Relevant First-Order Logic LP# and Curry’s Paradox Resolution

Abstract

In 1942 Haskell B. Curry presented what is now called Curry's paradox which can be found in a logic independently of its stand on negation. In recent years there has been a revitalised interest in non-classical solutions to the semantic paradoxes. In this article the non-classical resolution of Curry's Paradox and Shaw-Kwei's paradox without rejection any contraction postulate is proposed. In additional relevant paraconsistent logic # , 1 , in fact, provide an effective way of circumventing triviality of da Costa's paraconsistent Set Theories

The Solution of the Invariant Subspace Problem in Complex Hilbert Space: External non-Archimedean eld *Rc by Cauchy Completion of the internal Non-Archimedean eld *R (Part I)

“…BN ST + has nontrivial consequences for standard set theory; for example, it implies existence of inner models with measurable cardinals.It has been proved in [21]- [24] that any set theory wich implies existence of inner models with measurable cardinals is inconsistent. However hyper Infinitary first-order logic 2 L # ∞ # with restricted rules of conclusions obviously can save BN ST + from a triviality.Note that logic with restricted MP originally has been proposed in [25].…”

Set Theory INC# Based on Intuitionistic Logic with Restricted Modus Ponens Rule (Part. I)

“…It can then be argued that NC leads directly, not merely to an isolated contradiction, but to triviality. For the argument that this is so,see Curry's paradox [7].…”

“…Curry's paradox violated N C since any Φ statement is provable.Therefore, in a consistent set theory, the set {x | x ∈ x → Φ} does not exist for false Φ such that 0 = 1,etc.Some proposals for set theory have attempted to deal with Curry's paradox not by restricting the rule of comprehension, but by restricting the deduction rules of canonical logic [7]. The existence of proofs like the one above…”

Internal Set Theory IST# Based on Hyper Infinitary Logic with Restricted Modus Ponens Rule: Nonconservative Extension of the Model Theoretical NSA

“…In this paper we look for nonconservative extensions of ZF C in which one can prove statements related to number theory and analysis. In this paper we deal with set theory NC # ∞ # based on hyper infinitary logic 2 L # ∞ # with restricted modus ponens rule [4]- [7].Non trivial applications of the set theory NC # ∞ # to transcendental number theory and functional analysis has been recently obtained in my papers [7]- [10].However all results obtained in [7]- [10] based on a small part of the set theory NC # ∞ # and in fact are obtained using an nonconservative extension of the canonical internal set theory IST [11]- [13]. The main goal of this paper is to present an nonconservative extension IST # of the canonical internal set theory IST.Nonconservative extension of the model theoretical nonstandard analysis also is considered.…”