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.
- For the theorem about the sparseness of prime numbers, see Rosser's theorem. For a general introduction to the incompleteness theorems, see Gödel's incompleteness theorems.
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".
Background
Rosser's trick begins with the assumptions of Gödel's incompleteness theorem. A theory T 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 ProofT(x,y) which has the intended meaning that y is a natural number code (a Gödel number) for a formula and x is the Gödel number for a proof, from the axioms of T, of the formula encoded by y. (In the remainder of this article, no distinction is made between the number y and the formula encoded by y, and the number coding a formula φ is denoted #φ). Furthermore, the formula PvblT(y) is defined as ∃x ProofT(x,y). It is intended to define the set of formulas provable from T.
The assumptions on T also show that it is able to define a negation function neg(y), with the property that if y is a code for a formula φ then neg(y) is a code for the formula ¬φ. The negation function may take any value whatsoever for inputs that are not codes of formulas.
The Gödel sentence of the theoryT is a formula φ, sometimes denoted GT such that T proves φ ↔ ¬PvblT(#φ). Gödel's proof shows that if T 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 T=PA+¬GPA proves ¬GT. 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 sentence
For a fixed arithmetical theory T, let ProofT(x,y) and neg(x) be the associated proof predicate and negation function.
A modified proof predicate ProofRT(x,y) is defined as:
which means that
This modified proof predicate is used to define a modified provability predicate PvblRT(y):
Informally, PvblRT(y) is the claim that y is provable via some coded proof x such that there is no smaller coded proof of the negation of y. Under the assumption that T is consistent, for each formula φ the formula PvblRT(#φ) will hold if and only if PvblT(#φ) holds. However, these predicates have different properties from the point of view of provability in T.
Using the diagonal lemma, let ρ be a formula such that T proves ρ ↔ ¬ PvblRT(#ρ). The formula ρ is the Rosser sentence of the theory T.
Rosser's theorem
Let T be an effective, consistent theory including a sufficient amount of arithmetic, with Rosser sentence ρ. Then the following hold (Mendelson 1977, p. 160):
- T does not prove ρ
- T does not prove ¬ρ.
The proof of (1) is as in Gödel's proof of the first incompleteness theorem. The proof of (2) is more involved. Assume that T proves ¬ρ and let e be a natural number coding for a proof of ¬ρ in T. Because T is consistent, there is no code for a proof of ρ in T, so ProofRT(e,neg(#ρ)) will hold (because there is no z < e that codes a proof of ρ).
The assumption that T includes enough arithmetic ensures that T will prove
and (using the assumption of consistency and the fact that e is a natural number)
From the latter formula, the assumptions on T show that it proves
Thus T proves
But this last formula is provably equivalent to ρ in T, by definition of ρ, which means that T proves ρ. This is a contradiction, as T was assumed to prove ¬ρ and assumed to be consistent. Thus T cannot prove ¬ρ under the assumption T is consistent.
References
- 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
- Rosser (1936), "Extensions of some theorems of Gödel and Church", Journal of Symbolic Logic, v. 1, pp. 87–91.
External links
- Avigad (2007), "Computability and Incompleteness", lecture notes.