Abstract
In this paper, we develop the system LZF of set theory with the unrestricted comprehension in full linear logic and show that LZF is a conservative extension of ZF− i.e., the Zermelo-Fraenkel set theory without the axiom of regularity. We formulate LZF as a sequent calculus with abstraction terms and prove the partial cut-elimination theorem for it. The cut-elimination result ensures the subterm property for those formulas which contain only terms corresponding to sets in ZF−. This implies that LZF is a conservative extension of ZF− and therefore the former is consistent relative to the latter.
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Shirahata, M. A linear conservative extension of Zermelo-Fraenkel set theory. Stud Logica 56, 361–392 (1996). https://doi.org/10.1007/BF00372772
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DOI: https://doi.org/10.1007/BF00372772