Set Theory Inc# Based on Infinitary Intuitionistic Logic with Restricted Modus Ponens Rule (Part.Ii) Hyper Inductive Definitions [Enter Paper Title]

2021 ◽  
Author(s):  
Jaykov Foukzon
Keyword(s):  
Set Theory ◽  
Intuitionistic Logic ◽  
Modus Ponens ◽  
Inductive Definitions

scholarly journals Set Theory INC# ∞# Based on Infinitary Intuitionistic Logic with Restricted Modus Ponens Rule (Part.II) Hyper Inductive Definitions

2021 ◽  
pp. 90-112
Author(s):  
Jaykov Foukzon
Keyword(s):  
Number Theory ◽  
Set Theory ◽  
Intuitionistic Logic ◽  
Theoretical Language ◽  
Modus Ponens ◽  
Inductive Definitions ◽  
Induction Principle ◽  
Intuitionistic Set Theory ◽  
Intuitionistic Arithmetic

In this paper intuitionistic set theory INC# ∞# in infinitary set theoretical language is considered. External induction principle in nonstandard intuitionistic arithmetic were derived. Non trivial application in number theory is considered.The Goldbach-Euler theorem is obtained without anyreferences to Catalan conjecture.

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

2021 ◽  
pp. 73-88
Author(s):  
Jaykov Foukzon
Keyword(s):  
Set Theory ◽  
Intuitionistic Logic ◽  
Modus Ponens ◽  
Russell’S Paradox ◽  
Russell's Paradox

In this article Russell’s paradox and Cantor’s paradox resolved successfully using intuitionistic logic with restricted modus ponens rule.

scholarly journals Set Theory INC# ∞# Based on Innitary Intuitionistic Logic with Restricted Modus Ponens Rule. Hyper Inductive Denitions. Application in Transcendental Number Theory. Generalized Lindemann-Weierstrass Theorem

2021 ◽  
pp. 70-119
Author(s):  
Jaykov Foukzon
Keyword(s):  
Number Theory ◽  
Set Theory ◽  
Intuitionistic Logic ◽  
Theoretical Language ◽  
Modus Ponens ◽  
Transcendental Number ◽  
Weierstrass Theorem ◽  
Induction Principle ◽  
Intuitionistic Set Theory ◽  
Intuitionistic Arithmetic

In this paper intuitionistic set theory INC#∞# in infinitary set theoretical language is considered. External induction principle in nonstandard intuitionistic arithmetic were derived. Non trivial application in number theory is considered.The Goldbach-Euler theorem is obtained without any references to Catalan conjecture. Main results are: (i) number ee is transcendental; (ii) the both numbers e + π and e − π are irrational.

On undecidable statements in enlarged systems of logic and the concept of truth

1939 ◽  
Vol 4 (3) ◽  
pp. 105-112 ◽  
Author(s):  
Alfred Tarski
Keyword(s):  
Set Theory ◽  
Modus Ponens ◽  
Logical System ◽  
Natural Numbers ◽  
System L ◽  
Concept Of Truth ◽  
Made In

It is my intention in this paper to add somewhat to the observations already made in my earlier publications on the existence of undecidable statements in systems of logic possessing rules of inference of a “non-finitary” (“non-constructive”) character (§§1–4).I also wish to emphasize once more the part played by the concept of truth in relation to problems of this nature (§§5–8).At the end of this paper I shall give a result which was not touched upon in my earlier publications. It seems to be of interest for the reason (among others) that it is an example of a result obtained by a fruitful combination of the method of constructing undecidable statements (due to K. Gödel) with the results obtained in the theory of truth.1. Let us consider a formalized logical system L. Without giving a detailed description of the system we shall assume that it possesses the usual “finitary” (“constructive”) rules of inference, such as the rule of substitution and the rule of detachment (modus ponens), and that among the laws of the system are included all the postulates of the calculus of statements, and finally that the laws of the system suffice for the construction of the arithmetic of natural numbers. Moreover, the system L may be based upon the theory of types and so be the result of some formalization of Principia mathematica. It may also be a system which is independent of any theory of types and resembles Zermelo's set theory.

Some properties of epistemic set theory with collection

1986 ◽  
Vol 51 (3) ◽  
pp. 748-754 ◽  
Author(s):  
Andre Scedrov
Keyword(s):  
Set Theory ◽  
Intuitionistic Logic ◽  
Existential Quantifier ◽  
Modal Interpretation ◽  
Conservative Extension ◽  
Disjunction Property ◽  
First Order ◽  
Slight Extension ◽  
Order Arithmetic ◽  
Epistemic Systems

Myhill [12] extended the ideas of Shapiro [15], and proposed a system of epistemic set theory IST (based on modal S4 logic) in which the meaning of the necessity operator is taken to be the intuitive provability, as formalized in the system itself. In this setting one works in classical logic, and yet it is possible to make distinctions usually associated with intuitionism, e.g. a constructive existential quantifier can be expressed as (∃x) □ …. This was first confirmed when Goodman [7] proved that Shapiro's epistemic first order arithmetic is conservative over intuitionistic first order arithmetic via an extension of Gödel's modal interpretation [6] of intuitionistic logic.Myhill showed that whenever a sentence □A ∨ □B is provable in IST, then A is provable in IST or B is provable in IST (the disjunction property), and that whenever a sentence ∃x.□A(x) is provable in IST, then so is A(t) for some closed term t (the existence property). He adapted the Friedman slash [4] to epistemic systems.Goodman [8] used Epistemic Replacement to formulate a ZF-like strengthening of IST, and proved that it was a conservative extension of ZF and that it had the disjunction and existence properties. It was then shown in [13] that a slight extension of Goodman's system with the Epistemic Foundation (ZFER, cf. §1) suffices to interpret intuitionistic ZF set theory with Replacement (ZFIR, [10]). This is obtained by extending Gödel's modal interpretation [6] of intuitionistic logic. ZFER still had the properties of Goodman's system mentioned above.

scholarly journals Logics with Impossibility as the Negation and Regular Extensions of the Deontic Logic D2

2017 ◽  
Vol 46 (3/4) ◽  
Author(s):  
Krystyna Mruczek-Nasieniewska ◽  
Marek Nasieniewski
Keyword(s):  
Intuitionistic Logic ◽  
Deontic Logic ◽  
Modus Ponens ◽  
Modal Logics ◽  
Point Of View ◽  
Modal Operator ◽  
Modal Interpretation ◽  
Semantical Point ◽  
The Right ◽  
Normal Modal Logics

In [1] J.-Y. Bèziau formulated a logic called Z. Bèziau’s idea was generalized independently in [6] and [7]. A family of logics to which Z belongs is denoted in [7] by K. In particular; it has been shown in [6] and [7] that there is a correspondence between normal modal logics and logics from the class K. Similar; but only partial results has been obtained also for regular logics (see [8] and [9]). In (Došen; [2]) a logic N has been investigated in the language with negation; implication; conjunction and disjunction by axioms of positive intuitionistic logic; the right-to-left part of the second de Morgan law; and the rules of modus ponens and contraposition. From the semantical point of view the negation used by Došen is the modal operator of impossibility. It is known this operator is a characteristic of the modal interpretation of intuitionistic negation (see [3; p. 300]). In the present paper we consider an extension of N denoted by N+. We will prove that every extension of N+ that is closed under the same rules as N+; corresponds to a regular logic being an extension of the regular deontic logic D21 (see [4] and [13]). The proved correspondence allows to obtain from soundnesscompleteness result for any given regular logic containing D2, similar adequacy theorem for the respective extension of the logic N+.

The consistency of classical set theory relative to a set theory with intu1tionistic logic

1973 ◽  
Vol 38 (2) ◽  
pp. 315-319 ◽  
Author(s):  
Harvey Friedman
Keyword(s):  
Set Theory ◽  
Classical Theory ◽  
Intuitionistic Logic ◽  
Classical Logic ◽  
Predicate Calculus ◽  
Basic Tool ◽  
Axiom Scheme ◽  
Power Set ◽  
Image Position

Let ZF be the usual Zermelo-Fraenkel set theory formulated without identity, and with the collection axiom scheme. Let ZF−-extensionality be obtained from ZF by using intuitionistic logic instead of classical logic, and dropping the axiom of extensionality. We give a syntactic transformation of ZF into ZF−-extensionality.Let CPC, HPC respectively be classical, intuitionistic predicate calculus without identity, whose only homological symbol is ∈. We use the ~ ~-translation, a basic tool in the metatheory of intuitionistic systems, which is defined byThe two fundamental lemmas about this ~ ~ -translation we will use areFor proofs, see Kleene [3, Lemma 43a, Theorem 60d].This - would provide the desired syntactic transformation at least for ZF into ZF− with extensionality, if A− were provable in ZF− for each axiom A of ZF. Unfortunately, the ~ ~-translations of extensionality and power set appear not to be provable in ZF−. We therefore form an auxiliary classical theory S which has no extensionality and has a weakened power set axiom, and show in §2 that the ~ ~-translation of each axiom of Sis provable in ZF−-extensionality. §1 is devoted to the translation of ZF in S.

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