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Axiom vs Undecidable - What's the difference?

axiom | undecidable |

As a noun axiom

is (philosophy) a seemingly which cannot actually be proved or disproved.

As an adjective undecidable is

(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.

axiom

English

(wikipedia axiom)

Noun

  • (en noun); also axiomata (though, becoming less common and sometimes considered archaic)
  • (philosophy) A seemingly which cannot actually be proved or disproved.
  • * '>citation
  • (mathematics, logic, proof theory) A fundamental of theorems. Examples: "Through a pair of distinct points there passes exactly one straight line", "All right angles are congruent".
  • *
    • The axioms read as follows. For every composable pair f'' and ''g'' the composite f \circ g goes from the domain of ''g'' to the codomain of ''f''. For each object ''A'' the identity arrow 1_A goes from ''A'' to ''A . Composing any arrow with an identity arrow (supposing that the two are composable) gives the original arrow. And composition is associative.
  • An established principle in some artistic practice or science that is universally received.
    • The axioms of political economy cannot be considered absolute truths.

    Synonyms

    * (now rare)

    Hypernyms

    * (in logic) well-formed formula, wff, WFF

    Hyponyms

    * (in mathematics) * (in mathematics) * (in mathematics)

    Holonyms

    * (in logic) formal system

    Derived terms

    *

    Related terms

    * axiomatize * axiomatization

    See also

    (other terms of interest) * conjecture * corollary * demonstration * hypothesis * law * lemma * porism * postulate * premise * principle * proof * proposition * theorem * theory * truism

    External links

    * * ----

    undecidable

    English

    Adjective

    (-)
  • (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.
  • *
    • The first-order 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 first-order 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 '''semi-decision'''
    '     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. [...]
  • (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.)

    • Antonyms

    * decidable

    Related terms

    * noncomputable