The problem of consistencyThe notion of consistency involved in Lucas's argument runs into difficulties for humans and/or machines. Note: see Gödel second theorem. Important Properties of Formal SystemsConsistency A system is consistent if it is impossible, within the system, to derive both a statement and its negation. A system is inconsistent if a statement and its negation are both derivable. In an inconsistent system, every possible statement (of its language) can be derived as a theorem, because everything can be logically derived from a contradiction. In a consistent system, that isn't the case. Consistency is also referred to as the correctness or soundness of the system. Completeness A complete system will have a derivable theorem to correspond to every true formula in its language. An incomplete system will not be able to derive some true formula. Decidability A system is decidable if every true formula of the system has a proof and every untrue formula of the system has a disproof. A system is undecidable if there is some statement that it can neither prove nor disprove. Note: The properties described here are stated in terms of logical systems; they can also be stated for formal systems more generally. For discussion of formal systems and their properties see Smullyan (1961). For further discussion of the properties of logical systems, see Hunter (1973), Anderton (1972), or Smulyan (1961).
CONTEXT(Help)
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Artificial Intelligence »
Are thinking computers mathematically possible?  »
No: computers are limited by Gödel's theorems »
The problem of consistency
Belief in one's own consistency leads to inconsistency »
Gödel’s theorems don't apply to inconsistent machines »
Inconsistency without explosion of belief »
The mechanist's dilemma »
We don't know that mathematics is consistent »