Gödel showed moreover that the proof relation of first-order logic is recursive.
A reader of this volume will acquire a broad acquaintance with the history of the theory of computation in the twentieth century, and with ways in which this theory will continue to develop in the twenty-first century.
At the heart of the twentieth-century revolution of computation are Gödel's incompleteness theorems of 1931, which assert the existence of arithmetic sentences that are true in the standard natural numbers but unprovable in any formalized axiomatic theory of those natural numbers.
The cogency of this argument comes down, of course, to whether one accepts Hilbert's thesis.
One might hold that an informal deduction cannot be expressed in a first-order way, since informal notions are too rich for their meanings to be settled by any single finite expression.
He claims that computations are specific forms of mathematical deductions, since they are sets of instructions whose output is supposed to follow deductively from those instructions.
Suppose, Kripke says, that the steps of a given deduction are fully expressible in first-order logic (he calls this supposition "Hilbert's thesis").The content of these results was revolutionary for the foundations of mathematics, but their proof is more directly relevant to the theory of computation.Gödel's method for demonstrating these theorems involves coding the syntax of formal theory using purely arithmetic resources.The question is whether these sharpenings were somehow "tacit" in our original pre-theoretic concept, or whether these revisions are instead replacing the original concept with something new.The advocate of open texture holds that the original concept was not precise enough to fix any particular revision as being right.That the notion chosen seemed "right" hinges upon choices that the advocate of open texture stresses are not determined by the pre-theoretic concept.We could come across procedures in the future that we want to count as computations that we do not right now: Dorit Aharonov and Umesh Vazirani discuss one such possibility, quantum computation, in their chapter.Such a view seems to have been part of Brouwer's belief that mathematical thought is essentially unformalizable.One might instead maintain the need for higher-order logic to formalize adequately the concepts and inferences of branches of mathematics that implicate the infinite, such as real analysis.It would be no exaggeration to say that computation changed the world in the twentieth century.Implicated in nearly all contemporary technology, today's computers empower so-called "intelligent systems" to negotiate the world of human reasoners, even driving cars.