Western University. Imants Barušs King's University College, Robert Woodrow

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1 Western University Psycholoy Psycholoy 2013 A reduction theorem or the Kripke-Joyal semantics: Forcin over an arbitrary cateory can always be replaced by orcin over a complete Heytin alebra Imants Barušs Kin's University Collee, baruss@uwo.ca Robert Woodrow Follow this and additional works at: Part o the Loic and Foundations Commons Citation o this paper: Barušs, Imants and Woodrow, Robert, "A reduction theorem or the Kripke-Joyal semantics: Forcin over an arbitrary cateory can always be replaced by orcin over a complete Heytin alebra" (2013). Psycholoy. 3.

2 A Reduction Theorem or the Kripke-Joyal Semantics: Forcin over an arbitrary cateory can always be replaced by orcin over a complete Heytin alebra Imants Barušs and Robert Woodrow Abstract. It is assumed that a Kripke-Joyal semantics A = C, Cov, F, has been deined or a irst-order lanuae L. To transorm C into a Heytin alebra C on which the orcin relation is preserved, a standard construction is used to obtain a complete Heytin alebra made up o cribles o C. A pretopoloy Cov is deined on C usin the pretopoloy on C. A shea F is made up o sections o F that obey unctoriality. A orcin relation is deined and it is shown that A = C, Cov, F, is a Kripke-Joyal semantics that aithully preserves the notion o orcin o A. That is to say, an object a o COb orces a sentence with respect to A i and only i the maximal a-crible orces it with respect to A. This reduces a Kripke-Joyal semantics deined over an arbitrary site to a Kripke-Joyal semantics deined over a site which is based on a complete Heytin alebra. We will bein by recapitulatin the deinition o the Kripke-Joyal semantics since the details o the deinition will be needed to prove the reduction theorem in the second part o this paper. 1. The Kripke-Joyal Semantics Robert Goldblatt s version o a site rom [1] and A. Kock and G.E. Reyes notion o orcin over a site [2] are used to establish the deinition o the Kripke-Joyal semantics used in this paper. Alternative expositions o orcin in cateorical contexts can be ound in [3], [4], [5], and [6]. In this paper, except or set membership and inclusion, all compositions o arrows are written in the order o composition. Also, a cateory is considered to be small i its collection o arrows is a set. This allows us to orm the irst deinition. Deinition 1.1. A stack or preshea o sets over a small cateory C is a contravariant unctor C F S where S is the cateory o sets. The unctor 2013 Imants Barušs and Robert Woodrow

3 2 Imants Barušs and Robert Woodrow cateory S Cop is the cateory o all stacks over C. For a COb, s af is called a erm. I A is a set then AP is its power set. The capital letters, I, X, Y, and so on, represent index sets. Deinition 1.2. A pretopoloy on a small cateory C is an assinment COb Cov ((CAr)P)P which takes a COb to a collection o sets o arrows in C with codomain a satisyin the ollowin conditions: (i) the empty set φ acov. (ii) the sinleton a 1a acov. (iii) i (iv) i a x X acov and or each x X y Cov,then y x y a y Y x and x X acov. a x X acov and b the pullback b a x b o alon y y Y x a CAr then or each x X b a x b P b a b a a x x b x x bcov. exists and Both Goldblatt and Kock and Reyes leave condition (i) out o their deinitions o pretopoloies. However, it is necessary or the development o a semantics or a ormal lanuae. A collection a x X acov is called a cover o a. I C is the cateory o open sets o a topoloical space (with a (unique) arrow rom U to V i and only i U V or U and V open sets) then the assinment U Cov = open covers o U satisies the above conditions. The unction Cov can be extended to a contravariant unctor by assinin to b a the unction that takes a cover o a to its correspondin cover on b by (iv) o the above deinition. Thus Cov (S Cop )Ob. Deinition 1.3. A site C, Cov is a small cateory C with a pretopoloy Cov. Let C, Cov be a site and a : x X acov. Then the pullback o alon y is called a a y x a y and the pullback o y alon

4 A Reduction Theorem or the Kripke-Joyal Semantics 3 is called a a y y or each x, y X as shown in the diaram: a a y x a y y P b a y I F is a stack over C then the imae o a a y y is called F F x y x F ( a a y )F and the imae o a is called af x ax F, or all x, y X. Deinition 1.4. A stack F is a shea over the site C, Cov i it satisies the compatibility condition: Given any cover a x X acov or a COb and any collection o elements s x F x X that are pairwise compatible, i.e., s x Fy x = s y Fx y or all x, y X, there is exactly one s af so that sf x = s x or all x X. The ull subcateory o S C0p enerated by those objects that are sheaves over the site C, Cov is called CSh. A Grothendieck topos is a cateory that is equivalent to one o the orm CSh or a site C, Cov. Let L be a irst order ormal lanuae deined in the usual way [7]. We let v, v 1, v 2, and so on represent the variables o the lanuae, w, w 1, w 2, and so on represent the constants o the lanuae and u, u 1, u 2, and so on be metavariables or variables and constants. For an inteer m, the set mw t is the collection o all constants and variables that are contained in the list u 1,..., u m. An inteer m is appropriate to a well-ormed ormula Ξ i all o the variables and constants o Ξ appear in the list u 1,..., u m. In the ollowin deinition, an A-valuation at a or a well-ormed ormula o L with m appropriate and a COb, is a unction mw t t af that has action v i t i af or variables v i with i m and w w a af or constants. The notation t(i/s) indicates the valuation obtained rom t by replacin t i by the element s af. An element in the imae o t is also written as t. The A-valuations satisy the ollowin closure condition: Let t be an A-valuation at a COb or a well-ormed ormula Φ with m appropriate to Φ and (b a) CAr with codomain a. Then w a (F ) is the imae o w or any constant w o L and any A-valuation at b or any well-ormed ormula Ψ with k m appropriate to Ψ. By an a-evaluated well-ormed ormula is meant a well-ormed ormula Ξ o L, m appropriate to Ξ and an A-valuation at a or Ξ. An a-evaluated well-ormed ormula Ξ with A-valuation t is written as Ξ[t]. Now:

5 4 Imants Barušs and Robert Woodrow Deinition 1.5. Let L be a irst-order lanuae as beore and let A = C, Cov, F, be an ordered quadruple where C, Cov is a site, F is a shea and is a binary relation between objects a o C and a-evaluated well-ormed ormulae. Then A is a Kripke-Joyal model i any well-ormed ormula Ξ with m appropriate to Ξ, a COb and t an A-valuation at a, satisies the appropriate condition below: or Π atomic, i a Π[t] and b i a is an arrow o C, then b Π[t(F )]; a x X acov so that x X, Π[t( F )] then a Π[t]; a ( v i )Σ[t] i there is a cover a x X acov and a collection s x F x X so that Π[t( F )(i/s x )], x X; a (Φ Ψ)[t] i there is a cover a x X acov so that Φ[t( F )] or Ψ[t( F )], x X; a Φ[t] i (b a), not b Φ[t(F )]; a (Φ Ψ)[t] i (b a), i b Φ[t(F )] then b Ψ[t(F )]; a (Φ Ψ)[t] i or every b a, b Φ[t(F )] and b Ψ[t(F )]; a ( v i )Σ[t] i (b a) and any s bf, b Σ[t(F )(i/s)]. I Ξ has index n, let a Ξ[s 1,..., s n ] i a Ξ[t] or some A-valuation t or Ξ with m appropriate to Ξ, where t i1 = s 1,..., t in = s n. Then a Ξ i a Ξ[s 1,..., s n ] or all [s 1,..., s n ] (af ) n. Finally, A is a model o Ξ, written A = Ξ i or all a COb, a Ξ. We will need the property that truth persists in time. More speciically: Lemma 1.6. Let A = C, Cov, F, be a Kripke-Joyal model or L and Ξ a well-ormed ormula o L. I a COb, m is appropriate to Ξ and t is an A-valuation o Ξ at a so that a Ξ[t], then b Ξ[t(F )] or all b a. Proo. The proposition is true by deinition i Ξ is an atomic ormula. The remainin cases are proved by induction on the complexity or Ξ. The technique o the proo is illustrated here by establishin the cases where Ξ is ( v i )Σ(v i ) and where Ξ is (Φ Ψ). Let a ( v i )Σ[t]. Then there is a cover a x X acov and a collection s x F x X so that Σ[t( F )(i/s n )]. Let b x x b be the pullback o alon or each x X, as shown in the diaram: x b P b a b x x Note that ( F )( x F ) = ( x )F = ( x)f = (F )( xf ) or each x X because F is a contravariant unctor. By the induction hypothesis,

6 A Reduction Theorem or the Kripke-Joyal Semantics 5 b x Σ[t(F )(F )(i/s x ( x F ))] or each x X. Since b x x b x X bcov by condition (iv) o Deinition 1.2, rom Deinition 1.5 it can be seen that b ( v i )Σ[t(F )]. Let a (Φ Ψ)[t]. Then or every arrow c a, c Φ[t(hF )] and h c Ψ[t(hF )]. In particular, i c a actors throuh b a, i.e. i h is the composition c b a or some c b, then c Φ[t(()F )] and c Ψ[t(( )F )]. And since F is a contravariant unctor, t(()f ) = (t(f ))(F ). But then c Φ[(t(F ))(F )] and c Ψ[(t(F ))(F )]. Since c b is any arrow with codomain b, rom the deinition o conjunction, b (Φ Ψ)[t(F )]. h 2. A Reduction Theorem or the Kripke-Joyal Semantics It is not clear at irst what should be done to successully turn the domain C o the model A = C, Cov, F, into a Heytin alebra. I the uture state o an object a in C is deined to be the domain o an arrow with codomain a, then it may seem that ideals o objects closed with respect to the uture should enerate an appropriate Heytin alebra. That stratey works successully where C is a preorder IP, so that or p, q IP Ob there is at most a sinle arrow p q. In the case o a, b COb, however, there may be parallel arrows a b. Now, the topoloy enerated by principal ideals made up o objects o C is too coarse. Given a well-ormed ormula Ξ and a valuation t bf o its ree variables or which b Ξ[t] there is no way o distinuishin between a Ξ[t(F )] and a Ξ[t(F )]. This becomes a problem when tryin to deine orcin over the resultin Heytin alebra. To enable these distinctions to be made, it turns out to be more ruitul to take collections o arrows closed under preextension. That is to say, C is deined to be the collection o all the cribles o C. Deinition 2.1. A collection p o arrows o C is a crible i and only i ( p)( CAr)( = p). A crible p is a b-crible i every arrow o p has codomain b. A crible p is an -crible i every arrow o p actors throuh. The maximal -crible is denoted by and is said to be the crible enerated by. These constitute the basis o a topoloy which is ine enouh to keep individual cases o orcin distinct. Deinition 2.2. Let C be a cateory. Then C is the topoloy enerated by the choice o CAr or a basis. Notice in the above deinition that only arbitrary unions are needed, and that the resultin topoloy is the collection o all the cribles o C. We know that every topoloical space is a Heytin alebra closed under arbitrary joins. In this case C is a complete Heytin alebra since it is closed under arbitrary meets as well.

7 6 Imants Barušs and Robert Woodrow The problem arises o deinin a pretopoloy or C. The passae rom C to C determines to a lare extent the deinition o Cov. In particular, or p COb, a cover o p will be a collection o inclusions p i p i I whose domains are themselves cribles. So that the resultant semantics matches the oriinal, only those collections are accepted that have the property, that or any arrow a b p there is a cover a x X a Cov in the oriinal pretopoloy so that or each x X, p i or some i I. More ormally: Deinition 2.3. Given a site C, Cov, let Cov be the assinment COb (CAr)P 2 p p i p i I ( (a b) p) ( ) a x X acov ( x X)( i I)( p i ). Lemma 2.4. Cov is a pretopoloy or C. Proo. Deinition 1.2 needs to be checked: (i) φ pcov since φ acov or any a COb. (ii) Let p COb. Then 1 p pcov since, or a a 1a a acov ives 1 a p. b p, the cover For (iii) assume that D = p i p i I p Cov and that or each p i, D i is a cover D i = p i j p i j J i pi Cov. It must be shown that D = p i j p j J i i I p Cov. Let a b p. Then there is a cover a x X a Cov so that x X there is an i I so that p i. Because D i is a cover o p i, or p i there is a cover y y y Y x Cov so that y Y x there is j J i so that y x p i j. Because Cov is a pretopoloy or C it satisies condition (ii) and so y x y a y Y x a Cov which, by the precedin arument has the property that or each y Y x and x X there is i I and j J i so that y x p i j. But then D is a cover o p. To see that (iv) is satisied, let r p and D = p i p i I p Cov. The pullback o r p with p i p is the lattice meet r p i, i I. It must be shown that D r = r p i i I is a cover o r. Let a b r. Then there is a cover a x X a Cov so that, or each x X there is an i I with the property that p i. But r and so r p i. This is true or every x X. Hence D is a cover o r and property (iv) is satisied. Hence Cov is a pretopoloy or C. Next comes the question o what the riht translation or a shea F is over C, Cov. Here one is uided by tradition. That is to say, one looks or

8 A Reduction Theorem or the Kripke-Joyal Semantics 7 a shea whose imae o p COb is a collection o sections o F and whose arrows are restrictions. By a section s o F is meant a unction p s S that chooses or each (a b) p a erm rom the stalk af over a. That is to say, s af. Now, these sections are used to evaluate variables in wellormed ormulae and in order to obtain a semantics that is compatible with the oriinal, the imae pf is deined to be the collection o all the sections over p that satisy unctoriality. More ormally: Deinition 2.5. Let F be a shea over the site C, Cov, then F is the assinment F COb S p p s p ( ( p)( (c q p (pf qf ) s s q Lemma 2.6. F is a shea over C, Cov. )F ( p)(s ( )F ) )()s = (s)(f ) Proo. Clearly F is a contravariant unctor. To check Deinition 1.4 let D = p i p i I p Cov and S = s i p i F i I be a collection o sections that aree on the intersections o the p i. Then it must be shown that there is a unique section over p whose restriction to each p i is just s i. First, a candidate s is deined. Let a b p. Then choose a cover a x X a Cov and or each x X choose i I with the property that p i. Let s x = ( )s i. Now p i and y p j. Let s x = ( )s i and s y = ( y )s j. By ormin the pullback o with y it can be seen that s x and s y are compatible. For suppose that is the pullback o alon y and y is the pullback o y alon. Then, because the pullback square commutes, and because o the deinin property or s i and s j and the act that they aree on the intersection o p i and p j : s x ( yf ) = ( y x )s i = ( y )sj = s y (F ). This is true or any x X, y X and appropriate i I, j I, and so, because F is a shea over C, Cov there is a unique s af so that s x = s( F ), x X. Let s = s. Now observe that the deinition o s is independent o the choice o end, let a x X a x X a Cov and p i s x F x X ive the value s at and B = with p i. To this a Cov and i x with p ix p Cov and y c y a y Y a Cov and j y with y p jy p Cov and t y a y F y Y ive the value t at, where s x and t y are deined above. Now pull back B alon each or x y x X. This ives a new cover c x y a y Y and x X acov

9 8 Imants Barušs and Robert Woodrow x y where c y is the pullback o y alon or each y Y and x X. Then s((y x )F ) = (y x )s ix or each y Y and x X. By a similar arument t((x y y )F ) = (x y y )s jy, y Y and x X. But then s(( x y )F ) = ( x y )s ix = ( y x y )s jy = t(( y x y )F ) = t(( x y )F ) or all x X and y Y. Because F is a shea, s = t. Now let s be the choice unction over p that ives s = s where s is deined above or p. Does s satisy the conditions o Deinition 2.5? To see that it does, let (a b) p and c a be an arrow with codomain a. Then p. Let s = s and t = ()s where s and t are deined by the construction iven above. It must be shown that t = s(f ). To see that, let A = a x X a Cov and or x X let i x I be such that p ix. Because covers are stable under pullbacks (that is to say, because o property (iv) o a pretopoloy), A can be pulled back alon to ive C = c x x c x X c Cov. Because o the way that t is deined, ()s ix = t(f ) or all x X since = x and x p ix or all x X. But then, because each s i satisies the deinin property, ( x )s ix = s( x )F, x X, where x is the pullback o alon, x X. But s(( x )F ) = s(()f ) = s(f )(F ), x X and so s(f )(F ) = s(( x )F ) = ( x )s ix = ()s ix = t(f ) or all x X. But C is a cover o c and the t(f ) are clearly compatible. Thereore there is only one t cf so that t(f ) = ()s ix, or all x X. Hence t = s q F. Next, to see that s p i = s i, i I, let (a b) p i and choose a 1a a a Cov. Then by deinition s = s i. To see that s is unique, assume that there is a section t on p that obeys the unctoriality condition with the property that t p i = t i, i I. Let (a b) p. It must be shown that s = t. There is a cover a x X a Cov so that x X there is i I with the property that p i. But then ( )s = ( )s i = ( )t, x X and appropriate i I. As shown beore, the ( )s i are compatible. Hence, because F is a shea, there is a unique s af so that s( F ) = s x, x X. And so s = s = t. Finally, it remains to deine a orcin relation over the site C, Cov, with respect to the shea F, so that a Kripke-Joyal model can be obtained that is compatible with the oriinal. There is a natural candidate: Deinition 2.7. Let A = C, Cov, F, be a Kripke-Joyal model or the irstorder lanuae L. Let Ξ be a well-ormed ormula o L. Then or p COb and t a valuation or Ξ at p, consistin o sections o F, deine p Ξ[t] i (a b) p, a Ξ[t ].

10 A Reduction Theorem or the Kripke-Joyal Semantics 9 Given Ξ with m appropriate and p COb, the valuation t is a choice unction mw t t pf that has action v j t j pf or variables v j with j m and w w pf or constants w where w = w ( )F or all p, and where w is the A-valuation o w at. To see that w obeys unctoriality, recall that or c, w = w (F ) interprets w at c. As a result, there is no problem with the treatment o constants o L in the transition rom A to A. Theorem 2.8. Given A = C, Cov, F,, the derived quadruple A = C, Cov, F, is a Kripke-Joyal semantics. Proo. It remains only to show that satisies the properties o the Kripke- Joyal semantics in Deinition 1.5. The proo proceeds by induction on the complexity o a well-ormed ormula Ξ. Let p COb and t (pf ) m where m is appropriate to Ξ. The atomic case is an easy consequence o unctoriality. Let Ξ be the well-ormed ormula ( v i )Σ(v i ) where Σ contains at least the ree variable v i, and assume that the hypothesis holds or Σ. It must be shown that p ( v i )Σ[ t ] i and only i there is a cover p i p i I p Cov and a collection s i p i F i I so that p i [Σ(t pi )(i/s i )], i I. To show this last equivalence, assume irst that p ( v i )Σ[t ]. It is necessary to construct an appropriate cover o p. To do this, irst observe that (a b) p, a ( v i )Σ[t ]. There is a cover a x X a Cov and a collection s x F x X so that Σ((t )( F ))(i/s x )F, x X. But then p, x X. Let the cover o p be the collection x X p. Clearly this belons to p Cov. For each let the section s x associated with it be the section enerated by s x. By that is meant that i c is any arrow then ( )s x = s x (F ). Then s x satisies the deinin property o a erm in the stalk ( )F. But now, since Σ((t )( F ))(i/s x ), by Lemma 1.6 i c is an arrow o C then c Σ[((t)( F )(F ))(i/s x (F ))]. Since (t)( F )(F ) = ( )t or all x X then c Σ[( )t(i/( )s x )]. But then Σ[t (i/s x )]. This is true or every x X and p. Hence there is a cover x x X p p Cov and a collection s x x X p so that Σ[ t (i/s x )] or all x X and p. To show the converse, assume that there is a cover p i p i I p Cov and a collection s i p i F i I so that p i Σ[ t pi (i/s i )], i I. Let (a b) p. Then there is a cover a x X a Cov x X there is an i I with the property that p i. But Σ[( )t(i/( )s i )] or every x X and appropriate i I. Since ( )t = (t)( F ), x X then a ( v i )Σ[t ]. But then p ( v i )Σ[ t].

11 10 Imants Barušs and Robert Woodrow The case or disjunction is similar to that or the existential quantiier. Neation, implication and conjunction are similar to the case or the universal quantiier which is demonstrated below: It must be shown that p ( v i )Σ[ t ] i or every q p and or every s i qf, q Σ[ t q(i/s i )]. Assume that p ( v i )Σ[ t ], q p, s i qf and let (a b) q. Then p and a ( v i )Σ[t]. But then a Σ[(t)(i/s i )]. Hence q Σ[ t q(i/s i )]. Now, to show the converse, assume or every q p and s i qf that q Σ[ t q(i/s i )]. Let (a b) p, let c a be an arrow with codomain a and let s cf. Let s i ()F be the section enerated by s. Since p then Σ[ t q(i/s i )]. But then c Σ[()t(i/s)] and so a ( v i )Σ[t]. Now, with any a COb is associated the maximal crible 1 a, and with any t af can be associated the section t enerated by the erm t. This leads immediately to: Corollary 2.9. Given a Kripke-Joyal model A = C, Cov, F, and derived Kripke-Joyal model A = C, Cov, F, then, or well-ormed ormulae Ξ o the irst-order lanuae L, or any a COb and any valuation t or Ξ at a, a Ξ[t] i and only i 1 a Ξ[ t]. 3. Conclusion This paper bean by considerin a Kripke-Joyal model A = C, Cov, F, deined over an arbitrary cateory C. By takin the collection o cribles on C, a complete Heytin alebra C was obtained. A pretopoloy Cov was deined or C that preserved the essential eatures o Cov. Furthermore, in the transition rom C, Cov, to C, Cov, the pretopoloy Cov has acquired the property that any collection p i p i I with i I p i = p is a cover o p. By takin sections o F that obey unctoriality, a shea F was obtained whose erms are used to interpret the constants and ree variables o well-ormed ormulae o L in a manner that is compatible with the oriinal A-valuation. Finally, a orcin relation was deined over C, Cov in terms o the oriinal relation. It was shown that A = C, Cov, F, is a Kripke-Joyal model and that it orces precisely the same well-ormed ormulae as the oriinal model A. As a result, without loss o enerality, orcin over an arbitrary cateory can always be replaced by orcin over a complete Heytin alebra. On the ace o it, this result seems impossible. How is it that the enerality o a cateory can be replaced by the speciicity o a complete Heytin alebra as a domain o orcin? The answer lies in the construction o the derived cateory. What we see is that, in the derived cateory, each object o the oriinal cateory indexes all uture states o that object, so that any pathway rom the past to the uture in the oriinal cateory can be traced alon any o a number o pathways in the derived cateory. This also reveals that the oriinal cateory is not really embedded in the derived cateory or, more

12 A Reduction Theorem or the Kripke-Joyal Semantics 11 accurately perhaps, that it is embedded in multiple distorted variations. In that sense this is an embeddin theorem whereby a eneral construction is multiply enolded in a construction with more structure. What is the siniicance o this result? This work arose in the course o attemptin to internalize Cohen s orcin in a topos. Given that Cohen s orcin is an instance o a Kripke-Joyal semantics, it remained only to internalize a Kripke-Joyal semantics. The idea was to try to orce alon the lattice o subobjects o 1 o a topos and then to o rom there. That could only be done i the domain o orcin o a Kripke-Joyal semantics had more structure to it than it appeared to have, namely, that the domain o orcin could always be replaced by a complete Heytin alebra. That is how we set about lookin or a proo, which we ound, as detailed in this paper. The remainder o our project was less successul in that we could only internalize a eneralized version o orcin or very restricted lanuaes [8]. There is also a philosophical point that could be o siniicance. Given that a complete Heytin alebra is isomorphic to the lattice o subobjects o 1 o the cateory o sheaves over it, the domain o orcin can always be thouht o as a collection o possible truth values. And not just arbitrary truth values, but truth values that are situated between true and alse. Thus, this theorem not only provides a reduction o the enerality o the domain o orcin o a Kripke-Joyal semantics, but also reduces the enerality o truth values so that they can always be said to lie between true and alse or any semantics that are instances o a Kripke-Joyal semantics. Authors Note This paper is Section 1 o Chapter 2 and all o Chapter 5, with some editorial chanes, rom the irst author s master s thesis in mathematics at the University o Calary, titled An Examination o Cohen s Forcin in a Topos-Theoretic Framework, which he successully deended in The irst author would like to thank his thesis advisor, Verena Huber-Dyson or teachin him loic, cateory theory, and topos theory, and or assumin responsibility or the thesis. He would also like to thank Robert Woodrow, the second author o this paper, who assumed the role o interim thesis advisor while Proessor Dyson was on sabbatical and under whose uidance and with whose ull participation the work reported in this paper was done. The authors thank the University o Calary or inancial support that allowed or this work to be done, and they thank Kin s University Collee or a research rant and Medical Technoloy (W.B.) Inc. or inancial support that was used to prepare this paper or publication. The authors are also rateul to Pat Malone or typin the manuscript.

13 12 Imants Barušs and Robert Woodrow Reerences [1] R. Goldblatt, Topoi the Cateorial Analysis o Loic, Volume 98 o Studies in Loic and the Foundations o Mathematics, North-Holland Publishin Company, Amsterdam, New York, Oxord, [2] A. Kock and G.E. Reyes, Doctrines in Cateorical Loic, paes o the Handbook o Mathematical Loic edited by J. Barwise, rom the series Studies in Loic and the Foundations o Mathematics, North-Holland Publishin Company, Amsterdam, New York, Oxord, [3] R.J. Grayson, Forcin in Intuitionistic Systems Without Power-Set, The Journal o Symbolic Loic, Volume 48, Number 3, paes , [4] A. S cedrov, Forcin and Classiyin Topoi, Number 295 o Memoirs o the American Mathematical Society, American Mathematical Society, Providence, Rhode Island, USA, [5] J. Lambek and P.J. Scott, Introduction to Hiher Order Cateorical Loic, Number 7 in the series Cambride Studies in Advanced Mathematics, Cambride University Press, Cambride, UK, [6] S. Mac Lane and I. Moerdijk, Sheaves in Geometry and Loic: A First Introduction to Topos Theory, Spriner-Verla, New York, [7] V. Huber-Dyson, Gödel s Theorems; a Workbook on Formalization, B.G. Teubner Verlasesellschat, Stuttart, Germany [8] I. Barušs, An Examination o Cohen s Forcin in a Topos-Theoretic Framework, Master s Thesis, Department o Mathematics and Statistics, University o Calary, Calary, Alberta, Canada. Imants Barušs Department o Psycholoy Kin s University Collee at The University o Western Ontario baruss@uwo.ca Robert Woodrow Department o Mathematics and Statistics University o Calary woodrow@ucalary.ca