Number Theory

Tableau Systems for First Order Number Theory and Certain by Dr. Sue Toledo (auth.) PDF By Dr. Sue Toledo (auth.)

ISBN-10: 3540071490

ISBN-13: 9783540071495

ISBN-10: 3540374426

ISBN-13: 9783540374428

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Additional info for Tableau Systems for First Order Number Theory and Certain Higher Order Theories

Example text

6. Let where b By the induction hypothesis of the required type for S'. ,tn')(b) X(t I, .... tn)(b) is a closed tableau for Proposition S T s = t~ S 3. A~ A' If S not using cut or induction rules. is a set of formulas containing where A and A' are equivalent formulas relative to s = t, has a closed tableau not using the cut and induction rules. Proof. Again it is clear that A and (and thus have the same logical structures the proposition we see that tion c similarly to proposition IT s = t, A, A'} and the tableau 2.

Is not of degree 0, one of the other a proposition, S A ~-~ and case, A' is an for i = 1,2 not using cut and induction rules. Si Thus, sk J \ ~i y-~ the has a closed S~ The or a also satisfies the assumptions of the so that by the induction hypothesis is a closed tableau for ~ ~2 S case is similar. not using the proscribed rules. 46 Corollary. a finite If set ~is S a tableau of formulas tically unsatisfiable, where A and A' T s = t, A, A' , s = t, then ~ each of ~hose branches such that either a) or b) S contains are equivalent, where A and A' can be extended use of cut or induction or c) however closed a formula tableau putting ~, \$a' \$b' \$c' or \$d satisfies condition to a set of closed formulas relative tableau without tableau; to the we will, indicate at the end of a branch branches).

The system ~ is syntactically consistent. For assume you could prove both A and tableaux ~i and ~2" Then you could prove the tableau. FO=O' T'~A FA -~A FIA ! 2 in with 0 = O' with 60 But is a false numerical 0 = 0' formula, so such a proof is impossible. Now let us turn to the proof of Theorem arbitrary provable numerical ticular normal derivation v a t i o n must have rank formula for it. only (2) all the formulas (3) every branch isfying ~,8, of (i) Lemma P and let us be given a parthis deri- (i), from the normal derivation of and cut rules are used; in the tableau contains are numerical; a numerically false atomic formula.