1
Васил Пенчев1
ТЕОРЕМАТА НА МАРТИН ЛЬОБ ВЪВ ФИЛОСОФСКА ИНТЕРПРЕТАЦИЯ
Abstract: A necessary and sufficient condition that a given proposition to be provable in such
a theory that allows to be assigned to the proposition a Gödel number for containing Peano...
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1
Васил Пенчев1
ТЕОРЕМАТА НА МАРТИН ЛЬОБ ВЪВ ФИЛОСОФСКА ИНТЕРПРЕТАЦИЯ
Abstract: A necessary and sufficient condition that a given proposition to be provable in such
a theory that allows to be assigned to the proposition a Gödel number for containing Peano
arithmetic is that Gödel number itself.
This is the sense of Martin Löb’s theorem (1955).
Now
we can put several philosophical questions.
Whether is the Gödel number of a propositional
formula necessarily finite? What would the Gödel number of a theorem itself containing
Peano arithmetic be? That is the case of the so-called first incompleteness theorem (Gödel
1931).
What would the Gödel number of a self-referential statement be? What would the
Gödel number of such a proposition be: its Peano arithmetic expression after coding contains
itself as an operand? What is the Gödel number of Gödel’s proposition [R(q); q] that states its
proper unprovability? It is the key statement for his proving of the first incompleteness
theorem.
Wh
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