Definitions
from The Century Dictionary.
 Incapable of being decided, settled, or solved.
from Wiktionary, Creative Commons Attribution/ShareAlike License.
 adjective mathematics, computing theory Incapable of being
algorithmically decided infinite time. For example, a set ofstrings is undecidable if it is impossible to program acomputer (even one with infinitememory ) to determine whether or not specified strings are included.  adjective mathematics (of a
WFF ) logically independent from theaxioms of a given theory; i.e., that it can never be either proved or disproved (i.e., have its negation proved) on the basis of the axioms of the given theory. (Note: this latter definition is independent of any time bounds or computability issues, i.e., more Platonic.)
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Examples

Whether this difference is only perspectival or whether we deal with an interactive manyworlds "world" even at the global level may remain undecidable, too.
Chaosmic Orders: Nonclassical Physics, Allegory, and the Epistemology of Blake's Minute Particulars. 2001

We should remark that this problem is nontrivial since deciding whether a finite set of equations provides a basis for Boolean algebra is undecidable, that is, it does not permit an algorithmic representation; also, the problem was attacked by Robbins, Huntington, Tarski and many of his students with no success.
Automated Reasoning Portoraro, Frederic 2005

But the "undecidable" stuff needs a lot more care.

But the "undecidable" stuff needs a lot more care.

But the "undecidable" stuff needs a lot more care.

However, undecidable statements which are free from selfreference have been found in various branches of mathematics.
Archive 20090601 Gordon McCabe 2009

Crucially, however, whilst the theory of a model, Th (M), may be undecidable, it is guaranteed to be complete, and it is the models of a theory which purport to represent physical reality.
Archive 20090601 Gordon McCabe 2009

Moreover, whilst Peano arithmetic is axiomatizable, there is a particular model of Peano arithmetic, whose theory is typically referred to as Number theory, which Godel demonstrated to be undecidable and nonaxiomatizable.
Archive 20090601 Gordon McCabe 2009

What matters is that the winner deserves it, not the undecidable question of whether the most deserving wins.

It transpires that the theory of arithmetic (technically, Peano arithmetic) is both incomplete and undecidable.
Archive 20090601 Gordon McCabe 2009
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