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In [[mathematical logic]], '''Löb's theorem''' states that in a theory with [[Peano arithmetic]], for any formula ''P'', if it is provable that "if ''P'' is provable then ''P'' is true", then ''P'' is provable. I.e.
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:<math>\mathrm{if}\ PA \vdash (Bew(\# P) \rightarrow P)\mathrm{, then}\ PA \vdash P</math>
 
where ''Bew''(#''P'') means that the formula ''P'' with [[Gödel number]] #''P'' is provable.
 
Löb's theorem is named for [[Martin Löb|Martin Hugo Löb]], who formulated it in 1955.
 
==Löb's theorem in provability logic==
[[Provability logic]] abstracts away from the details of encodings used in [[Gödel's incompleteness theorems]] by expressing the provability of <math>\phi</math> in the given system in the language of [[modal logic]], by means of the modality {{tooltip|<math>\Box \phi</math>|The box is intended to be a box; it is not your browser failing to display another glyph.}}.
 
Then we can formalize Löb's theorem by the axiom
 
:<math>\Box(\Box P\rightarrow P)\rightarrow \Box P,</math>
 
known as axiom GL, for Gödel-Löb. This is sometimes formalised by means of an inference rule that infers
 
:<math> P</math>
 
from
 
:<math>\Box P\rightarrow P.</math>
 
The provability logic GL that results from taking the [[modal logic]] ''K4'' and adding the above axiom GL is the most intensely investigated system in provability logic.
 
==Modal Proof of Löb's theorem==
Löb's theorem can be proved within modal logic using only some basic rules about the provability operator plus the existence of modal fixed points.
 
===Modal Formulas===
We will assume the following grammar for formulas:
# If <math>X</math> is a propositional variable, then <math>X</math> is a formula.
# If <math>K</math> is a propositional constant, then <math>K</math> is a formula.
# If <math>A</math> is a formula, then <math>\Box A</math> is a formula.
# If <math>A</math> and <math>B</math> are formulas, then so are <math>\neg A</math>, <math>A \rightarrow B</math>, <math>A \wedge B</math>, <math>A \vee B</math>, and <math>A \leftrightarrow B</math>
 
A modal sentence is a modal formula that contains no propositional variables. We use <math>\vdash A</math> to mean <math>A</math> is a theorem.
 
===Modal Fixed Points===
If <math>F(X)</math> is a modal formula with only one propositional variable <math>X</math>, then a modal fixed point of <math>F(X)</math> is a sentence <math>\Psi</math> such that
:<math>\vdash \Psi \leftrightarrow F(\Box \Psi)</math>
 
We will assume the existence of such fixed points for every modal formula with one free variable. This is of course not an obvious thing to assume, but if we interpret <math>\Box</math> as provability in Peano Arithmetic, then the existence of modal fixed points is in fact true.
 
===Modal Rules of Inference===
In addition to the existence of modal fixed points, we assume the following rules of inference for the provability operator <math>\Box</math>:
 
# From <math>\vdash A</math> conclude <math>\vdash \Box A</math>: Informally, this says that if A is a theorem, then it is provable.
# <math>\vdash \Box A \rightarrow \Box \Box A</math>: If A is provable, then it is provable that it is provable.
# <math>\vdash \Box (A \rightarrow B) \rightarrow (\Box A \rightarrow \Box B)</math>:  This rule allows you to do modus ponens inside the provability operator. If it is provable that A implies B, and A is provable, then B is provable.
 
===Proof of Löb's theorem===
# Assume that there is a modal sentence <math>P</math> such that <math>\vdash \Box P \rightarrow P</math>. Roughly speaking, it is a theorem that if <math>P</math> is provable, then it is, in fact true. This is a claim of <em>soundness</em>.
# Let <math>\Psi</math> be a sentence such that <math>\vdash \Psi \leftrightarrow (\Box \Psi \rightarrow P)</math>. The existence of such a sentence follows the existence of a fixed point of the formula <math>F(X) == X \rightarrow P</math>.
# From 2, it follows that <math>\vdash \Psi \rightarrow (\Box \Psi \rightarrow P)</math>.
# From rule of inference 1, it follows that <math>\vdash \Box(\Psi \rightarrow (\Box \Psi \rightarrow P))</math>.
# From 4 and rule of inference 3, it follows that <math>\vdash \Box\Psi \rightarrow \Box(\Box \Psi \rightarrow P)</math>.
# From rule of inference 3, it follows that <math>\vdash \Box(\Box \Psi \rightarrow P) \rightarrow \Box\Box\Psi \rightarrow \Box P</math>
# From 5 and 6, it follows that <math>\vdash \Box \Psi \rightarrow \Box\Box\Psi \rightarrow \Box P</math>
# From rule of inference 2, it follows that <math>\vdash \Box \Psi \rightarrow \Box \Box \Psi</math>
# From 7 and 8, it follows that <math>\vdash \Box \Psi \rightarrow \Box P</math>
# From 1 and 9, it follows that <math>\vdash \Box \Psi \rightarrow P</math>
# From 10 and 2, it follows that <math>\vdash \Psi</math>
# From 11 and rule of inference 1, it follows that <math>\vdash \Box \Psi</math>
# From 12 and 10, it follows that <math>\vdash P</math>
 
==References==
* {{cite book | author = Hinman, P. | title = Fundamentals of Mathematical Logic | publisher = A K Peters | year = 2005 | isbn = 1-56881-262-0}}
* {{cite book | author = Boolos, George S. | title = The Logic of Provability | publisher = Cambridge University Press | year = 1995 | isbn = 0-521-48325-5}}
 
==External links==
* [http://planetmath.org/?op=getobj&from=objects&id=9381 Löb's theorem at PlanetMath]
* [http://yudkowsky.net/rational/lobs-theorem The Cartoon Guide to Löb’s Theorem, by Eliezer Yudkowsky]
* [http://arxiv.org/abs/1301.5340 Generalized Löb's Theorem.Jaykov Foukzon]
* [http://www.ams.org/amsmtgs/2210_abstracts/1089-03-60.pdf]Spring Western Sectional Meeting 
University of Colorado Boulder, Boulder,CO April 13-14, 2013 (Saturday-Sunday) Meeting #1089
* [http://www.ams.org/amsmtgs/2208_abstracts/1092-03-13.pdf]
* [http://www.ams.org/meetings/sectional/2208_program_saturday.html#2208:AMSCP1]Fall Southeastern Sectional Meeting University of Louisville, Louisville, KY October 5-6, 2013 (Saturday - Sunday) Meeting #1092
{{DEFAULTSORT:Lob's theorem}}
[[Category:Mathematical logic]]
[[Category:Axioms of modal logic]]
[[Category:Theorems in the foundations of mathematics]]
[[Category:Metatheorems]]

Latest revision as of 12:32, 5 May 2014

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