General theory of $ abla$-models
Petr Vopěnka · Czech digital mathematics library · 1967
Syntactic models of the set theory in the set theory constructed in [l]-[63 (so called V -models) depend on several parameters* By a specification of the parameters, models in which the continuum hypothesis (or other statements) does not hold may be constructed* It was proved in C7]-[8] that the number of parameters may be limited to two, namely to a complete Boolean algebra and an ultrafilter on it* With respect to this fact it seems to be reasonable to present the whole theory of V -models once more in a quite simpler form* This new form enables to find a general method of finding a set-theoretical formula c/C&) with one free variable to every closed normal set theoretical formula * (called the translation by the variables P,R ) is defined in the following recursive way: 1) if cf is atomic, say, .xe nf, then y p ^R is € R, 2) if Cf is cf^ it c/^ f -i * is Cf ?> K & cf ?>* t -r cf ?>* respectively, 3) if cf is (3X)Y, then Cf ?> R is (3x )Cxe P* y, P > R ).If V is a formula equivalpnt to an elementary formula Cf in the set theory then ijr ?> R is defined as Cf ?> * .We shall write cf ?instead of e * Remark.The formula X » ^ is understood aa an abbreviation of n(3z)(zety£ ~i(ze x )) $ R ).R is said to be a model-relation on P (shortly, Wc CP P R) ) iff 1) (VxeP)C3y~& P)C$ PjR Cx) s y pR (ty)) 2) (V*sPK3^€ P)Cx*Q PfR Cry,)) 3) (V«x,/y.€P)fJ^€ PKGs-rgr CXiy))**) tor 4~ 1f», 8 ( #£ are the Godel's operations) 4) (VX€ P)CCX + O^Ca^^*** * n^.* 0» P > R ) -146 -5) fC3x)** be variables aenoting elements of P. Define C&*OOs,,X€ P.v.X*P&X~% 9 t(X)8 e R,V, X* p&y*p&