Rosser's trick

In mathematical logic, Rosser's trick is a method for proving Gödel's incompleteness theorems without the assumption that the theory being considered is ω-consistent (Smorynski 1977, p. 840; Mendelson 1977, p. 160). This method was introduced by J. Barkley Rosser in 1936, as an improvement of Gödel's original proof of the incompleteness theorems that was published in 1931.

While Gödel's original proof uses a sentence that says (informally) "This sentence is not provable", Rosser's trick uses a formula that says "If this sentence is provable, there is a shorter proof of its negation".

BackgroundEdit

Rosser's trick begins with the assumptions of Gödel's incompleteness theorem. A theory   is selected which is effective, consistent, and includes a sufficient fragment of elementary arithmetic.

Gödel's proof shows that for any such theory there is a formula   which has the intended meaning that   is a natural number code (a Gödel number) for a formula and   is the Gödel number for a proof, from the axioms of  , of the formula encoded by  . (In the remainder of this article, no distinction is made between the number   and the formula encoded by  , and the number coding a formula   is denoted  ). Furthermore, the formula   is defined as  . It is intended to define the set of formulas provable from  .

The assumptions on   also show that it is able to define a negation function  , with the property that if   is a code for a formula   then   is a code for the formula  . The negation function may take any value whatsoever for inputs that are not codes of formulas.

The Gödel sentence of the theory   is a formula  , sometimes denoted  , such that   proves   ↔ . Gödel's proof shows that if   is consistent then it cannot prove its Gödel sentence; but in order to show that the negation of the Gödel sentence is also not provable, it is necessary to add a stronger assumption that the theory is ω-consistent, not merely consistent. For example, the theory  , in which PA is Peano axioms, proves  . Rosser (1936) constructed a different self-referential sentence that can be used to replace the Gödel sentence in Gödel's proof, removing the need to assume ω-consistency.

The Rosser sentenceEdit

For a fixed arithmetical theory  , let   and   be the associated proof predicate and negation function.

A modified proof predicate   is defined as:

 

which means that

 

This modified proof predicate is used to define a modified provability predicate: :

 

Informally,   is the claim that   is provable via some coded proof   such that there is no smaller coded proof of the negation of  . Under the assumption that   is consistent, for each formula   the formula   will hold if and only if   holds, because if there is a code for the proof of  , then (following the consistency of  ) there is no code for the proof of  . However,   and   have different properties from the point of view of provability in  .

An immediate consequence of the definition is that if   includes enough arithmetic, then it can prove that for every formula  ,   implies  . This is because otherwise, there are two numbers  , coding for the proofs of   and  , respectively, satisfying both   and  . (In fact   only needs to prove that such a situation cannot hold for any two numbers, as well as to include some first-order logic)

Using the diagonal lemma, let   be a formula such that   proves ρ ↔ ¬ PvblR
T
(#ρ).   ↔ . The formula   is the Rosser sentence of the theory  .

Rosser's theoremEdit

Let   be an effective, consistent theory including a sufficient amount of arithmetic, with Rosser sentence  . Then the following hold (Mendelson 1977, p. 160):

  1.   does not prove  
  2.   does not prove  

In order to prove this, one first shows that for a formula   and a number  , if   holds, then   proves  . This is shown in a similar manner to what is done in Gödel's proof of the first incompleteness theorem:   proves  , a relation between two concrete natural numbers; one then goes over all the natural numbers   smaller than   one by one, and for each  ,   proves  , again, a relation between two concrete numbers.

The assumption that   includes enough arithmetic (in fact, what is required is basic first-order logic) ensures that   also proves   in that case.

Furthermore, if   is consistent and proves  , then there is a number   coding for its proof in  , and there is no number coding for the proof of the negation of   in  . Therefore   holds, and thus   proves  .

The proof of (1) is similar to that in Gödel's proof of the first incompleteness theorem: Assume   proves  ; then it follows, by the previous elaboration, that   proves  . Thus   also proves  . But we assumed   proves  , and this is impossible if   is consistent. We are forced to conclude that   does not prove  .

The proof of (2) also uses the particular form of  . Assume   proves  ; then it follows, by the previous elaboration, that   proves  . But by the immediate consequence of the definition of Rosser's provability predicate, mentioned in the previous section, it follows that   proves  . Thus   also proves  . But we assumed   proves  , and this is impossible if   is consistent. We are forced to conclude that   does not prove  .

ReferencesEdit

  • Mendelson (1977), Introduction to Mathematical Logic
  • Smorynski (1977), "The incompleteness theorems", in Handbook of Mathematical Logic, Jon Barwise, Ed., North Holland, 1982, ISBN 0-444-86388-5
  • Barkley Rosser (September 1936). "Extensions of some theorems of Gödel and Church". Journal of Symbolic Logic. 1 (3): 87–91. doi:10.2307/2269028. JSTOR 2269028.

External linksEdit