same theny=v. So(b)of Def. Definition2.1A set Y is calledinductiveif (a)∅PY (b)@xPY(S(x)PY). Reflexive xPX→xRx The material is mostly elementary. to set theory. haveyPx. (i) A partial ordering R on a set A,(A,R)is awellfounded relationif for any subset YĎA, if Y≠ picture is thus: At the bottom isV=df∅;V=dfP(V)=P(∅);V=dfP(V);Vn+=dfP(Vn).. .The question AˆB=d f{⟨x,y⟩∣xPA∧yPB}. this is not all just fantasy. numbers etc., from analysis, or other mathematical construct, can be constructed fromsets. Lemma1.34(Lemma onTC)For any set x (i) xĎTC(x)andTrans(TC(x)); (ii) IfTrans(t)∧xĎt→ ∅,then Y has an R-minimal element. He defined a “first Definition1.29A set x istransitive,Trans(x), iff@yPx(yĎ x). We formally state this as a principle about inductive sets given by some property Φ: Copyright © 2020 StudeerSnel B.V., Keizersgracht 424, 1016 GC Amsterdam, KVK: 56829787, BTW: NL852321363B01. Exercise1.8 LetI=Q∩(, /)be the set of rationalspwith .

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