# Dictionary Definition

unprovable adj : not provable; "it was both
unproved and unprovable"

# User Contributed Dictionary

## English

# Extensive Definition

In mathematical
logic, a
sentence σ is called independent of a given
first-order theory T if T neither proves nor refutes
σ; that is, it is impossible to prove σ from T,
and it is also impossible to prove from T that σ is
false.

Sometimes, σ is said (synonymously) to
be undecidable from T; however, this usage risks confusion with the
distinct notion of the undecidability of a decision
problem.

Many interesting statements in set theory are
independent of
Zermelo-Fraenkel set theory (ZF). It is possible for the
statement "σ is independent from T" to be itself
independent from T. This reflects the fact that statements about
proofs of mathematical statements when represented in mathematics
become themselves mathematical statements.

## Usage note

Some authors say that σ is independent of T if T simply cannot prove σ, and do not necessarily assert by this that T cannot refute σ. These authors will sometimes say "σ is independent of and consistent with T" to indicate that T can neither prove nor refute σ.## Independence results in set theory

The following statements in set theory are known to be independent of ZF, granting that ZF is consistent (see also the list of statements undecidable in ZFC):- The axiom of choice
- The continuum hypothesis and the generalised continuum hypothesis
- The Souslin conjecture
- The existence of a Kurepa tree

The following statements (none of which have been
proved false) cannot be proved in ZFC to be independent of ZFC,
even if the added hypothesis is granted that ZFC is consistent.
However, they cannot be proved in ZFC (granting that ZFC is
consistent), and few working set theorists expect to find a
refutation of them in ZFC.

- The existence of strongly inaccessible cardinals
- The existence of large cardinals

The following statements are inconsistent with
the axiom of choice, and therefore with ZFC. However they are
probably independent of ZF, in a corresponding sense to the above:
They cannot be proved in ZF, and few working set theorists expect
to find a refutation in ZF. However ZF cannot prove that they are
independent of ZF, even with the added hypothesis that ZF is
consistent.

unprovable in French: Décidabilité et
indécidabilité

unprovable in Italian: Decidibilità (logica
matematica)

unprovable in Russian: Независимость системы
аксиом

unprovable in Swedish: Oavgörbar

# Synonyms, Antonyms and Related Words

agnostic, ambiguous, capricious, chancy, changeable, controvertible, dicey, doubting, equivocal, erratic, fickle, hesitant, hesitating, incalculable, indecisive, indemonstrable, irresolute, polysemous, skeptical, touch-and-go,
unaccountable,
unattestable,
uncertain, unconfirmable, unconvinced, undemonstrable, undemonstratable,
undivinable,
unforeseeable,
unpersuaded,
unpredictable,
unsubstantiatable,
unsupportable,
unsure, unsustainable, unverifiable, variable, wavering, whimsical