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undecidable
μη αποφασίσιμος

reprinted in "the undecidable", m. davis ed., 1965.
reprinted in "the undecidable", m. davis ed., 1965.

even more difficult are the undecidable problems, such as the halting problem.
Ακόμα πιο δύσκολα είναι τα μη αποφασίσιμα προβλήματα, όπως το πρόβλημα τερματισμού.

* the unification problem for formula_85-equivalence is undecidable.
* Το πρόβλημα ενοποίησης για την formula_64-ισοδυναμία είναι μη-αποφασίσιμο.

the list of undecidable problems gives additional examples of problems with no computable solution.
Ο κατάλογος των μη λυμένων προβλημάτων παρέχει επιπλέον παραδείγματα των προβλημάτων χωρίς υπολογίσιμη λύση.

"on formally undecidable propositions of principia mathematica and related systems", tr.
"on formally undecidable propositions of principia mathematica and related systems", tr.

"typability and type checking in the second-order lambda-calculus are equivalent and undecidable.
"typability and type checking in the second-order lambda-calculus are equivalent and undecidable.

reprinted in "the undecidable", m. davis ed., 1965.:* m. davis, ed., 1965.
reprinted in "the undecidable", m. davis ed., 1965.:* m. davis, ed., 1965.

"on formally undecidable propositions of "principia mathematica" and related systems," 596–616.
"on formally undecidable propositions of "principia mathematica" and related systems," 596-616.

"the undecidable—basic papers on undecidable propositions, unsolvable problems and computable functions", raven, new york.
"the undecidable—basic papers on undecidable propositions, unsolvable problems and computable functions", raven, new york.

reprinted in "the undecidable", m. davis ed., 1965.:* t. slaman and w. h. woodin, 1986.
reprinted in "the undecidable", m. davis ed., 1965.:* shore, richard a.; slaman, theodore a.

in fact kruskal's tree theorem (or its finite form) is undecidable in a much stronger system codifying the principles acceptable based on a philosophy of mathematics called predicativism.
in fact kruskal's tree theorem (or its finite form) is undecidable in a much stronger system codifying the principles acceptable on the basis of a philosophy of mathematics called predicativism.

kirby and paris later showed goodstein's theorem, a statement about sequences of natural numbers somewhat simpler than the paris-harrington principle, to be undecidable in peano arithmetic.
kirby and paris later showed goodstein's theorem, a statement about sequences of natural numbers somewhat simpler than the paris-harrington principle, to be undecidable in peano arithmetic.

in 1977, paris and harrington proved that the paris-harrington principle, a version of the ramsey theorem, is undecidable in the first-order axiomatization of arithmetic called peano arithmetic, but can be proven in the larger system of second-order arithmetic.
in 1977, paris and harrington proved that the paris-harrington principle, a version of the ramsey theorem, is undecidable in the first-order axiomatization of arithmetic called peano arithmetic, but can be proven to be true in the larger system of second-order arithmetic.