Disjunction and existence under implication in elementary intuitionistic formalisms

1962 ◽  
Vol 27 (1) ◽  
pp. 11-18 ◽  
Author(s):  
S. C. Kleene
Keyword(s):  
Number Theory ◽  
Predicate Calculus ◽  
Elementary Number Theory ◽  
Formal Systems ◽  
Elementary Number

Let Pp, Pd, and N be the intuitionistic formal systems of prepositional calculus, predicate calculus, and elementary number theory, respectively.1 Consider the following six propositions.8(1) ├A V B only if ├A or ├B.(2) ├∋xA(x) only if ├Ã(t) for some formula Ã(x) congruent to A(x) and some term t free for x in Ã(x).

Concerning formulas of the types A→B ν C,A →(Ex)B(x) in intuitionistic formal systems

1960 ◽  
Vol 25 (1) ◽  
pp. 27-32 ◽  
Author(s):  
Ronald Harrop
Keyword(s):  
Number Theory ◽  
Modus Ponens ◽  
Main Step ◽  
Closed Formula ◽  
Elementary Number Theory ◽  
Formal Systems ◽  
Elementary Number

In a previous paper [1] it was proved, among other results, that a closed disjunction of intuitionistic elementary number theory N can be proved if and only if at least one of its disjunctands is provable and that a closed formula of the type (Ex)B(x) is provable in N if and only if B(n) is provable for some numeral n. The method of proof was to show that, as far as closed formulas are concerned, N is equivalent to a calculus N1 for which the result is immediate. The main step in the proof consisted in showing that the set of provable formulas of N1 is closed under modus ponens. This was done by obtaining a subset of the set which is closed under modus ponens and contains all members of the original set, with which it is therefore identical.

Undecidable sentences generated by semantic paradoxes

1955 ◽  
Vol 20 (1) ◽  
pp. 31-43 ◽  
Author(s):  
Hao Wang
Keyword(s):  
Number Theory ◽  
Set Theory ◽  
Predicate Calculus ◽  
Simple Extension ◽  
Elementary Number Theory ◽  
Semantic Paradoxes ◽  
Elementary Number ◽  
Diagonal Argument ◽  
Strong System

In applying the method of arithmetization to a proof of the completeness of the predicate calculus, Bernays has obtained a result which, when applied to set theories formulated in the predicate calculus, may be stated thus.1.1. By adding an arithmetic sentence Con(S) (expressing the consistency of a set theory S) as a new axiom to the elementary number theory Zμ (HB II, p. 293), we can prove in the resulting system arithmetic translations of all theorems of S.It then follows that things definable or expressible in S have images in a simple extension of Zμ, if S is consistent. Since S can be a “strong” system, this fact has interesting consequences. Some of these are discussed by me and some are discussed by Kreisel. Kreisel finds an undecidable sentence of set theory by combining 1.1 and the Cantor diagonal argument. I shall prove below, using similar methods, a few further results, concerned with the notions of truth and designation. The method of numbering sets which I use (see 3.1 below) is different from Kreisel's. While the method used here is formally more elegant, Kreisel's method is much more efficient if we wish actually to calculate the numerical values.

p-Adic Analysis and Elementary Number Theory

1937 ◽  
Vol 38 (2) ◽  
pp. 451 ◽  
Author(s):  
Max Zorn
Keyword(s):  
Number Theory ◽  
Elementary Number Theory ◽  
Elementary Number

scholarly journals On new theorems for elementary number theory.

1967 ◽  
Vol 8 (4) ◽  
pp. 353-356 ◽  
Author(s):  
Albert A. Mullin
Keyword(s):  
Number Theory ◽  
Elementary Number Theory ◽  
Elementary Number

Elementary Number Theory

2016 ◽  
pp. 1-32
Author(s):  
Gary L. Mullen ◽  
James A. Sellers
Keyword(s):  
Number Theory ◽  
Elementary Number Theory ◽  
Elementary Number

Some Elementary Number Theory

2019 ◽  
pp. 239-244
Author(s):  
Richard Evan Schwartz
Keyword(s):  
Number Theory ◽  
Elementary Number Theory ◽  
Result Section ◽  
Structural Result ◽  
Elementary Number

This chapter proves some number-theoretic results about the sequences defined in Chapter 23. It proceeds as follows. Section 24.2 proves Lemma 24.1, a multipart structural result. Section 24.3 takes care of several number-theoretic details left over from Section 23.6 and Section 23.7.

Infinite Orbits Revisited

2019 ◽  
pp. 227-238
Author(s):  
Richard Evan Schwartz
Keyword(s):  
Number Theory ◽  
Elementary Number Theory ◽  
Elementary Number ◽  
Model Section

This is the first of four chapters giving a self-contained proof of Theorem 0.7. Section 23.2 describes a sequence of even rationals {pn/qn} that converges to A. Section 23.3 states the two main technical results, the Box Theorem and the Copy Theorem. Section 23.4 shows how to choose a sequence {cn}. Section 23.5 states three auxiliary results about arc copying in the plaid model. Section 23.6 deduces the Box Theorem from one of these auxiliary lemmas. Section 23.7 deduces the Copy Theorem from the auxiliary lemmas and some elementary number theory. Thus, after this chapter ends, the only remaining task is to prove the auxiliary copy lemmas and prove a few lemmas in elementary number theory.

Some Elementary Number Theory

2019 ◽  
pp. 239-244
Keyword(s):  
Number Theory ◽  
Elementary Number Theory ◽  
Elementary Number
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