Introduction This book and its predecessor The Emperor's New Mind argue that natural minds cannot be understood and artificial minds cannot be constructed without new physics, about which the book gives some ideas. We have no objection to new physics but don't see it as necessary for artificial intelligence. We see artificial intelligence research as making definite progress on difficult scientific problems. I take it that students of natural intelligence also see present physics as adequate for understanding mind. This review concerns only some problems with the first part of the book. 1 2 Awareness and Understanding Penrose discusses awareness and understanding briefly and concludes (with no references to the AI literature) that AI researchers have no idea of how to make computer programs with these qualities. I substantially agree with his characterizations of awareness and unders

In AI, consciousness of self consists in a program having certain kinds of facts about its own mental processes and state of mind. We discuss what consciousness of its own mental structures a robot will need in order to operate in the common sense world and accomplish the tasks humans will give it. It's quite a lot. Many features of human consciousness will be wanted, some will not, and some abilities not possessed by humans have already been found feasible and useful in limited contexts. We give preliminary fragments of a logical language a robot can use to represent information about its own state of mind. A robot will often have to conclude that it cannot decide a question on the basis of the information in memory and therefore must seek information externally. Godel's idea of relative consistency is used to formalize non-knowledge. Programs with the kind of consciousness discussed in this article do not yet exist, although programs with some components of it exist. Thinking about c...

A logic for specification and verification is derived from the axioms of Zermelo-Fraenkel set theory. The proofs are performed using the proof assistant Isabelle. Isabelle is generic, supporting several different logics. Isabelle has the flexibility to adapt to variants of set theory. Its higher-order syntax supports the definition of new binding operators. Unknowns in subgoals can be instantiated incrementally. The paper describes the derivation of rules for descriptions, relations and functions, and discusses interactive proofs of Cantor’s Theorem, the Composition of Homomorphisms challenge [9], and Ramsey’s Theorem [5]. A generic proof assistant can stand up against provers dedicated to particular logics. Key words. Isabelle, set theory, generic theorem proving, Ramsey’s Theorem,

1.2. The axioms

Contents 1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 2 The expressive power of second order Logic : : : : : : : : : : : 3 2.1 The language of second order logic : : : : : : : : : : : : : 3 2.2 Expressing size : : : : : : : : : : : : : : : : : : : : : : : : 4 2.3 Defining data types : : : : : : : : : : : : : : : : : : : : : 6 2.4 Describing processes : : : : : : : : : : : : : : : : : : : : : 8 2.5 Expressing convergence using second order validity : : : : : : : : : : : : : : : : : : : : : : : : : 9 2.6 Truth definitions: the analytical hierarchy : : : : : : : : 10 2.7 Inductive definitions : : : : : : : : : : : : : : : : : : : : : 13 3 Canonical semantics of higher order logic : : : : : : : : : : : : 15 3.1 Tarskian semantics of second order logic : : : : : : : : : 15 3.2 Function and re

. Although set theory is the most popular foundation for mathematics, not many mechanized mathematics systems are based on set theory. Zermelo-Fraenkel (zf) set theory and other traditional set theories are not an adequate foundation for mechanized mathematics. stmm is a version of von-Neumann-Bernays-Godel (nbg) set theory that is intended to be a Set Theory for Mechanized Mathematics. stmm allows terms to denote proper classes and to be undened, has a denite description operator, provides a sort system for classifying terms by value, and includes lambdanotation with term constructors for function application and function abstraction. This paper describes stmm and discusses why it is a good foundation for mechanized mathematics. Keywords: Set theory, nbg, higher-order logic, mechanized mathematics, theorem proving systems, partial functions, undenedness, sorts. 1.

We investigate modal deduction through translation into standard logic and set theory. Derivability in the minimal modal logic is captured precisely by translation into a weak, computationally attractive set theory \Omega\Gamma This approach is shown equivalent to working with standard first-order translations of modal formulas in a theory of general frames. Next, deduction in a more powerful second-order logic of general frames is shown equivalent with set-theoretic derivability in an `admissible variant' of \Omega\Gamma Our methods are mainly model-theoretic and set-theoretic, and they admit extension to richer languages than that of basic modal logic. 1 Introduction We are interested in analyzing general deduction for modal formulae [4]. The standard systems used for this purpose are the so-called "minimal modal logic" K or, if one wants to work over general frames (as we do), the system K s obtained from K adding a substitution rule. We do not consider special purpose calculi for ...

Most current research on automated theorem proving is concerned with proving theorems of first-order logic. We discuss two examples which illustrate the advantages of using higher-order logic in certain contexts. For some purposes type theory is a much more convenient vehicle for formalizing mathematics than axiomatic set theory. Even theorems of first-order logic can sometimes be proven more expeditiously in higher-order logic than in first-order logic. We also note the need to develop automatic theorem-proving methods which may produce proofs which do not have the subformula property. 1. Introduction In some ways it appears that the field of automated deduction is ahead of its time. We have increasingly good methods for proving theorems, but the hardware available is still not adequate for many of the problems to which we would like to apply our techniques. However, this will change. Radically new computers based on exotic technologies such as DNA computing, quantum computing, and ...

this article I will be looking at the leading question from the point of view of the logician, and for a substantial part of that, from the perspective of one supremely important logician: Kurt Godel. From the time of his stunning incompleteness results in 1931 to the end of his life, Godel called for the pursuit of new axioms to settle undecided arithmetical problems. And from 1947 on, with the publication of his unusual article, "What is Cantor's continuum problem?" [11], he called in addition for the pursuit of new axioms to settle Cantor's famous conjecture about the cardinal number of the continuum. In both cases, he pointed primarily to schemes of higher infinity in set theory as the direction in which to seek these new principles. Logicians have learned a great deal in recent years that is relevant to Godel's program, but there is considerable disagreement about what conclusions to draw from their results. I'm far from unbiased in this respect, and you'll see how I come out on these matters by the end of this essay, but I will try to give you a fair presentation of other positions along the way so you can decide for yourself which you favor.

Partial functions can be easily represented in set theory as certain sets of ordered pairs. However, classical set theory provides no special machinery for reasoning about partial functions. For instance, there is no direct way of handling the application of a function to an argument outside its domain as in partial logic. There is also no utilization of lambda-notation and sorts or types as in type theory. This paper introduces a version of von-Neumann-Bernays-Gödel set theory for reasoning about sets, proper classes, and partial functions represented as classes of ordered pairs. The underlying logic of the system is a partial first-order logic, so classvalued terms may be nondenoting. Functions can be specified using lambda-notation, and reasoning about the application of functions to arguments is facilitated using sorts similar to those employed in the logic of the imps Interactive Mathematical Proof System. The set theory is intended to serve as a foundation for mechanized mathematics systems.