absoluteness$278$ - translation to ελληνικό
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absoluteness$278$ - translation to ελληνικό

IN MATHEMATICAL LOGIC, PROPERTY OF FORMULA THAT HAS THE SAME TRUTH VALUE IN EACH OF SOME CLASS OF STRUCTURES
Shoenfield's absoluteness theorem; Absoluteness (mathematical logic); Absolute Formula; Shoenfield absoluteness theorem; Shoenfield absoluteness

absoluteness      
n. απόλυτο

Ορισμός

Absolute
·adj Authoritative; peremptory.
II. Absolute ·adj Pure; unmixed; as, absolute alcohol.
III. Absolute ·adj Positive; clear; certain; not doubtful.
IV. Absolute ·adj Capable of being thought or conceived by itself alone; unconditioned; non-relative.
V. Absolute ·adj Loosed from, or unconnected by, dependence on any other being; self-existent; self-sufficing.
VI. Absolute ·adj Complete in itself; perfect; consummate; faultless; as, absolute perfection; absolute beauty.
VII. Absolute ·noun In a plane, the two imaginary circular points at infinity; in space of three dimensions, the imaginary circle at infinity.
VIII. Absolute ·adj Not immediately dependent on the other parts of the sentence in government; as, the case absolute. ·see Ablative absolute, under Ablative.
IX. Absolute ·adj Viewed apart from modifying influences or without comparison with other objects; actual; real;
- opposed to relative and comparative; as, absolute motion; absolute time or space.
X. Absolute ·adj Loosed from any limitation or condition; uncontrolled; unrestricted; unconditional; as, absolute authority, monarchy, sovereignty, an absolute promise or command; absolute power; an absolute monarch.

Βικιπαίδεια

Absoluteness

In mathematical logic, a formula is said to be absolute to some class of structures (also called models), if it has the same truth value in each of the members of that class. One can also speak of absoluteness of a formula between two structures, if it is absolute to some class which contains both of them.. Theorems about absoluteness typically establish relationships between the absoluteness of formulas and their syntactic form.

There are two weaker forms of partial absoluteness. If the truth of a formula in each substructure N of a structure M follows from its truth in M, the formula is downward absolute. If the truth of a formula in a structure N implies its truth in each structure M extending N, the formula is upward absolute.

Issues of absoluteness are particularly important in set theory and model theory, fields where multiple structures are considered simultaneously. In model theory, several basic results and definitions are motivated by absoluteness. In set theory, the issue of which properties of sets are absolute is well studied. The Shoenfield absoluteness theorem, due to Joseph Shoenfield (1961), establishes the absoluteness of a large class of formulas between a model of set theory and its constructible universe, with important methodological consequences. The absoluteness of large cardinal axioms is also studied, with positive and negative results known.