An Excursion into Set Theory using a Constructivist Approach

Transcription

1 An Excursion ino Se Theory using a Consrucivis Approach Miderm Repor Nihil Pail under supervision of Ksenija Simic Fall 2005 Absrac Consrucive logic is an alernaive o he heory of classical logic ha draws a direc relaion beween an asserion and is exisence. Some of he few imporan differences beween a classical proof and a consrucionis proof include he avoidance of proof by conradicion and he rejecion of he law of he excluded middle. However, given he possibiliy ha consrucive logic is an alernaive in proof heory, is his approach o proof an effecive alernaive o undersanding imporan heorems ha have already been classically proven? To moivae an answer, his projec shall inroduce he basic foundaions of Se Theory, a plaform upon which we inend o apply consrucivis logic. We will hen reconsruc imporan definiions from a consrucivis poin of view.

2 Informal and Formal Noions of Consrucive Mahemaics The birh of modern consrucive logic is ofen aribued o he docoral disseraion of L.E.J Brower (88-966) iled Over de Grondslagen der Wisunde which was published in 907. In his paper, Brouwer inroduced he idea of inuiionism, a fac when applied o mahemaics ha demands a direc relaion beween he validiy of an absrac mahemaical srucure o he exisence of is consrucion. To inroduce, informally, imporan conceps in consrucive mahemaics, one only needs o begin wih fundamenal conceps of classical logic ha form he basis of many heorems in classical mahemaics. For example classically, i is possible o prove ha P Q holds wihou explicily showing ha P holds or ha Q holds. However, an inuiionis approach o he disjuncion rule would require ha i is no enough o asser P Qwihou eiher proving he exisence of P or proving he exisence of Q. Keeping he rule of disjuncion in mind, how would we acle a siuaion when Q = P? This siuaion leads o he well nown Law of he Excluded Middle or LEM ha saes ha for any proposiion P i is rue ha ( P P). While a firs glance he proposiion seems ordinary and unineresing, one should realize ha mos ofen we will no now eiher P or P. A famous example is Ferma s Las Theorem. If we were o consider his open mahemaics problem before is soluion was inroduced in 994 by Andrew Wiles, could we really asser ( F F ) wihou having proven F or F? I is his idea ha led consrucivis mahemaician L.E.J Brouwer o rejec LEM from any form of consrucivis proof. I is imporan a his sage o emphasize ha a consrucivis proof ofen requires a finie rouine wih a se of rules ha will no rely on he idealisic noion of disjuncion as implied by LEM and solely be resed on he fundamenal noion of exisence. Thus he following rules are an inroducion o wha possibly one can use o prove an asserion consrucively: (a) (or) o prove P Q we mus eiher consruc a proof of P or a proof of Q (b) (and) o prove P Q we mus prove boh exisence of P and Q ( implies) P Q (d) (no) o prove P we mus show ha P implies 0= (e) (here exiss) o prove xp( x) (f) (for all) a proof of xp( x) (c) a proof of is an algorihm ha convers a proof of P ino a proof of Q holds. we mus consruc an objec x and prove ha P(x) holds, is an algorihm, applied o any objec x, proves ha P(x) Consrucive Inerpreaion of Logic, Sanford Encyclopedia of Philosophy

3 Brief Inroducion o he Theory of Ses In his projec, se heory will be inroduced in is naïve form. Naïve se heory is simply he heory of ses ha are inroduced and undersood using wha is aen o be he self eviden concep of ses as collecions of objecs considered as a whole 2. On he oher hand, axiomaic se heory is he heory of ses ha are defined and inroduced by firs defining he axioms ha define heir properies. This form of se heory, ofen nown as ZFC or Zarmelo-Fraenel-Choice Se Theory, shall no be considered during he course of his projec. The concep of a se is fundamenal o mahemaics and is defined by he group elemens ha consiue ha se. A funcion f is defined as a rule f: A B ha associaes an elemen in b f(a) of a se B o each elemen a of a se A. Thus he se of all funcions from a se A o a se B will be denoed F(A,B). I is imporan o undersand ha non-counable ses are of lile pracical imporance, since i would be impossible o validae he exisence of such a srucure. Definiion Le A and B be ses. Thus, A is considered o be a subse of B if and only if every elemen in A is an elemen of se B. This is denoed in abbreviaed form as A B. If on he oher hand, he se A is indeed a subse of B and here is a leas one elemen in B ha is no conained in A, hen A is called a proper subse of B. Ses A and B are said o be equal if and only if A Band B A. Definiion 2 Le A and B be ses. Thus, he union of ses A and B is he se of elemens ha are no shared by boh ses and is ofen denoed A B. The inersecion of ses A and B, denoed A B, is he se of all elemens ha are shared by boh ses A and B. Definiion 3 Suppose here exiss a non empy se Λ conaining for each λ a se A λ, hen ξ = { A λ : λ Λ } is called an indexed family of ses wih Λ as he index se. Thus, he union of all ses conained in ξ is given by { A λ : λ Λ } = { x : x A λ for some λ Λ } Liewise, he inersecion of all ses conained in ξ is given by I { A λ : λ Λ} = {x : x A λ for all λ Λ } Definiion 4 Le A and B be ses. Then he Caresian produc (also called cross produc) of A and B, denoed A Bis he se of all ordered pairs (a,b) such ha a A and b B. This can be wrien as A B= {(a,b) : a A and b B} 2 Se Theory, Wiipedia, hp\\en.wiipedia.org

4 Definiion 5 The complemen of se B in A, denoed A \ B, is he se of all elemens in A and no in B. If he inersecion of ses A and B consiue he empy se i.e. A B =, hen boh A and B are disjoin Definiion 6 Le S R. If by definiion S S, hen S is called a closed se. If, S R \ S, hen S is called an open se. The following are also rue: (a) he union of any collecion of open ses is an open se (b) he inersecion of any finie collecion of open ses is an open se Consrucive Se Theory To inroduce imporan ideas in consrucive se heory, le us consider he noion of complemened ses. In paricular we shall inroduce, by example, excepions which are based on he idea of complemened ses. The idea of complemenaion is imporan since mos of is proofs can be defined by proof of conradicion which is no allowed from consrucivis poin of view. Thus, i is necessary o define complemenaion using he noion of complemened ses, and using is properies in such a way as o avoid any usage of proof by conradicion. One such way is use he noion of inequaliy wih complemened ses. Definiion 7 3 Suppose X is a se and F a non-empy se of funcions from X o. A complemened se in X consiss of an ordered pair (A, B) of subses of X such ha for each x in A and each y in B here is exiss f in F wih inequaliy f(x) f(y). Thus a union of such a pair denoed ( A, B ) is defined by T A A B T I T While he inersecion denoed I ( A, B ) is defined by T A I A B T T The following rules are valid for complemened ses: ( A) =A (a) An = I ( A n ) (b) IAn = ( A n ) (c) Thus given he noion of complemened ses, he following inroduces excepions o some laws involving complemened ses and wha i means o consrucively decide an asserion of a mahemaical saemen. The infinie disribuive law defined in Bishop & Bridges 985 4, is shown on he following page: B B

5 3 Definiion 5, Se Theory, Foundaions of Consrucive Analysis, Erre Bishop, McGraw Hill, 967 I I ) = ( n () n is no valid for complemened ses and n. To show his, one can consider he se defined as: = Now, le he se Γ and a sequence ( n) denoe a sequence of inegers consiuing eiher a or a 0 for each. We also do no now wheher n is a 0 for a paricular. Now we define he h complemened se wih an ordered pair = (, ) by he following: = (, ) if n = = (, ) if n = 0 Now he union of ses and gives rise o he ordered pair (, ) as shown below: sing (), we have from LHS: Since = (, ) for each. Now, I = I{ } = (, ) (, ) = I = ( ) using rule (c) = ( ) using rule (a) As one can see, his is analogous o he LEM which is no allowed as par of a consrucive proof. Thus i is no possible o find an elemen in belonging o I n. Thus, he relaion defined in () is no valid. I mus be menioned here ha he reason we demand a consrucive basis is simply o avoid usage of negaion which is ofen used in operaions involving complemened ses. And as a resul, while mos laws w.r. o complemened ses are derived from classical laws, here are ofen he odd one or wo ha need o be redefined from a consrucivis poin of view. 4 Equaion (2.3), Se Theory, Consrucive Analysis, Erre Bishop & Douglas Bridges, 985

6 Las bu no leas, a consrucive subsiue 5 o he infinie disribuive law (), is defined below as follows: ( ) = ( ) ( ) Projec Ouline and Inended Oucome Having consruced a plaform based on fundamenal ideas of se heory, we shall now embar on an excursion ha will aemp o reconsruc imporan heorems/definiions using consrucive se heory. Of ineres, and a possible research opic as a moivaion o consrucive mahemaics, I inend o explore a possible consrucive proof o he Burali-Fori Paradox which is saed below: The Burali-Fori Paradox shows ha by naively consrucing 6 a se of all ordinal numbers, one arrives a a conradicion wih respec o is consrucion. The reason behind his is ha by consrucing a se of ordinal numbers ζ, his se auomaically qualifies as having a successor ( ζ + ) which is sricly greaer han ζ, given by he following relaion: ( ζ ) ζ < + However, a conradicion arises ha since ζ is he se of all ordinal numbers hen ( ζ + ) being iself an ordinal number, here by belongs o he se ζ. If his is so, hen he following relaion mus also hold: And hus, we have a conradicion. ( ) ζ < ζ + ζ The aim for he res of he semeser is o explore possible ways o circumnavigae his paradox consrucively if a all his paradox exiss from a consrucive view poin and his remains o be seen. 5 Equaion (2.3), Se Theory, Consrucive Analysis, Erre Bishop & Douglas Bridges, 985

7 6 Equaion (2.3), Se Theory, Consrucive Analysis, Erre Bishop & Douglas Bridges, 985