Number Theory

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

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By Dr. Sue Toledo (auth.)

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Sample text

C, constructively: is an e-number, and b ~e implies a6eb ©~, a~b~, b a (b) ~<~'~e, n a #b~l~. b. 1 ~ a ~J e We will often associate tableaux and also with as follows. tableau. X Let If X is of rank If for each ~n' X n ordinals the tableaux there Let X is a then the rank of X tableau with origin branch = a e = a (~) = E. n with formulas themselves. be an occurrence is an end point ~ ~. a ~ ) ~ = a ~#c implies We will usually of a formula of a branch, and in finite-branched in a finite-branched ~ is any ordinal, be a point w i t h successors ~n < ~ is ~ ~.

If X is an end point of ~ X ~ in and assume the ~ if there are the proposition is true by assumption. If X noting that ' and is not an end point, we consider individual cases, X is above an end point that has a different rank in ~f" iff at least one of its successors is either such an end point or lies above one. If X is not above such an end point (it cannot be equal to one), neither are any of its successors. Since by the induction hypothesis all the successors of the same ranks in ~ since the rank of X ' and ~", X then have in the two trees, is computed by the same rule from the ranks of its successors in the two trees.

Let X be a point of proposition true for all points below any. If X is an end point of ~ X ~ in and assume the ~ if there are the proposition is true by assumption. If X noting that ' and is not an end point, we consider individual cases, X is above an end point that has a different rank in ~f" iff at least one of its successors is either such an end point or lies above one. If X is not above such an end point (it cannot be equal to one), neither are any of its successors. Since by the induction hypothesis all the successors of the same ranks in ~ since the rank of X ' and ~", X then have in the two trees, is computed by the same rule from the ranks of its successors in the two trees.

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