consistency

the aggregate of rheological properties of a viscous liquid or of a viscoplastic or viscoelastic body. The concept of consistency does not always have a clearly defined meaning in physics.

Unlike other rheological concepts with a similar connotation (for example, viscosity, fluidity, and plasticity), “consistency” as a technological term denotes the mobility (thickness) of liquid (”semiliquid”) and solid (”semisolid”) products and materials. It is usually used in describing systems whose flow (or deformation) characteristics vary with changes in applied stress.

Consistency is sometimes estimated qualitatively, by comparison of a given system with that of commonly known products, such as “consistency of honey” or “consistency of butter.” Special instruments called consistometers, which express consistency in arbitrary units or in indexes of viscosity and strength, are more frequently used.

a property of a deductive theory (or system of axioms by means of which the theory is specified) such that it is impossible to derive from it a contradiction, that is, any two sentences A and ┐A, each of which is the negation of the other. Consistency for a broad class of formal theories, including the axiom A & ┐A 3 B (“a contradiction implies any assertion”), is equivalent to the existence in the given theory of at least one unprovable sentence.

Consistency, which is necessary in order that a system be considered a description of an “intuitive situation,” does not by any means guarantee the existence of such a situation. However, abstract models can be indicated in each case for any consistent system of axioms. Consistency therefore serves for representatives of “classical” trends in the foundations of mathematics and logic (and even more so for adherents of model theory) if not as the foundation for the “existence” of sets of abstract objects described by the axioms, then at least as a sufficient reason for an intuitive examination and study of such objects. Insofar as a “situation” that can be described by a theory lies outside the theory itself, the aforementioned concept of consistency, which may be called “internal” (in other words, syntactic, or logical) consistency, is closely related to what is called “external” (semantic) consistency. External consistency consists in the unprovability in a given theory of any sentence contradicting (in the ordinary intuitive sense) the facts of the “reality” described by the theory. In spite of this relation, syntactic and semantic consistency are equivalent only for such “poor” logical theories as, for example, the propositional calculus. However, in general, internal consistency is stronger than external consistency. The role of the “reality” reflected by a given concrete theory can also be played by some other deductive theory, so that the external consistency of the original theory can be understood as relative consistency, and the proof of this consistency is the indication of a system of corresponding semantic rules for translating concepts, expressions, and assertions from the second theory into the first, yielding an interpretation (model) of the original theory.

In classical mathematics, set theory serves in the final analysis as a source for the construction of models for such proofs. However, the discovery of paradoxes (antinomies) in set theory has led to the necessity of searching for new, in a sense, “absolute,” methods of proving consistency that are fundamentally different from the method of interpretations. The need for new methods of proving consistency also arises from the noncoincidence of the concepts of internal and external consistency.

One may choose an intermediate method, requiring an absolute proof of consistency only for the axiomatic theory of sets (to which it would be possible to reduce the consistency problem of specific mathematical theories by using purely model-theoretic means) or even for such a relatively simple part of it as the formalized arithmetic of natural numbers, since a set-theoretic universe of discourse of the principal divisions of classical mathematics can be constructed by means of this formalized arithmetic. Such a path was taken by D. Hubert, who proposed a broad program, in the carrying out of which the theories for which a foundation was to be provided would first undergo formalization and the resulting formal systems (calculi) would be checked for syntactic consistency by finitary, that is, intuitive, means, but not using questionable set-theoretic abstractions.

Such absolute consistency proofs constituted the fundamental content of the metamathematics (proof theory) developed by Hilbert’s school. But by 1931, K. Godel had demonstrated the fundamental unfeasability of Hilbert’s program and thereby the limitations of the axiomatic method, within the framework of which the requirements of consistency and completeness, for sufficiently rich formal theories, are inconsistent. The requirement of completeness loses its meaning in the case of intuitive, or “nonformal,” deductive (including mathematical) theories, so that their consistency remains, as before, the most important necessary criterion for their having meaning and being applicable in practice.