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proposition, statement, (assertive) sentence

A narrative communication which by virtue of its meaning is true or false. In a narrower sense, the term assertion in mathematical logic is understood to be a closed formula in a logico-mathematical language which by virtue of the semantics of the language can be classified as true or false.

Thus, in axiomatic set theory, various mathematical statements, for example, the axiom of choice or the continuum hypothesis, are written in the form of formulas that for general semantic reasons can be interpreted as expressing the contents of the statement. It is by no means necessary that there exists a method of recognizing which assertions in the language are true and which false. Moreover, the semantics need not actually be sufficiently developed, or may present fundamental difficulties in the solution of the question of whether certain assertions are true. The unsolvability of assertions within the framework of a theory is explained by the formalization method (for examples, see Axiomatic set theory).

How to Cite This Entry:
Assertion. A.G. Dragalin (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098