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mathematical logic Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic com ...
, Löb's theorem states that in
Peano arithmetic In mathematical logic, the Peano axioms (, ), also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th-century Italian mathematician Giuseppe Peano. These axioms have been used nea ...
(PA) (or any
formal system A formal system is an abstract structure and formalization of an axiomatic system used for deducing, using rules of inference, theorems from axioms. In 1921, David Hilbert proposed to use formal systems as the foundation of knowledge in ma ...
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'') is the assertion that the formula ''P'' is provable in PA, we may express this more formally as :If :\mathit \vdash :then :\mathit \vdash P. An immediate corollary (the
contrapositive In logic and mathematics, contraposition, or ''transposition'', refers to the inference of going from a Conditional sentence, conditional statement into its logically equivalent contrapositive, and an associated proof method known as . The contrap ...
) 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 1+1=3 is provable in PA, then 1+1=3" 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 Curry's paradox is a paradox in which an arbitrary claim ''F'' is proved from the mere existence of a sentence ''C'' that says of itself "If ''C'', then ''F''". The paradox requires only a few apparently-innocuous logical deduction rules. Since '' ...
.


Löb's theorem in provability logic

Provability logic Provability logic is a modal logic, in which the box (or "necessity") operator is interpreted as 'it is provable that'. The point is to capture the notion of a proof predicate of a reasonably rich formal theory, such as Peano arithmetic. Examples ...
abstracts away from the details of encodings used in
Gödel's incompleteness theorems Gödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of in formal axiomatic theories. These results, published by Kurt Gödel in 1931, are important both in mathematical logic and in the phi ...
by expressing the provability of \phi in the given system in the language of
modal logic Modal logic is a kind of logic used to represent statements about Modality (natural language), necessity and possibility. In philosophy and related fields it is used as a tool for understanding concepts such as knowledge, obligation, and causality ...
, by means of the modality . That is, when \phi is a logical formula, another formula can be formed by placing a box in front of \phi, and is intended to mean that \phi is provable. Then we can formalize Löb's theorem by the axiom :\Box(\Box P\rightarrow P)\rightarrow \Box P, known as axiom GL, for Gödel–Löb. This is sometimes formalized by means of the inference rule: :If :\vdash \Box P \rightarrow P :then :\vdash P. The provability logic GL that results from taking the
modal logic Modal logic is a kind of logic used to represent statements about Modality (natural language), necessity and possibility. In philosophy and related fields it is used as a tool for understanding concepts such as knowledge, obligation, and causality ...
K4 (or K, since the axiom schema 4, \Box A\rightarrow\Box\Box A, then becomes redundant) 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
normal modal logic In logic, a normal modal logic is a set ''L'' of modal formulas such that ''L'' contains: * All propositional tautology (logic), tautologies; * All instances of the Kripke_semantics, Kripke schema: \Box(A\to B)\to(\Box A\to\Box B) and it is closed ...
using only some basic rules about the provability operator (the K4 system) plus the existence of modal fixed points.


Modal formulas

We will assume the following
grammar In linguistics, grammar is the set of rules for how a natural language is structured, as demonstrated by its speakers or writers. Grammar rules may concern the use of clauses, phrases, and words. The term may also refer to the study of such rul ...
for formulas: # If X is a
propositional variable In mathematical logic, a propositional variable (also called a sentence letter, sentential variable, or sentential letter) is an input variable (that can either be true or false) of a truth function. Propositional variables are the basic building ...
, then X is a formula. # If K is a propositional constant, then K is a formula. # If A is a formula, then \Box A is a formula. # If A and B are formulas, then so are \neg A, A \rightarrow B, A \wedge B, A \vee B, and A \leftrightarrow B A modal sentence is a formula in this syntax that contains no propositional variables. The notation \vdash A is used to mean that A is a theorem.


Modal fixed points

If F(X) is a modal formula with only one propositional variable X, then a modal fixed point of F(X) is a sentence \Psi such that :\vdash \Psi \leftrightarrow F(\Box \Psi) 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 \Box as provability in Peano Arithmetic, then the existence of modal fixed points follows from the
diagonal lemma In mathematical logic, the diagonal lemma (also known as diagonalization lemma, self-reference lemma or fixed point theorem) establishes the existence of self-referential sentences in certain formal theories. A particular instance of the diagonal ...
.


Modal rules of inference

In addition to the existence of modal fixed points, we assume the following rules of inference for the provability operator \Box, known as
Hilbert–Bernays provability conditions In mathematical logic, the Hilbert–Bernays provability conditions, named after David Hilbert and Paul Bernays, are a set of requirements for formalized provability predicates in formal theories of arithmetic (Smith 2007:224). These conditions are ...
: # (necessitation) From \vdash A conclude \vdash \Box A: Informally, this says that if A is a theorem, then it is provable. # (internal necessitation) \vdash \Box A \rightarrow \Box \Box A: If A is provable, then it is provable that it is provable. # (box distributivity) \vdash \Box (A \rightarrow B) \rightarrow (\Box A \rightarrow \Box B): 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

Much of the proof does not make use of the assumption \Box P \to P, so for ease of understanding, the proof below is subdivided to leave the parts depending on \Box P \to P until the end. Let P be any modal sentence. # Apply the existence of modal fixed points to the formula F(X) = X \rightarrow P. It then follows that there exists a sentence \Psi such that
\vdash \Psi \leftrightarrow (\Box \Psi \rightarrow P). # \vdash \Psi \rightarrow (\Box \Psi \rightarrow P), from 1. # \vdash \Box(\Psi \rightarrow (\Box \Psi \rightarrow P)), from 2 by the necessitation rule. # \vdash \Box\Psi \rightarrow \Box(\Box \Psi \rightarrow P), from 3 and the box distributivity rule. # \vdash \Box(\Box \Psi \rightarrow P) \rightarrow (\Box\Box\Psi \rightarrow \Box P), box distributivity rule " \vdash \Box(A \rightarrow B) \rightarrow (\Box A \rightarrow \Box B) " with A = \Box \Psi and B= P. # \vdash \Box \Psi \rightarrow (\Box\Box\Psi \rightarrow \Box P), from 4 and 5. # \vdash \Box \Psi \rightarrow \Box \Box \Psi, internal necessitation rule. # \vdash \Box \Psi \rightarrow \Box P, from 6 and 7.

Now comes the part of the proof where the hypothesis is used.

# Assume that \vdash \Box P \rightarrow P. Roughly speaking, it is a theorem that if P is provable, then it is, in fact true. This is a claim of ''soundness''. # \vdash \Box \Psi \rightarrow P, from 8 and 9. # \vdash (\Box \Psi \rightarrow P) \rightarrow \Psi, from 1. # \vdash \Psi, from 10 and 11. # \vdash \Box \Psi, from 12 by the necessitation rule. # \vdash P, from 13 and 10. More informally, we can sketch out the proof as follows. # Since \mathit \vdash by assumption, we also have \mathit \vdash , which implies \ \vdash . # Now, the hybrid theory \ can reason as follows: ## Suppose \ is inconsistent, then PA proves \neg P \to \bot , which is the same as P . ## However, \ already knows that \neg \mathrm_ (P) , a contradiction. ## Therefore, \ is consistent. # By Godel's second incompleteness theorem, this implies \ is inconsistent. # Thus, PA proves \neg P \to \bot , which is the same as P .


Examples

An immediate corollary 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. Given we know PA is consistent (but PA does not know PA is consistent), here are some simple examples: * "If 1+1=3 is provable in PA, then 1+1=3" is not provable in PA, as 1+1=3 is not provable in PA (as it is false). * "If 1+1=2 is provable in PA, then 1+1=2" is provable in PA, as is any statement of the form "If X, then 1+1=2". * "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 Doxastic logic is a type of logic concerned with reasoning about beliefs. The term ' derives from the Ancient Greek (''doxa'', "opinion, belief"), from which the English term ''doxa'' ("popular opinion or belief") is also borrowed. Typically, a ...
, 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 \bot for ''P''.


Converse: Löb's theorem implies the existence of modal fixed points

Not only does the existence of modal fixed points imply Löb's theorem, but the converse is valid, too. When Löb's theorem is given as an axiom (schema), the existence of a fixed point (up to provable equivalence) p\leftrightarrow A(p) for any formula ''A''(''p'')'' modalized in p'' can be derived. Thus in
normal modal logic In logic, a normal modal logic is a set ''L'' of modal formulas such that ''L'' contains: * All propositional tautology (logic), tautologies; * All instances of the Kripke_semantics, Kripke schema: \Box(A\to B)\to(\Box A\to\Box B) and it is closed ...
, Löb's axiom is equivalent to the conjunction of the axiom schema 4, (\Box A\rightarrow \Box\Box A), and the existence of modal fixed points.


Notes


References

* * * * * *


External links

* * {{DEFAULTSORT:Lob's theorem Mathematical logic Theorems in the foundations of mathematics Metatheorems Provability logic Mathematical axioms Modal logic