# Advances in Discrete and Computational Geometry by Chazelle B., Goodman J.E., Pollack R. (eds.) By Chazelle B., Goodman J.E., Pollack R. (eds.)

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8 : The last corollary could have been established directly by adapting the completeness proof of ch. 5 § 10. 9 : For a set of formulas U, by T F U we mean T F X for all X e U. 10 (strong completeness) : Let U be any set of unsigned formulas with no parameters. Then UrI X if and only if in any model (C§, fll, P, �>, for any T e C§, if T F U, T F X. Proof' UrI X if and only if { TY l Ye U} u {FX} is inconsistent. 11 : (cut elimination, Gentzen's Hauptsatz) : If S is a set of signed formulas with no constants and {S, TX} and {S, FX} are in­ consistent, so is {S}.

Similarly TYEr. Hence TX1\ YEr :::. TXEr and TYEr . • Similarly we may show :::. FX E r and FY E r , FX v Y E r :::. TX v Y E r TX E r or T Y E r , :::. FX 1\ Y Er FX E r or FYE r , :::. T ""' X E r FX E r , :::. TX => Y E r FX E r or T Y E r , T (Vx) X (X) E r :::. TX (a ) E r for every a E P (r) , F (3x) X (X) E r :::. FX (a) E r for every a E P (r) . Moreover T (3x) X (X) E r :::. TX (a) E r for some a E P (r) , since r is good with respect to P". M, d1 63 Suppose F"",Xer. Since r is consistent, rT,TX is consistent.

Xis called valid if Xis valid in all models. § 3. Motivation The intuitive interpretation given in ch. 1 § 3 for the propositional case may be extended to this first order situation. CH. 4 § 4 SOME PROPERTIES OF MODELS 47 In one's usual mathematical work, parameters may be introduced as one proceeds, but having introduced a parameter, of course it remains introduced. P is intended to represent. p (r) is the set of all parameters introduced to reach r. ) Since parameters, once introduced, do not disappear, we have QO.

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