“Löb’s axiom”的概念、定义、翻译、参考文献-科学参考

Löb’s axiom

Logic

In modal logic, the axiom scheme named after mathematician Martin Löb (1921–2006):
- • $□ (□ φ \to φ) \to □ φ$
When $□$ is interpreted as a provability operator, i.e., when $□ φ$ is read ‘ $φ$ is provable’, Löb’s axiom aligns with Löb’s theorem in arithmetic:
- • If $PA ⊢ Prv (⌜ φ ⌝) \to φ$ then $PA ⊢ φ$
From a semantic perspective, Löb’s axiom corresponds to transitive and converse well-founded Kripke frames. That is, the accessibility relation, $R$ , is such that if $x R y$ and $y R z$ then $x R z$ , and there is no infinite sequence of worlds such that $w_{1} R w_{2} R w_{3} R . . .$ .

单词	Löb’s axiom
释义	Löb’s axiom Logic In modal logic, the axiom scheme named after mathematician Martin Löb (1921–2006): • $□ (□ φ \to φ) \to □ φ$ When $□$ is interpreted as a provability operator, i.e., when $□ φ$ is read ‘ $φ$ is provable’, Löb’s axiom aligns with Löb’s theorem in arithmetic: • If $PA ⊢ Prv (⌜ φ ⌝) \to φ$ then $PA ⊢ φ$ From a semantic perspective, Löb’s axiom corresponds to transitive and converse well-founded Kripke frames. That is, the accessibility relation, $R$ , is such that if $x R y$ and $y R z$ then $x R z$ , and there is no infinite sequence of worlds such that $w_{1} R w_{2} R w_{3} R . . .$ .
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