Per Lindström

Aspects of Incompleteness

This thoroughly revised second edition of a classic book on the main ideas and results of general meta-mathematics contains new results and simplified proofs, as well as an up to date bibliography. In addition to the standard results of Goedel and others on incompleteness, (non) finite axiomatizability, interpretability, etc.., it contains a thorough treatment of partial conservativity and degrees of interpretability. The reader should be familiar with the widely used method of arithmetization and with the elements of recursion theory.

Penrose's New Argument

It has been argued, by Penrose and others, that Gödels proof of his first incompleteness theorem shows that human mathematics cannot be captured by a formal system F: the Gödel sentence G(F) of F can be proved by a (human) mathematician but is not provable in F. To this argment it has been objected that the mathematician can prove G(F) only if (s)he can prove that F is consistent, which is unlikely if F is complicated. Penrose has invented a new argument intended to avoid this objection. In the paper I try to show that Penroses new argument is inconclusive.

On Characterizing Elementary Logic

This is an expository paper. My aim is to explain in a way accessible to the non-specialist a number of results that amount to characterizations of elementary logic (EL), i.e. first order predicate logic with identity. The characterizations that I have in mind can be described as follows: First I define the very general concept abstract logic (AL). Then I define an inclusion relation ⊆ between ALs. Intuitively L⊆L′ means that everything that can be expressed in L can also be expressed in L′ although perhaps in a different way. For example, in EL it can be said of a binary relation that it is transitive. Thus if EL⊆L, this can also be expressed in L. (I assume, of course, that EL is an AL) L and L′ are equivalent, L≡L’ if L⊆L′ and L′⊆L. The next step consists in choosing a number of properties P 1,…,P n of ALs such that EL has P 1,…, P n . An example of an interesting property of this type is the following which may be called the Lowenheim property, since it means that the original Lowenheim theorem holds for L: If a sentence of L has a model, then it has a countable (finite or denumerable) model. Finally, a characterization of EL is a (true) statement of the form: If L is an AL, EL⊆L, and L has P 1,…, P n , then L≡EL. Since all reasonable ALs contain EL, this may also be expressed by saying that EL is the strongest AL having P 1,…, P n . Of course, one can also speak of characterizations of logics other than EL and there are, in fact, results of this type but they will not be discussed here.

Remarks on Penrose’s “New Argument”

It is commonly agreed that the well-known Lucas–Penrose arguments and even Penrose’s ‘new argument’ in [Penrose, R. (1994): Shadows of the Mind, Oxford University Press] are inconclusive. It is, perhaps, less clear exactly why at least the latter is inconclusive. This note continues the discussion in [Lindström, P. (2001): Penrose’s new argument, J. Philos. Logic30, 241–250; Shapiro, S.(2003): Mechanism, truth, and Penrose’s new argument, J. Philos. Logic32, 19–42] and elsewhere of this question.

Note on Some Fixed Point Constructions in Provability Logic

We present a quite simple proof of the fixed point theorem for GL. We also use this proof to show that Sambins algorithm yields a fixed point.