Vopěnka, Petr

Institute of Mathematics AS CR, v. v. i.

Commentationes Mathematicae Universitatis CarolinaePetr VopěnkaThe lattice of indiscernibility equivalencesCommentationes Mathematicae Universitatis Carolinae, Vol. 20 (1979), No. 4, 631--638Persistent URL: http://dml.cz/dmlcz/105957Terms of use:© Charles University in Prague, Faculty of Mathematics and Physics, 1979Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document mustcontain these Terms of use.This paper has been digitized, optimized for electronic delivery andstamped with digital signature within the project DML-CZ: The Czech DigitalMathematics Library http://project.dml.czCOMMENTATIONES MATHEMATICAE UNIVERSITATIS CAROLINAE 20, 4 (1979) THE LATTICE OF INDISCERNIBILITY EQUIVALENCES Petr VOPENKA Abstract: In this paper we prove that the class of all indiscernibility equivalences creates a lattice in alternati-ve set theory* Every subclass of this lattice has supremum; every countable subclass of it has infimum. Descending sequen-ces ordered by type XL have no infimum* Key words: Alternative set theory, equivalences of in-discernibility . Classification: Primary 02KL0, 02K99 Secondary 02H20, 06A20 Indiscernibility equivalences (see LV1) play a very im-portant role when we are building mathematics in alternative set theory* The class of all indiscernibility equivalences is ordered by inclusion* In this article it is shown that this class creates a lattice and some properties of this lattice are described* The whole paper can be considered as immediate continua-tion of the book LV3. When referring to that book we shall cite only the section and the chapter in question* In the § 1 Ch. III. the notion of an indiscernibility equivalence is defined* The fact that the intersection of countably many indiscernibility equivalences is an indiscer-- 631 -nibility equivalence is also proved in the mentioned chapter* Every indiscernibility equivalence is a iff -class and that is the reason why the class of all indiscernibility equi-valences is coda ble. In this paper we shall deal with various indiscernibi-lity equivalences. Thus we shall denote e.g. the figure of the class X in the equivalences »', =f etc. by Fig*(X)f Fig+(X)t etc. respectively. We also use the notation Mon*(x)» Mon+(x) etc. We use the notation Fig**(X)f Mon+(x) also in the case if the equivalence - is not an indiscernibility equivalence. We use this notation in the following natural sense Moxf(x)*\y\y*x\9 Fig+(X)-4y;(a xeX)(ytx)i. We say that a codable class A is a closed base for the indiscernibility equivalence s if 1tfl has the following pro-perties : (1) M is at most countable. (2) Every X e M is closed in =. (3) For every X,Y e M , XnY e M * (4) If X is closed in -- then X=-fUY;X£ Ye fl&J. Theorem. If * is an indiscernibility equivalence then ,4. there is a Wl which is a closed base for =. Proof: In the § 2 Ch. Ill it is proved that there is a class Z which is at most countable and such that for every closed figure X the following holds: X-nCFig"**(u);XsFig+"(u)8< &ueZf. We put $l0 = 4Fig*(u);us Z]. Now it is sufficient to put 131 to be the least class such that ^tlQ £ ^t and such that for Y,X & ffll the formula XnY & OfH holds. 632 Theorem. Let - be an indiscernibility equivalence. Let Ktl be a closed base for -• For every disjoint closed fi-gures Xlt X2 (in = ) there are X.-,Y2 e W such that the for-mulas X-jSY-i, X 2SY 2 > X-.nY2=0, X2nX1^09 YjU Y2=V hold. Proof; X-,, X2 are disjoint ar-classes. Thus there is & set-theoretically definable cla83 Z 3uch that X,£Z, Z n X g ^ hold. Evidently Fig*(Z)nX2=-0. Let us put Iftftt; Fig*(Z) s £ 1 6 ^ 5 , We have Fig*(Z)= O Wt Q. If for every Y & ^ t Q the formula YnXg-i-^ holds, then f. 29t nX2#:0 - a contra-diction. Thus there is an Y-̂  * 1tl0 such that Fig*(Z)£ Y^ & %cY-Lr.X2«0 holds. The proof of the existence of the class Y2 e W having the properties Fig*(V-Z)£.Y2f Y2 A X - ^ 0 is a-mlogous. Evidently X XE Z£Fig*(Z)^Y-L and X 2£ X 2 hold. We have V=Z u (V-Z) £ Fig*(Z) u Fig*(V-Z) £ Y ^ Y2. Remember that if for any x, y the formula x^y ==> x-y holds then the equivalence = is called finer than =• Theorem. Let * be an indiscernibility equivalence. Let -* be finer than =. If for any u the class Fig*(u) is a en*-class then s is an indiscernibility equivalence. Proof; - is compact because it is coarser than -?. Let W be a closed base for -. Let X-^Xg c Wtl be such that X, u uX2=V. We put R=(Fig+(X1)>2u(Fig4"(X2))2. Aa Xx s Ml the-re ie a u euch that X-^Fig^u). Thu3 Fig+(X1)=Fig*t(u) and Fig4(X1)> Fig+(X2) are tf-elaases. It follows that R is a jr*-elas3. As there are at most count ably many pairs X-, ,X2 e & Ifitl 7 there are at most countably many relations R having the mentioned properties. We prove that the equivalence -is the intersection of all such relations R and thus we pro-- 633 -ve that $ i s a -W-equivalence. Let x£ y . I f xeX, then y c ePig^tX^), thus <x,y>e R. Por xeX-^ the proof i s analo-gous. To prove the opposite implication, we suppose Mon+(x)AMon+(y)*0. Using the previous theorem we obtain X1 , X2 6 W having the following properties X^uXg^V, Mon4(x)£X l f Mon*(y)£X2, X^n Mon+(y)*0, X2nMon4'(x)*0. Thus we have x^Pig+fX^), y ^ F i g + U ^ and <xfy> £ (Fig+tt-^A u t P i g ^ ) ) 2 . Theorem. l e t = be an indiscernibil ity equivalence. Let 33t be a closed base for * . If - i s finer than s and i f for every x the class Mon*(x) i s a JT-class then the following holds . x * y*s lVXeM)(x€P ig+(X)sy€P ig + (X) ) . Proof: The implication -=-> is evident. To prove an opposite implication let us suppose Mon+(x)nMon4"(y)a-0. The classes Mon"*"(x)f Mon+(y) are figures in the equivalence -and are also iPt-classes. Thus there is X e W, such that Mon+(x)£Xf XnMon+(y)=0. It follows that xeFig+(X) and y4Pig+(X). Theorem. Let \ «<# ; & e £1} be a class of indiscerni-bility equivalences.If there is an indiscernibility equiva-lence • such that * is finer than «^ for every 06 then there is r « - i such that Ci<*<ci<x> & 1 H * D< ^ . oc e ?n &*. In this case O •£ s^ ; cc e II? is an indiscernibility equiva-lence. Proof: We can suppose that co<. (I -*-=̂  »^ is finer than «£ • (Remember that the intersection of at most count-ably many indiscernibility equivalences is an indiscernibi-- 634 -l i t y equivalence.) Let tat be a cloaed baae for • • Let ma choose X e M , then -{Fig^CX); cc e XL 5 ia a descending ee-quence of closed figures in the equivalence * and thus using § 2 Ch. I l l , we obtain a ¥ e XL such that for -y-c o& e -Q- the formula Figo6(X)*Fig^. (X) holds. Because HfV i s at most coun-table! we can suppose that there i s a f e U such that for every X* 37t and for every y<< oc € H the formula Fig^X)* 'Fig^, (X) holds. Using the previous theorem for y ^ o u c H we obtain that MonoC(x)« O-C FigoC(X);X c ^ t &'xeFig^OCft* • O t F i g r ( X ) ; X € ftt ft xsFig r (XH*Mon a . (x ) . Thus ** , « r are identical . We have proved H -C a^ ; oc e fl\ » « ^ . Theorem. For a class 4 "^ f <*• 6 XX 3; of indiscernibi l i ty equivalences there i s an indiscernibi l i ty equivalence « ha-ving the following properties. (a) For every cc e XI we have *£ i s finer than *. (b) If £ i s an indiscernibi l i ty equivalence such that for every cc c.0. the equivalence «<- i s finer than * then *" i s finer than *. Proof; Let Hfl be the class of a l l indiscernibi l i ty e-quivalencea = such that for every cc 9 the equivalence moc i s finer than - . C W i s codable as i t i s a subclass of the c lass of a l l m-classes and V & 131 .) DTK i s an ind i s -cernibi l i ty equivalence. (To prove i t we use the previous theorem.) HWt has evidently the properties required in the theorem. In the l a s t part of the paper we suppose the reader to be familiar with § 1 Ch.kV. - 635 -Theorem. Let F be an automorphism. If R is an indis-cernibility equivalence then F * R is an indiscernibility equivalence, too. Proof: F " R is evidently an equivalence. The image of a 3f-class is a sf-class because the image of a set-theore-tically definable class is a set-theoretically definable clas9. Now it suffices to prove the compactness of F w R. Let u be an infinite set, put v^F" "u. v is obviously an in-finite set. Using the compactness of R we obtain x,y£ v such that x#y and<x,y>eR. Thus F ( x ) , F(y)eu, F(x)4=F(y), < F(x), F(x) > • F( < x,y > ) € F " R. Let us put x ̂  y iff for any set-formula g?(z) of the language FL̂ . the formula ylxjsyly) holds. The set y is said to be definable using parameters from the class X iff there is a set-formula cp(z) of the language FLj, such that the formulas (3 tz)g?(z) and 9>(y) hold. We use the notation Defx for the class of all sets definable using parameters from the class X. If a class X is countable then for the class Defx and for the equivalence § the analogues of assertions concerning the class Def anft the equivalence s hold. Especially J is a totally disconnected indiscernibility equivalence. Theorem: (a) XeDefx. (b) Y£ DefxsDefy £ Def x . (c) For any set-formula of the language FL-^ there 1 X i s an equivalent set-formula of the language FL-̂ . (d) The equivalence ^ i s finer than ^ i f f Defy £ Defx. - 636 -(e) Let F be an automorphism. The image of the equi-valence j> i s the equivalence j . jf x and F * Def^Def^w^ holds. Theorem. For any class X which i s at most countable there i s a set u such that the formula X£Defjuj holds. Proof: Let a be a function such that X»G "FN. Let g be a prolongation of G. We prove XSDef - , . Let x c X , l e t ncFN be such that x=*G(n). Thus x«g(n). We have g€.Def, , and n e Def ̂  • Thus x 6 * Def.-«-. * Theorem. For any indiscernibi l i ty equivalence * there i s a set u such that jgi i s finer than *. Proof. Let {-^jneFNj be a generating sequence for ^. As ^ are set-theoretical ly definable c lasses , there i s a c lass C which i s at most countable and a sequence *f$?n(x,y); n6FN$ of set-formulas of the language FI^ such that l^35 a s{<x,y> ; cp n (x ,y ) | . Let «{vn|ne FNl be a sequence such that vn i s a maximal ^j-net on V. For every n the set v i s f i n i -t e . Using the previous theorem we obtain a set u such that CuiT^neFJflcDef # V e prove that .g, i s finer than * . O 4-Let x g y and - i x * y hold. In this case there i s an n such that <x f y>^ 1^. Thus there i s -5£vn + 1 such that <x ,z>e lL + 1 i <y ,z> #)%+-,. Thus y n + 1 ( x , z ) and " i y n + 1 ( y , « ) . But the formula 9 ^ + T / X - Z ) i s equivalent with a set formula tjr(x) of the language FL^-.. Thus ~i x^g>y - a contradiction. The following theorem i s a special case of the previous one. Theorem. For any indiscernibil i ty equivalence * there i s an indiscernibi l i ty equivalence - having the following - 637 -properties. (a) £ is finer than =* . (b) - is totally disconnected. (c) Every clopen monad in the equivalence - is a sing-leton. R e f e r e n c e tV] P. VOPŽNKA: Mathematics in the Alternative Set Theory, Teubner-Texte zur Mathematik, Leipzig 1979. Matematický ústav Universita Karlova Sokolovská 83, 18600 Praha 8 Československo (Oblátům 19.4. 1979) - 638 -

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The lattice of indiscernibility equivalences

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The lattice of indiscernibility equivalences

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Institute of Mathematics AS CR, v. v. i.

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