D. Prawitz. "Normalizations of Proofs in Set Theory." Atti Degli Incontri di Logica Matematica, Vol.2. Scuola di Specializzazione in Logica Matematica, Dipartimento di Matematica, Universit`a di Siena, 1984.  Home/Search   Document Not in Database   Summary   Related Articles

....The same method can be used to establish a partial cut elimination result for set theory, using a well ordering built in the standard model of set theory. However, the complete cut elimination is not possible even for a system of set theory in classical logic with only the principle of separation [14, 24]. CHAPTER 1. INTRODUCTION 13 Our proof of the cut elimination theorem for linear set theory is, however, very simple. This is precisely because the use of contraction is restricted in linear logic. In fact, we can define a measure of the size of a proof in such a way that it decreases along with ....

.... 2 u 8v(v 2 u v [ fvg 2 u) Regularity) 8u[u 6= 9v(v 2 u v u = Replacement) 8u[8w(w 2 u 8w 1 8w 2 (F (w; w 1 ) F (w; w 2 ) w 1 = w 2 ) 9v8w(w 2 v 9w 1 (w 1 2 u F (w 1 ; w) Our first task is to reformulate ZF in the language of linear logic with set terms fx : Ag [24, 14]. For the sake of simplicity, we will omit the regularity axiom. Hence our target theory is ZF without the regularity, or ZF Gamma . In ZF, the use of an set term is acceptable only when we can show the existence and uniqueness of the set denoted by the term. In the cases of (Empty) Pair) ....

D. Prawitz. "Normalizations of Proofs in Set Theory." Atti Degli Incontri di Logica Matematica, Vol.2. Scuola di Specializzazione in Logica Matematica, Dipartimento di Matematica, Universit`a di Siena, 1984.

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