undecidable
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undecidable (not comparable)
 (mathematics, computing theory) Incapable of being algorithmically decided in finite time. For example, a set of strings is undecidable if it is impossible to program a computer (even one with infinite memory) to determine whether or not specified strings are included.

1982, Wolfgang Bibel, Automated Theorem Proving, Braunschweig: Friedr. Vieweg & Sohn, ISBN 3528085207, page 83:
 The firstorder procedure SP differs from the proposi
tional procedure CP°_{1} in an essential feature. Namely, CP°_{1}
always terminates while SP may run forever as we have seen with
the example immediately after (3.7). This is not a specific
defect of SP. Rather it is known that firstorder logic is an
undecidable theory while propositional logic is a decidable
theory. This means that for the latter there are decision pro
cedures which for any formula decide whether it is valid or
not — and CP°_{1} in fact is such a decision procedure — while
for the former such decision procedures do not exist in princi
ple. Thus SP, according to these results for which the reader
is referred to any logic texts such as [End], [DrG] or [Lew],
is of the kind which we may expect, it is a semidecision
procedure which confirms if a formula is valid but may run
forever for invalid formulas. Therefore, termination by running
out of time or space after any finite number of steps will
leave the question for the validity of a formula unsettled. [...]
 The firstorder procedure SP differs from the proposi

 (mathematics) (of a WFF) logically independent from the axioms 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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incapable of being algorithmically decided
