Введение
Логика доказуемости
In mathematical logic, Löb's theorem states that in Peano arithmetic (PA) (or any formal system including PA), for any formula P, if it is provable in PA that "if P is provable in PA then P is true", then P is provable in PA. If Prov(P) means that the formula P is provable, we may express this more formally as
If
then
An immediate corollary (the contrapositive) of Löb's theorem is that, if P is not provable in PA, then "if P is provable in PA, then P is true" is not provable in PA. For example, "If is provable in PA, then " is not provable in PA.
Löb's theorem is named for Martin Hugo Löb, who formulated it in 1955. It is related to Curry's paradox.
В математической логике теорема Лёба гласит, что в арифметике Пеано (PA) (или любой формальной системе, включающей PA), для любой формулы P, если в PA доказуемо, что "если P доказуема в PA, то P истинна", то P доказуема в PA. Если Prov(P) означает, что формула P доказуема, то мы можем выразить это более формально как:
In mathematical logic, Löb's theorem states that in Peano arithmetic (PA) (or any formal system including PA), for any formula P, if it is provable in PA that "if P is provable in PA then P is true", then P is provable in PA. If Prov(P) means that the formula P is provable, we may express this more formally as
If
then
An immediate corollary (the contrapositive) of Löb's theorem is that, if P is not provable in PA, then "if P is provable in PA, then P is true" is not provable in PA. For example, "If is provable in PA, then " is not provable in PA.
Löb's theorem is named for Martin Hugo Löb, who formulated it in 1955. It is related to Curry's paradox.
Если
In mathematical logic, Löb's theorem states that in Peano arithmetic (PA) (or any formal system including PA), for any formula P, if it is provable in PA that "if P is provable in PA then P is true", then P is provable in PA. If Prov(P) means that the formula P is provable, we may express this more formally as
If
then
An immediate corollary (the contrapositive) of Löb's theorem is that, if P is not provable in PA, then "if P is provable in PA, then P is true" is not provable in PA. For example, "If is provable in PA, then " is not provable in PA.
Löb's theorem is named for Martin Hugo Löb, who formulated it in 1955. It is related to Curry's paradox.
то
In mathematical logic, Löb's theorem states that in Peano arithmetic (PA) (or any formal system including PA), for any formula P, if it is provable in PA that "if P is provable in PA then P is true", then P is provable in PA. If Prov(P) means that the formula P is provable, we may express this more formally as
If
then
An immediate corollary (the contrapositive) of Löb's theorem is that, if P is not provable in PA, then "if P is provable in PA, then P is true" is not provable in PA. For example, "If is provable in PA, then " is not provable in PA.
Löb's theorem is named for Martin Hugo Löb, who formulated it in 1955. It is related to Curry's paradox.
Непосредственным следствием (контрапозицией) теоремы Лёба является то, что если P не доказуема в PA, то "если P доказуема в PA, то P истинна" также не доказуема в PA. Например, "Если доказуема в PA, то " не доказуема в PA.
In mathematical logic, Löb's theorem states that in Peano arithmetic (PA) (or any formal system including PA), for any formula P, if it is provable in PA that "if P is provable in PA then P is true", then P is provable in PA. If Prov(P) means that the formula P is provable, we may express this more formally as
If
then
An immediate corollary (the contrapositive) of Löb's theorem is that, if P is not provable in PA, then "if P is provable in PA, then P is true" is not provable in PA. For example, "If is provable in PA, then " is not provable in PA.
Löb's theorem is named for Martin Hugo Löb, who formulated it in 1955. It is related to Curry's paradox.
Теорема Лёба названа в честь Мартина Гюго Лёба, который сформулировал её в 1955 году. Она связана с парадоксом Карри.
In mathematical logic, Löb's theorem states that in Peano arithmetic (PA) (or any formal system including PA), for any formula P, if it is provable in PA that "if P is provable in PA then P is true", then P is provable in PA. If Prov(P) means that the formula P is provable, we may express this more formally as
If
then
An immediate corollary (the contrapositive) of Löb's theorem is that, if P is not provable in PA, then "if P is provable in PA, then P is true" is not provable in PA. For example, "If is provable in PA, then " is not provable in PA.
Löb's theorem is named for Martin Hugo Löb, who formulated it in 1955. It is related to Curry's paradox.
Модальное доказательство теоремы Лёба
Теорема Лёба может быть доказана в модальной логике, используя лишь некоторые базовые правила об операторе доказуемости (система K4) и существование модальных неподвижных точек.
Примеры
Непосредственным следствием теоремы Лёба является то, что если P не доказуемо в PA, то утверждение "если P доказуемо в PA, то P истинно" также не доказуемо в PA. Учитывая, что мы знаем о непротиворечивости PA (но сама PA не знает о своей непротиворечивости), приведем несколько простых примеров: "Если доказуемо в PA, то " не доказуемо в PA, поскольку не доказуемо в PA (поскольку оно ложно). "Если доказуемо в PA, то " доказуемо в PA, как и любое утверждение вида "Если X, то ". "Если усиленная конечная теорема Рамзи доказуема в PA, то усиленная конечная теорема Рамзи истинна" не доказуемо в PA, поскольку "Усиленная конечная теорема Рамзи истинна" также не доказуемо в PA (несмотря на то, что это утверждение истинно). В доксастической логике теорема Лёба показывает, что любая система, классифицируемая как рефлексивный рассудитель "типа 4", должна также быть "скромной": такой рассудитель никогда не может поверить, что "моя вера в P влечет истинность P", не веря при этом в истинность P. Вторая теорема о неполноте Гёделя вытекает из теоремы Лёба путем подстановки ложного утверждения вместо P.
"If is provable in PA, then " is not provable in PA, as is not provable in PA (as it is false). "If is provable in PA, then " is provable in PA, as is any statement of the form "If X, then ". "If the strengthened finite Ramsey theorem is provable in PA, then the strengthened finite Ramsey theorem is true" is not provable in PA, as "The strengthened finite Ramsey theorem is true" is not provable in PA (despite being true). In Doxastic logic, Löb's theorem shows that any system classified as a reflexive "type 4" reasoner must also be "modest": such a reasoner can never believe "my belief in P would imply that P is true", without also believing that P is true. Gödel's second incompleteness theorem follows from Löb's theorem by substituting the false statement for P.
Обратная: теорема Лёба предполагает существование модальных фиксированных точек
Существование модальных фиксированных точек не только влечет за собой теорему Лёба, но и обратное также верно. Если теорема Лёба принимается как аксиома (схема), то можно вывести существование фиксированной точки (в смысле доказуемой эквивалентности) для любой формулы A(p), модализованной относительно p. Таким образом, в нормальной модальной логике аксиома Лёба эквивалентна сочетанию аксиоматической схемы 4, и существования модальных фиксированных точек.