Completeness of the functional calculus of first order

Studia Logica ◽  
1955 ◽  
Vol 2 (1) ◽  
pp. 245-250 ◽  
Author(s):  
J. Reichbach
Keyword(s):  
Functional Calculus ◽  
First Order

On Theses of the First-Order Functional Calculus

1961 ◽  
Vol 7 (11-14) ◽  
pp. 175-184
Author(s):  
Juliusz Reichbach
Keyword(s):  
Functional Calculus ◽  
First Order

A derivation of number theory from ancestral theory

1952 ◽  
Vol 17 (3) ◽  
pp. 192-197 ◽  
Author(s):  
John Myhill
Keyword(s):  
Number Theory ◽  
Functional Calculus ◽  
Natural Numbers ◽  
Class Inclusion ◽  
First Order ◽  
Free Variables

Martin has shown that the notions of ancestral and class-inclusion are sufficient to develop the theory of natural numbers in a system containing variables of only one type.The purpose of the present paper is to show that an analogous construction is possible in a system containing, beyond the quantificational level, only the ancestral and the ordered pair.The formulae of our system comprise quantificational schemata and anything which can be obtained therefrom by writing pairs (e.g. (x; y), ((x; y); (x; (y; y))) etc.) for free variables, or by writing ancestral abstracts (e.g. (*xyFxy) etc.) for schematic letters, or both.The ancestral abstract (*xyFxy) means what is usually meant by ; and the formula (*xyFxy)zw answers to Martin's (zw; xy)(Fxy).The system presupposes a non-simple applied functional calculus of the first order, with a rule of substitution for predicate letters; over and above this it has three axioms for the ancestral and two for the ordered pair.

The completeness of the first-order functional calculus

1949 ◽  
Vol 14 (3) ◽  
pp. 159-166 ◽  
Author(s):  
Leon Henkin
Keyword(s):  
Functional Calculus ◽  
Propositional Calculus ◽  
Higher Order ◽  
New Method ◽  
Completeness Proof ◽  
New Approach ◽  
First Order ◽  
Formal Systems ◽  
The Subject ◽  
Image Position

Although several proofs have been published showing the completeness of the propositional calculus (cf. Quine (1)), for the first-order functional calculus only the original completeness proof of Gödel (2) and a variant due to Hilbert and Bernays have appeared. Aside from novelty and the fact that it requires less formal development of the system from the axioms, the new method of proof which is the subject of this paper possesses two advantages. In the first place an important property of formal systems which is associated with completeness can now be generalized to systems containing a non-denumerable infinity of primitive symbols. While this is not of especial interest when formal systems are considered as logics—i.e., as means for analyzing the structure of languages—it leads to interesting applications in the field of abstract algebra. In the second place the proof suggests a new approach to the problem of completeness for functional calculi of higher order. Both of these matters will be taken up in future papers.The system with which we shall deal here will contain as primitive symbolsand certain sets of symbols as follows:(i) propositional symbols (some of which may be classed as variables, others as constants), and among which the symbol “f” above is to be included as a constant;(ii) for each number n = 1, 2, … a set of functional symbols of degree n (which again may be separated into variables and constants); and(iii) individual symbols among which variables must be distinguished from constants. The set of variables must be infinite.

A relative consistency proof

1954 ◽  
Vol 19 (1) ◽  
pp. 21-28 ◽  
Author(s):  
Joseph R. Shoenfield
Keyword(s):  
Set Theory ◽  
Functional Calculus ◽  
Direct Method ◽  
Axiom System ◽  
Similar Process ◽  
Consistency Proof ◽  
First Order ◽  
Definition Of ◽  
A Direct Method ◽  
Exact Definition

LetCbe an axiom system formalized within the first order functional calculus, and letC′ be related toCas the Bernays-Gödel set theory is related to the Zermelo-Fraenkel set theory. (An exact definition ofC′ will be given later.) Ilse Novak [5] and Mostowski [8] have shown that, ifCis consistent, thenC′ is consistent. (The converse is obvious.) Mostowski has also proved the stronger result that any theorem ofC′ which can be formalized inCis a theorem ofC.The proofs of Novak and Mostowski do not provide a direct method for obtaining a contradiction inCfrom a contradiction inC′. We could, of course, obtain such a contradiction by proving the theorems ofCone by one; the above result assures us that we must eventually obtain a contradiction. A similar process is necessary to obtain the proof of a theorem inCfrom its proof inC′. The purpose of this paper is to give a new proof of these theorems which provides a direct method of obtaining the desired contradiction or proof.The advantage of the proof may be stated more specifically by arithmetizing the syntax ofCandC′.

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