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

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No false numerical formula is provable in ~. This will imply: Theorem 2. 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.

We now associate with every formula of a normal derivation in a specific ordinal, to be called the ~-rank of the formula: 56 i) Every end formula of a branch has Y-rank i. 2) The predecessor of a replacement rule conclusion has the same T-rank as the conclusion. 3) YI' Y2 If the conclusion of an rule has the Y-rank the rank of its predecessor is yl~l. 4) rule have Y-ranks, If the conclusions of a B respectively, their predecessor is 5) ¥2 ~,y or ~ n and yl#Y2 . If the conclusions of a cut rule have Y-ranks and YI ¥i and is the difference between the depth of the conclusions and the depth of their predecessor, the predecessor has Y-rank Wn(Yl#Y2) • 6) y and If the conclusion of a complete induction rule has Y-rank n is the difference between the depth of the conclusion and the depth of its predecessor, the predecessor has Y-rank Throughout our discussion of the consistency of graph we will use "rank" to mean #-rank.

No false numerical formula is provable in ~. This will imply: Theorem 2. 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.

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