An attempt is made to apply information-theoretic computational complexity to metamathematics. The paper studies the number of bits of instructions that must be a given to a computer for it to perform finite and infinite tasks, and also the amount of time that it takes the computer to perform these tasks. This is applied to measuring the difficulty of proving a given set of theorems, in terms of the number of bits of axioms that are assumed, and the size of the proofs needed to deduce the theorems from the axioms. 2 G. J. Chaitin Key Words and Phrases: complexity of sets, computational complexity, difficulty of theoremproving, entropy of sets, formal systems, Godel's incompleteness theorem, halting problem, information content of sets, information content of axioms, information theory, information time trade-offs, metamathematics, random strings, recursive functions, recursively enumerable sets, size of proofs, universal computers CR Categories: 5.21, 5.25, 5.27, 5.6 1. Introduct...

. Generalized databases will be examined, in which attributes can be sets of attributes, or sets of sets of attributes, and other higher type constructs. A precise semantics will be developed for such databases, based on a higher type modal/intensional logic. 1 Introduction In some ways this is an eccentric paper---there are no theorems. What I want to do, simply stated, is present a semantics for relational databases. But the semantics is rich, powerful, and oddly familiar, and applies to databases that are quite general. It is a topic whose exploration I wish to recommend, rather than a finished product I simply present. Relational databases generally have entities of some kind as values of attributes, though it is a small stretch to allow sets of entities as well. I want to consider databases that stretch things further, allowing attributes to have as values sets of sets of entities, and so on, but further, I also want to allow sets of attributes, sets of sets of attributes, and so...

In 1973, Parikh proved a speed-up theorem conjectured by Gödel 37 years before: there exist arithmetical formulæ that are provable in first order arithmetic, but whose shorter proof in second order arithmetic is arbitrarily smaller than any proof in first order. On the other hand, resolution for higher order logic can be simulated step by step in a first order narrowing and resolution method based on deduction modulo, whose paradigm is to separate deduction and computation to make proofs clearer and shorter. We prove that i+1-th order arithmetic can be linearly simulated into i-th order arithmetic modulo some confluent and terminating rewrite system. We also show that there exists a speed-up between i-th order arithmetic modulo this system and i-th order arithmetic without modulo. All this allows us to prove that the speed-up conjectured by Gödel does not come from the deductive part of the proofs, but can be expressed as simple computation, therefore justifying the use of deduction modulo as an efficient first order setting simulating higher order.

. First-order modal logic, in the usual formulations, is not sufficiently expressive, and as a consequence problems like Frege's morning star/evening star puzzle arise. The introduction of predicate abstraction machinery provides a natural extension in which such di#culties can be addressed. But this machinery can also be thought of as part of a move to a full higher-order modal logic. In this paper we present a sketch of just such a higher-order modal logic: its formal semantics, and a proof procedure using tableaus. Naturally the tableau rules are not complete, but they are with respect to a Henkinization of the "true" semantics. We demonstrate the use of the tableau rules by proving one of the theorems involved in Godel's ontological argument, one of the rare instances in the literature where higher-order modal constructs have appeared. A fuller treatment of the material presented here is in preparation. 1 Introduction Standard first-order classical logic is so well behaved that co...

Solovay and Strassen, and Miller and Rabin have discovered fast algorithms for testing primality which use coin-flipping and whose con1 2 G. J. Chaitin clusions are only probably correct. On the other hand, algorithmic information theory provides a precise mathematical definition of the notion of random or patternless sequence. In this paper we shall describe conditions under which if the sequence of coin tosses in the Solovay-- Strassen and Miller--Rabin algorithms is replaced by a sequence of heads and tails that is of maximal algorithmic information content, i.e., has maximal algorithmic randomness, then one obtains an error-free test for primality. These results are only of theoretical interest, since it is a manifestation of the Godel incompleteness phenomenon that it is impossible to "certify" a sequence to be random by means of a proof, even though most sequences have this property. Thus by using certified random sequences one can in principle, but not in practice, convert proba...

We formulate a simple theory of deductive reasoning based on mental models. One prediction of the theory is experimentally tested and found to be incorrect. The bearing of our results on contemporary theories of mental models is discussed. We then consider a potential objection to current rule-theories of deduction. Such theories picture deductive reasoning as the successive application of inference-schemata from #rst-order logic. Relying on a theorem due to George Boolos, we show that under weak hypotheses #rst-order schemata cannot account for many people's abilitytoverify the validity of #rst-order arguments. The hypothesis that deductive reasoning is mediated by the construction of mental models has enjoyed predictive success across several studies. It has also proven to be a fertile source of ideas about other kinds of judgment, for example, temporal, spatial, and probabilistic. At the same time, the theory has su#ered from persistent criticism for ambiguity about the details of ...

Gottesbeweis“ ➢ Als Gottesbeweis bezeichnet man im Allgemeinen Versuche, die Existenz (eines) Gottes zu beweisen, bzw. Argumente für eine solche Existenz zu finden. [Wik04][Bec00]

As noted in Section 2.3, Gärdenfors ’ conceptual spaces theory of concepts is an example of a similarity space theory, which on the face of things puts it in company with prototype and exemplar theories of concepts, in the empirical tradition, and in contrast with e.g. theory theory and informational atomism, in the rationalist tradition – though that, I will argue, would at best be an over-simplification, and indeed, as is clear from passages in his book, Gärdenfors sees his theory as being compatible with and complementary to these other approaches. 1 Specifically, Gärdenfors sees his approach as a bridging account between different levels of explanation of cognition more generally, and different accounts of concepts more specifically. Indeed, I believe his theory is well placed to support the “toggling effect ” thesis I proposed in the last chapter. Sometimes with similarity-space-based theories, concepts that are more similar are grouped closer together (more literally or more metaphorically, depending on the theory), while those that are more dissimilar are grouped further apart. Fodor has argued, quite convincingly I think, that such an approach is doomed to failure, for invariably the measures of similarity that are being assumed depend upon an underlying layer of strict identity.(1998, p. 32) For example, if two prototypes are more similar or less similar depending on how many features they share, then those features must,