Online research in philosophy

In this paper I argue that it is more difficult to see how Godel's incompleteness theorems and related consistency proofs for formal systems are consistent with the views of formalists, mechanists and traditional intuitionists than it is to see how they are consistent with a particular form of mathematical realism. If the incompleteness theorems and consistency proofs are better explained by this form of realism then we can also see how there is room for skepticism about Church's Thesis and the claim that minds are machines.

How do we prove true but unprovable propositions? Gödel produced a statement whose undecidability derives from its ad hoc construction. Concrete or mathematical incompleteness results are interesting unprovable statements of formal arithmetic. We point out where exactly the unprovability lies in the ordinary ‘mathematical’ proofs of two interesting formally unprovable propositions, the Kruskal-Friedman theorem on trees and Girard's normalization theorem in type theory. Their validity is based on robust cognitive performances, which ground mathematics in our relation to space and time, such as symmetries and order, or on the generality of Herbrand's notion of ‘prototype proof’.

This work is a sequel to the author's Godel's Incompleteness Theorems, though it can be read independently by anyone familiar with Godel's incompleteness theorem for Peano arithmetic. The book deals mainly with those aspects of recursion theory that have applications to the metamathematics of incompleteness, undecidability, and related topics. It is both an introduction to the theory and a presentation of new results in the field.

While Gödel's (first) incompleteness theorem has been used to refute the main contentions of Hilbert's program, it does not seem to have been generally used to stress that a basic ingredient of that program, the concept of formal system as a closed system - as well as the underlying view, embodied in the axiomatic method, that mathematical theories are deductions from first principles must be abandoned. Indeed the logical community has generally failed to learn Gödel's lesson that Hilbert's concept of formal system as a closed system is inadequate and continues to use it as if there were no incompleteness theorem. In this paper I will stress the role of Gödel's incompleteness theorem in showing the inadequacy of such a concept of formal system and the need for a more articulated view of mathematical theories. More generally I will argue that Gödel's result entails that, as an alternative to mathematical logic, a new concept of logic is required: logic as the theory of communicating inference processes.

We prove that any 1-closed (see def 1.1) model of the Π 2 consequences of PA satisfies ¬Cons PA which gives a proof of the second Godel incompleteness theorem without the use of the Godel diagonal lemma. We prove a few other theorems by the same method.

Kurt Godel, the greatest logician of our time, startled the world of mathematics in 1931 with his Theorem of Undecidability, which showed that some statements in mathematics are inherently "undecidable." His work on the completeness of logic, the incompleteness of number theory, and the consistency of the axiom of choice and the continuum theory brought him further worldwide fame. In this introductory volume, Raymond Smullyan, himself a well-known logician, guides the reader through the fascinating world of Godel's incompleteness theorems. The level of presentation is suitable for anyone with a basic acquaintance with mathematical logic. As a clear, concise introduction to a difficult but essential subject, the book will appeal to mathematicians, philosophers, and computer scientists.

We show how second order logic incompleteness follows from incompleteness of arithmetic, as proved by Gödel.

It might seem that three of Godel’s results - the Completeness and the First and Second Incompleteness Theorems - assume so little that they are reasonably indisputable. A version of the Completeness Theorem, for instance, can be proven in RCA0, which is the weakest system studied extensively in Simpson’s encyclopaedic Subsystems of Second Order Arithmetic. And it often seems that the minimum requirements for a system just to express the Incompleteness Theorems are sufficient to prove them. However, it will be shown that a particular sub-system of Peano Arithmetic is powerful enough to express assertions about syntax, provability, consistency, and models, while being too weak to allow the standard proofs of the theorems to go through. An alternative proof is available for the First Incompleteness Theorem, but is of such a different nature that the import of the theorem changes. And there are no alternative proofs for (certainly) the Completeness and (apparently) the Second Incompleteness Theorems. It is therefore perfectly rational for someone to be skeptical about Godel’s results.

We give a new proof for Godel's second incompleteness theorem, based on Kolmogorov complexity, Chaitin's incompleteness theorem, and an argument that resembles the surprise examination paradox. We then go the other way around and suggest that the second incompleteness theorem gives a possible resolution of the surprise examination paradox. Roughly speaking, we argue that the flaw in the derivation of the paradox is that it contains a hidden assumption that one can prove the consistency of the mathematical theory in which the derivation is done; which is impossible by the second incompleteness theorem.

Gödel began his 1951 Gibbs Lecture by stating: “Research in the foundations of mathematics during the past few decades has produced some results which seem to me of interest, not only in themselves, but also with regard to their implications for the traditional philosophical problems about the nature of mathematics.” (Gödel 1951) Gödel is referring here especially to his own incompleteness theorems (Gödel 1931). Gödel’s first incompleteness theorem (as improved by Rosser (1936)) says that for any consistent formalized system F, which contains elementary arithmetic, there exists a sentence GF of the language of the system which is true but unprovable in that system. Gödel’s second incompleteness theorem states that no consistent formal system can prove its own consistency.