scholarly journals A finite axiom scheme for approach frames

2010 ◽  
Vol 17 (5) ◽  
pp. 899-909 ◽  
Author(s):  
Christophe Van Olmen ◽  
Stijn Verwulgen
Keyword(s):  
Axiom Scheme

scholarly journals A Remark on the Intersection of Tow Logics

1966 ◽  
Vol 26 ◽  
pp. 167-171 ◽  
Author(s):  
Satoshi Miura
Keyword(s):  
Intuitionistic Logic ◽  
General Feature ◽  
Minimal Logic ◽  
Axiom Scheme

The intuitionistic logic LJ and Curry’s LD (cf. [1], [2]) are logics stronger than Johansson’s minimal logic LM (cf. [3]) by the axiom schemes ⋏→x and y ∨ (y→⋏), respectively. However, LM can not be taken literally as the intersection of these two logics LJ and LD, which is stronger than LM by the axiom scheme (⋏ → x) VyV (y→⋏). In pointing out this situation, Prof. K. Ono suggested me to investigate the general feature of the intersection of any pair of logics. In this paper, I will show that the same situation occurs in general. I wish to express my thanks to Prof. K. Ono for his kind guidance.

scholarly journals A Theory of Mathematical Objects as a Prototype of Set Theory

1962 ◽  
Vol 20 ◽  
pp. 105-168 ◽  
Author(s):  
Katuzi Ono
Keyword(s):  
Set Theory ◽  
Axiom Of Choice ◽  
Simple System ◽  
Object Theory ◽  
Mathematical Objects ◽  
Axiom Scheme ◽  
The Axiom Of Choice ◽  
Primitive Notion

The theory of mathematical objects, developed in this work, is a trial system intended to be a prototype of set theory. It concerns, with respect to the only one primitive notion “proto-membership”, with a field of mathematical objects which we shall hereafter simply call objects, it is a very simple system, because it assumes only one axiom scheme which is formally similar to the aussonderung axiom of set theory. We shall show that in our object theory we can construct a theory of sets which is stronger than the Zermelo set-theory [1] without the axiom of choice.

On existence proofs of Hanf numbers

1974 ◽  
Vol 39 (2) ◽  
pp. 318-324 ◽  
Author(s):  
Harvey Friedman
Keyword(s):  
Existence Proofs ◽  
Axiom Scheme ◽  
Hanf Number ◽  
Full Separation ◽  
Open Question

This paper refines some results of Barwise [1] as well as answering the open question posed at the end of [1] about the Hanf number of positively. We conclude by showing that the existence of a Hanf bound for cannot be proved in the natural formally intuitionistic set theories with bounded predicates decidable of [3], [4] and [5].All notation not explained below is taken from [1]. In the Appendix, we give the axioms of ZF0, ZF1, and T in full. We remark that an important point about the axiom of foundation was not emphasized in [1]. This axiom was intended to be the axiom scheme (∀x)((∀y ∈ x)(A(y)) → A(x)) → (∀x)(A(x)), where y does not occur in A = A(x), instead of the more customary (∀x)(∀y)(y ∈ x → (∃z ∈ x)(∀w ∈ z) (w ∉ x)). This is of no consequence in the presence of full separation, but is vital when considering ZF0 and the T below, for with the customary form of foundation, these cannot even prove the existence of Rω+ω.In [1], a proof of the following is sketched.

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.

scholarly journals On a Theory Objects Based on a Single Axiom Scheme

1966 ◽  
Vol 26 ◽  
pp. 13-30
Author(s):  
Katuzi Ono
Keyword(s):  
Set Theory ◽  
Mathematical Theory ◽  
Axiom Of Choice ◽  
Fundamental Theory ◽  
Axiom Scheme ◽  
Single Axiom ◽  
The Axiom Of Choice ◽  
Formal Theories

There are some fundamental mathematical theories, such as the Fraenkel set-theory and the Bernays-Gödel set-theory, in which, I believe, all the actually important formal theories of mathematics can be embedded. Formal theories come into existence by being shown their consistency. As far as this is admitted, not all the axioms of set theory are necessary for a fundamental mathematical theory. The fundierung axiom is proved consistent by v. Neumann, the axiom of extensionality is proved consistent by Gandy, and even the axiom of choice is proved consistent by Göldel. Although it is not evident that a set-theory does not cease from being a fundamental theory of mathematics after abandoning these axioms all at once, the theory must be enough for being a fundamental theory of mathematics without some of them.

The Incompleteness of Peano Arithmetic with Exponentiation

1992 ◽  
Author(s):  
Raymond M. Smullyan
Keyword(s):  
Axiom System ◽  
Peano Arithmetic ◽  
Order Logic ◽  
First Order Logic ◽  
Inference Rules ◽  
First Order ◽  
Axiom Scheme

We shall now turn to a formal axiom system which we call Peano Arithmetic with Exponentiation and which we abbreviate “P.E.”. We take certain correct formulas which we call axioms and provide two inference rules that enable us to prove new correct formulas from correct formulas already proved. The axioms will be infinite in number, but each axiom will be of one of nineteen easily recognizable forms; these forms are called axiom schemes. It will be convenient to classify these nineteen axiom schemes into four groups (cf. discussion that follows the display of the schemes). The axioms of Groups I and II are the so-called logical axioms and constitute a neat formalization of first-order logic with identity due to Kalish and Montague [1965], which is based on an earlier system due to Tarski [1965]. The axioms of Groups III and IV are the so-called arithmetic axioms. In displaying these axiom schemes, F, G and H are any formulas, vi and vj are any variables, and t is any term. For example, the first scheme L1 means that for any formulas F and G, the formula (F ⊃ (G ⊃ F)) is to be taken as an axiom; axiom scheme L4 means that for any variable Vi and any formulas F and G, the formula . . . (∀vi (F ⊃ G) ⊃ (∀vi (F ⊃ ∀vi G) . . . is to be taken as an axiom.

A logic stronger than intuitionism

1971 ◽  
Vol 36 (2) ◽  
pp. 249-261 ◽  
Author(s):  
Sabine Görnemann
Keyword(s):  
Intuitionistic Logic ◽  
Classical Logic ◽  
Predicate Logic ◽  
Predicate Calculus ◽  
Theoretical Methods ◽  
Axiom Scheme ◽  
Simple Notion ◽  
Image Position ◽  
Quasi Ordering

S. A. Kripke has given [6] a very simple notion of model for intuitionistic predicate logic. Kripke's models consist of a quasi-ordering (C, ≤) and a function ψ which assigns to every c ∈ C a model of classical logic such that, if c ≤ c′, ψ(c′) is greater or equal to ψ(c). Grzegorczyk [3] described a class of models which is still simpler: he takes, for every ψ(c), the same universe. Grzegorczyk's semantics is not adequate for intuitionistic logic, since the formulawhere х is not free in α. holds in his models but is not intuitionistically provable. It is a conjecture of D. Klemke that intuitionistic predicate calculus, strengthened by the axiom scheme (D), is correct and complete with respect to Grzegorczyk's semantics. This has been proved independently by D. Klemke [5] by a Henkinlike method and me; another proof has been given by D. Gabbay [1]. Our proof uses lattice-theoretical methods.

Natural deduction based set theories: a new resolution of the old paradoxes

1986 ◽  
Vol 51 (2) ◽  
pp. 393-411 ◽  
Author(s):  
Paul C. Gilmore
Keyword(s):  
Set Theory ◽  
Classical Logic ◽  
Natural Deduction ◽  
Order Logic ◽  
First Order Logic ◽  
Truth Values ◽  
First Order ◽  
Axiom Scheme ◽  
Universe Of Sets ◽  
Comprehension Axiom

AbstractThe comprehension principle of set theory asserts that a set can be formed from the objects satisfying any given property. The principle leads to immediate contradictions if it is formalized as an axiom scheme within classical first order logic. A resolution of the set paradoxes results if the principle is formalized instead as two rules of deduction in a natural deduction presentation of logic. This presentation of the comprehension principle for sets as semantic rules, instead of as a comprehension axiom scheme, can be viewed as an extension of classical logic, in contrast to the assertion of extra-logical axioms expressing truths about a pre-existing or constructed universe of sets. The paradoxes are disarmed in the extended classical semantics because truth values are only assigned to those sentences that can be grounded in atomic sentences.

Solution to the P − W problem

1982 ◽  
Vol 47 (4) ◽  
pp. 869-887 ◽  
Author(s):  
E.P. Martin ◽  
R.K. Meyer
Keyword(s):  
Positive Solution ◽  
Axiom Scheme ◽  
System P ◽  
Two Systems

Anderson and Belnap asked in §8.11 of their treatise Entailment [1] whether a certain pure implicational calculus, which we will call P − W, is minimal in the sense that no two distinct formulas coentail each other in this calculus. We provide a positive solution to this question, variously known as The P − W problem, or Belnap's conjecture.We will be concerned with two systems of pure implication, formulated in a language constructed in the usual way from a set of propositional variables, with a single binary connective →. We use A, B,…, A1, B1, …, as variables ranging over formulas. Formulas are written using the bracketing conventions of Church [3].The first system, which we call S (in honour of its evident incorporation of syllogistic principles of reasoning), has as axioms all instances of (B) B → C →. A → B →. A → C (prefixing),(B) A → B →. B → C →. A → C (suffixing), and rules (BX) from B → C infer A → B →. A → C (rule prefixing),(B’X) from A → B infer B → C →. A → C (rule suffixing),(BXY) from A → B and B → C infer A → C (rule transitivity).The second system, P − W, has in addition to the axioms and rules of S the axiom scheme (I) A → A of identity.We write ⊢SA (⊣SA) to mean that A is (resp. is not) a theorem of S, and similarly for P − W.

scholarly journals The Axiom Scheme of Acyclic Comprehension

2014 ◽  
Vol 55 (1) ◽  
pp. 11-24
Author(s):  
Zuhair Al-Johar ◽  
M. Randall Holmes ◽  
Nathan Bowler
Keyword(s):  
Axiom Scheme
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