Diagonal lemma

Abstract: Introduction. In setting up his definition of analyticity, Carnap uses various non-constructive rules, of which we shall be concerned with only one. This one we shall call simply “Carnap's rule.” It can be roughly stated thus: If f(0), f(1), f(2), … are all provable, then (x)f(x) shall be provable. In this paper will be briefly considered the logics got by starting with the system of Principia plus Peano's axioms and allowing one, two, …, ω, ω+l, and so on up to, but not including ω uses of Carnap's rule (interspersed with uses of the ordinary rules), as well as the system which includes Principia and Peano's axioms and is closed under application of Carnap's rule and the ordinary rules. Under suitable assumptions as to consistency, it is shown that in each of these logics there occur undecidable propositions and that a formula which states the consistency of the logic exists in the logic but is not provable in the logic. An interesting side result is that the logic got by allowing co applications of Carnap's rule is not closed under Carnap's rule.

Abstract: This chapter contains sections titled: 41 The Previously Received View and More Recent Challenges, 42 Computation as a Special Form of Mathematical Argument, 43 Von Neumann's Problem of Characterizing and Proving Unsolvability and Godel's Theorem IX, 44 Some Clarificatory Remarks on the Present Characterization, 45 Conclusion, Notes, References

Abstract: We use ideas and machinery of effective algebra to investigate computable structures on the space C[0, 1] of continuous functions on the unit interval. We show that (C[0, 1], sup) has infinitely many computable structures non-equivalent up to a computable isometry. We also investigate if the usual operations on C[0, 1] are necessarily computable in every computable structure on C[0, 1]. Among other results, we show that there is a computable structure on C[0, 1] which computes + and the scalar multiplication, but does not compute the operation of pointwise multiplication of functions. Another unexpected result is that there exists more than one computable structure making C[0, 1] a computable Banach algebra. All our results have implications for the study of the number of computable structures on C[0, 1] in various commonly used signatures.