Penrose proved that a computational or formalizable theory of the brainís cognitive functioning is impossible, but suggested that a physical non-computational and non-formalizable one may be viable. Arguments as to why Penroseís program is unrealizable are presented. The main argument is that a non-formalizable theory should be verbal. However, verbal paradoxes based on Cantorís diagonal processes show the impossibility of a consistent verbal theory of the brain comprising its arithmetical cognition. It is suggested that comprehensive theories of the human brain (...) and of physical experience are Kantian rather than Platonic ideas. This suggestion is likewise based on arguments related to diagonal processes. (shrink)

This accessible book gives a new, detailed and elementary explanation of the Gödel incompleteness theorems and presents the Chaitin results and their relation to the da Costa-Doria results, which are given in full, but with no ...

In “Minds, Machines, and Gödel”, J. R. Lucas claims that Goedel's incompleteness theorem constitutes a proof “that Mechanism is false, that is, that minds cannot be explained as machines”. He claims further that “if the proof of the falsity of mechanism is valid, it is of the greatest consequence for the whole of philosophy”. It seems to me that both of these claims are exaggerated. It is true that no minds can be explained as machines. But it is not (...) true that Goedel's theorem proves this. At most, Goedel's theorem proves that not all minds can be explained as machines. Since this is so, Goedel's theorem cannot be expected to throw much light on why minds are different from machines. Lucas overestimates the importance of Goedel's theorem for the topic of mechanism, I believe, because he presumes falsely that being unable to follow any but mechanical procedures in mathematics makes something a machine. (shrink)

Gödel's Theorem is often used in arguments against machine intelligence, suggesting humans are not bound by the rules of any formal system. However, Gödelian arguments can be used to support AI, provided we extend our notion of computation to include devices incorporating random number generators. A complete description scheme can be given for integer functions, by which nonalgorithmic functions are shown to be partly random. Not being restricted to algorithms can be accounted for by the availability of an (...) arbitrary random function. Humans, then, might not be rule-bound, but Gödelian arguments also suggest how the relevant sort of nonalgorithmicity may be trivially made available to machines. (shrink)

"The Emperor's New Mind" by Roger Penrose has received a great deal of both praise and criticism. This review discusses philosophical aspects of the book that form an attack on the "strong" AI thesis. Eight different versions of this thesis are distinguished, and sources of ambiguity diagnosed, including different requirements for relationships between program and behaviour. Excessively strong versions attacked by Penrose (and Searle) are not worth defending or attacking, whereas weaker versions remain problematic. Penrose (like Searle) regards the notion (...) of an algorithm as central to AI, whereas it is argued here that for the purpose of explaining mental capabilities the architecture of an intelligent system is more important than the concept of an algorithm, using the premise that what makes something intelligent is not what it does but how it does it. What needs to be explained is also unclear: Penrose thinks we all know what consciousness is and claims that the ability to judge Go "del's formula to be true depends on it. He also suggests that quantum phenomena underly consciousness. This is rebutted by arguing that our existing concept of "consciousness" is too vague and muddled to be of use in science. This and related concepts will gradually be replaced by a more powerful theory-based taxonomy of types of mental states and processes. The central argument offered by Penrose against the strong AI thesis depends on a tempting but unjustified interpretation of Goedel's incompleteness theorem. Some critics are shown to have missed the point of his argument. A stronger criticism is mounted, and the relevance of mathematical Platonism analysed. Architectural requirements for intelligence are discussed and differences between serial and parallel implementations analysed. (shrink)

This junior/senior level text is devoted to a study of first-order logic and its role in the foundations of mathematics: What is a proof? How can a proof be justified? To what extent can a proof be made a purely mechanical procedure? How much faith can we have in a proof that is so complex that no one can follow it through in a lifetime? The first substantial answers to these questions have only been obtained in this century. The most (...) striking results are contained in Goedel's work: First, it is possible to give a simple set of rules that suffice to carry out all mathematical proofs; but, second, these rules are necessarily incomplete - it is impossible, for example, to prove all true statements of arithmetic. The book begins with an introduction to first-order logic, Goedel's theorem, and model theory. A second part covers extensions of first-order logic and limitations of the formal methods. The book covers several advanced topics, not commonly treated in introductory texts, such as Trachtenbrot's undecidability theorem. Fraissé's elementary equivalence, and Lindstroem's theorem on the maximality of first-order logic. (shrink)

The volume contains twenty essays devoted to the philosophy of mathematics and the history of logic. They have been divided into four parts: general philosophical problems of mathematics, Hilbert's program vs. the incompleteness phenomenon, philosophy of mathematics in Poland, mathematical logic in Poland. Among considered problems are: epistemology of mathematics, the meaning of the axiomatic method, existence of mathematical objects, distinction between proof and truth, undefinability of truth, Goedel's theorems and computer science, philosophy of mathematics in Polish mathematical and logical (...) schools, beginnings of mathematical logic in Poland, contribution of Polish logicians to recursion theory. (shrink)

Leibniz entertained the idea that, as a set of “striving possibles” competes for existence, the largest and most perfect world comes into being. The paper proposes 8 criteria for a Leibniz-world. It argues that a) there is no algorithm e.g., one involving pairwise compossibility-testing that can produce even possible Leibniz-worlds; b) individual substances presuppose completed worlds; c) the uniqueness of the actual world is a matter of theological preference, not an outcome of the assembly-process; and d) Goedel’s theorem implies (...) that there can be no algorithm for producing optimal worlds, assuming an optimal world contains truth-discerning creatures, though this is not to say that such worlds cannot arise naturally. (shrink)

This paper offers an elementary proof that formal arithmetic is consistent. The system that will be proved consistent is a first-order theory R♯, based as usual on the Peano postulates and the recursion equations for + and ×. However, the reasoning will apply to any axiomatizable extension of R♯ got by adding classical arithmetical truths. Moreover, it will continue to apply through a large range of variation of the un- derlying logic of R♯, while on a simple and straightforward translation, (...) the classical first-order theory P♯ of Peano arithmetic turns out to be an exact subsystem of R♯. Since the reasoning is elementary, it is formalizable within R♯ itself; i.e., we can actually demonstrate within R♯ (or within P♯, if we care) a statement that, in a natural fashion, asserts the consistency of R♯ itself. The reader is unlikely to have missed the significance of the remarks just made. In plain English, this paper repeals Goedel’s famous second theorem. (That’s the one that asserts that sufficiently strong systems are inadequate to demonstrate their own consistency.) That theorem (or at least the significance usually claimed for it) was a mis- take—a subtle and understandable mistake, perhaps, but a mistake nonetheless. Accordingly, this paper reinstates the formal program which is often taken to have been blasted away by Goedel’s theorems— namely, the Hilbert program of demonstrating, by methods that everybody can recognize as effective and finitary, that intuitive mathematics is reliable. Indeed, the present consistency proof for arithmetic will be recognized as correct by anyone who can count to 3. (So much, indeed, for the claim that the reliability of arithmetic rests on transfinite induction up to ε0, and for the incredible mythology that underlies it.). (shrink)

A review of Hao Wang's Reflections on Kurt Goedel, emphasising Goedel's reaction against his Vienna Circle background.

According to the received view, formalism – interpreted as the thesis that mathematical truth does not outrun the consequences of our maximal mathematical theory – has been refuted by Goedel's theorem. In support of this claim, proponents of the received view usually invoke an informal argument for the truth of the Goedel sentence, an argument which is supposed to reconstruct our reasoning in seeing its truth. Against this, Field has argued in a series of papers that the principles involved (...) in this argument – when applied to our maximal mathematical theory – are unsound. This paper defends the received view by showing that there is a way of seeing the truth of the Goedel sentence which is immune to Field's strategy. (shrink)

The focus of this paper is on two attempts Sainati made to renew neo-idealistic themes by means of suggestions drawn from the famous Goedel’s Incompleteness Theorems of 1931. Sainati’s remarks on the relationship between «logo astratto » and «logo concreto» are here pursued by reference to some of Goedel’s unpublished texts.

In The Consistency of Arithmetic and elsewhere, Meyer claims to “repeal” Goedel’s second incompleteness theorem. In this paper, I review his argument, and then consider two ways of understanding it: from the perspective of mathematical pluralism and monism, respectively. Is relevant arithmetic just another legitimate practice among many, or is it a rival of its classical counterpart—a corrective to Goedel, setting us back on the path to the (One) True Arithmetic? To help answer, I sketch a few worked examples (...) from relevant mathematics, to see what a non-classical (re)formulation of mathematics might look like in practice. I conclude that, while it is unlikely that relevant arithmetic describes past and present mathematical practice, and so might be most acceptable as a pluralist enterprise, it may yet prescribe a more monistic future venture. (shrink)

Roger Penrose is justly famous for his work in physics and mathematics but he is _notorious_ for his endorsement of the Gödel argument (see his 1989, 1994, 1997). This argument, first advanced by J. R. Lucas (in 1961), attempts to show that Gödel’s (first) incompleteness theorem can be seen to reveal that the human mind transcends all algorithmic models of it¹. Penrose's version of the argument has been seen to fall victim to the original objections raised against Lucas (see (...) Boolos (1990) and for a particularly intemperate review, Putnam (1994)). Yet I believe that more can and should be said about the argument. Only a brief review is necessary here although I wish to present the argument in a somewhat peculiar form. (shrink)

We propose that the sudden emergence of metazoans during the Cambrian was due to the appearance of a complex genome architecture that was capable of computing. In turn, this made defining recursive functions possible. The underlying molecular changes that occurred in tandem were driven by the increased probability of maintaining duplicated DNA fragments in the metazoan genome. In our model, an increase in telomeric units, in conjunction with a telomerase-negative state and consequent telomere shortening, generated a reference point equivalent to (...) a non-reversible counting mechanism. (shrink)

Critical study of Michael Potter, Reason's Nearest Kin. Philosophies of Arithmetic from Kant to Carnap. Oxford University Press, Oxford, 2000. x + 305 pages.

This paper develops in certain directions the work of Meyer in [3], [4], [5] and [6]. In those works, Peano’s axioms for arithmetic were formulated with a logical base of the relevant logic R, and it was proved finitistically that the resulting arithmetic, called R♯, was absolutely consistent. It was pointed out that such a result escapes incau- tious formulations of Goedel’s second incompleteness theorem, and provides a basis for a revived Hilbert programme. The absolute consistency result used as (...) a model arithmetic modulo two. Modulo arithmetics are not or- dinarily thought of as an extension of Peano arithmetic, since some of the propositions of the latter, such as that zero is the successor of no number, fail in the former. Consequently a logical base which, unlike classical logic, tolerates contradictory theories was used for the model. The logical base for the model was the three-valued logic RM3, which has the advantage that while it is an extension of R, it is finite valued and so easier to handle. The resulting model-theoretic structure is interesting in its own right in that the set of sentences true therein consti- tutes a negation inconsistent but absolutely consistent arithmetic which is an extension of R♯. In fact, in the light of the result of [6], it is an extension of Peano arithmetic with a base of a classical logic, P♯. A generalisation of the structure is to modulo arithmetics with the same logical base RM3, but with varying moduli. We first study the properties of these arithmetics in this paper. The study is then generalised by vary- ing the logical base, to give the arithmetics RMni, of logical base RMn and modulus i. Not all of these exist, however, as arithmetical properties and logical properties interact, as we will show. The arithmetics RMni give rise, on intersection, to an inconsistent arithmetic RMω which is not of modulo i for any i. We also study its properties, and, among other results, we show by finitistic means that the more natural relevant arithmetics R♯ and R♯♯ are incomplete. In the rest of the paper we apply these techniques to several topics, particularly relevant quantum arithmetic in which we are able to show that the law of distribution remains unprovable. Aside from its intrinsic interest, we regard the present exercise as a demonstration that inconsistent theories and models are of mathematical worth and interest. (shrink)

Historically, Ehrenfest’s theorem is the first one which shows that classical physics can emerge from quantum physics as a kind of approximation. We recall the theorem in its original form, and we highlight its generalizations to the relativistic Dirac particle and to a particle with spin and izospin. We argue that apparent classicality of the macroscopic world can probably be explained within the framework of standard quantum mechanics.

SummaryJ.R. Lucas has argued that it follows from Godel's Theorem that the mind cannot be a machine or represented by any formal system. Although this notorious argument against the mechanism thesis has received considerable attention in the literature, it has not been decisively rebutted, even though mechanism is generally thought to be the only plausible view of the mind. In this paper I offer an analysis of Lucas's argument which shows that it derives its persuasiveness from a subtle confusion. (...) In particular, I examine Benacerraf's reconstruction of Lucas's argument, thereby permitting a precise re‐statement of my own analysis and reinforcement of my conclusions regarding Lucas's mistake. This examination of Benacerraf's reconstruction permits us to evaluate certain radical and highly paradoxical conclusions concerning empirical psychology which Benacerraf draws, and this, in turn, permits us to consider certain suggestive implications for the mind which may, after all, follow from Godel's Theorem.RésuméJ.R. Lucas a dérivé du théorème de Gödel la conclusion que ľesprit ne peut pas être une machine ou être représenté par un systeme formel. Bien que cet argument célèbre contre la thése du mécanisme ait éveillé un interêt considérable dans la littérature, il ňa pas été réfuté de façon décisive, même si ľopinion générate prévaut que le mécanisme est la seule conception possible de ľesprit. Dans cet article, je propose une analyse du raisonnement de Lucas qui montre que sa force persuasive provient ?on;une subtile confusion. En particulier, j'examine la reconstruction, par Benacerraf, de la démonstration de Lucas, ce qui me pérmet de préciser ma propre analyse et de confirmer mes conclusions quant àľerreur de Lucas. Cet examen de la reconstruction de Benacerraf nous fournit ľoccasion de discuter certaines conclusions radicales et hautement paradoxals qu'il en tire concernant la psychologie expérimentale et nous conduit à certaines implications suggestives concernant ľesprit et qui pourraient finalement decouler du théorème de Gödel.ZusammenfassungJ.R. Lucas hat argumentiert, aus Gödels Theorem mUsse folgen, dass der menschliche Geist keine Maschine sein oder durch ein formales System dargestellt werden kann. Obschon dieses bekannte Argument gegen die mechanistische These in der Literatur beträchtliche Beachtung gefunden hat, ist es bisher noch nie eigentlich widerlegt worden, obgleich eine mechanistische Betrachtungsweise allgemein als die einzig plausible gilt. In dieser Arbeit biete ich eine Analyse von Lucas Argument an, die zeigen soil, dass seine Überzeugungskraft auf einer subtilen Konfusion beruht. Im besonderen untersuche ich Benacerrafs Rekonstruktion des Arguments, was rnirerlaubt, meine eigene Analyse in genauer Weise zu formulieren und meine Schlussfolgerung bezüglich des von Lucas gemachten Fehlers zu erharten. Diese Untersuchung von Benacerrafs Rekonstruktion ermöglicht es, gewisse radikale und im höchsten Grad paradoxe Schlüsse, die Benacerraf hinsichtlich der empirischen Psychologie zieht, zu beurteilen. Das wiederum erlaubt uns, gewisse den Geist betreffende Implikationen zu erörtern, die tatsächlich aus Gödels Theorem folgen kUnnten. (shrink)

The Goedelian approach is discussed as a prime example of a science towards the origins. While mere self­referential objectification locks in to its own by­products, self­releasing objectification informs the formation of objects at hand and their different levels of interconnection. Guided by the spirit of Goedel's work a self­reflective science can open the road where old tenets see only blocked paths. “This is, as it were, an analysis of the analysis itself, but if that is done it forms the fundamental (...) of human science, as far as this kind of things is concerned.” G. Leibniz, ('Methodus Nova ...', 1673) . (shrink)

In his classic paper on social costs, social scientist R. H. Coase has argued that in a world without transaction costs in the "buying and selling," of social benefits and damages, resource allocation would be unaffected by a change in the apportioning of liabilities. That is, whether or not a social nuisance-causer must pay damages to those to whom he is a nuisance, will not, in an efficient economy with no transaction costs, have any effect on resource allocation. In this (...) paper, the author intends to show that there is a certain class of nuisances that are not amenable to Coasean solutions even if one grants Coase 's intuitive notions of prudence and rationality, it is considered these situations only in the short term, and considerations of equity are ignored. This class of nuisances comprises certain of those instances in which the interests of a tenant are not identical with those of his landlord. The article also purposes to shed light on the merits of an "unearned increment" tax on the value of land. (shrink)

Einsteinian gravity, of which Newtonian gravity is a part, is fraught with the problem of singularity that has been established as a theorem by Hawking and Penrose. The _hypothesis_ that founds the basis of both Einsteinian and Newtonian theories of gravity is that bodies with unequal magnitudes of masses fall with the same acceleration under the gravity of a source object. Since, the Einstein’s equations is one of the assumptions that underlies the proof of the singularity theorem, therefore, (...) the above hypothesis is implicitly one of the founding pillars of the same. In this work, I demonstrate how one can possibly write a non-singular theory of gravity which manifests that the above mentioned hypothesis is only valid in an approximate sense in the “large distance” scenario. To mention a specific instance, under the gravity of the earth, a 5 kg and a 500 kg fall with accelerations which differ by approximately $113.148\times 10^{-32}$ meter/sec $^2$ and the more massive object falls with less acceleration. Further, I demonstrate why the concept of gravitational field is not definable in the “small distance” regime which automatically justifies why the Einstein’s and Newton’s theories fail to provide any “small distance” analysis. In course of writing down this theory, I demonstrate why the continuum hypothesis as spelled out by Goedel, is undecidable. The theory has several aspects which provide the following realizations: (i) Descartes’ self-skepticism concerning exact representation of numbers by drawing lines (ii) Born’s wish of taking into account “natural uncertainty in all observations” while describing “a physical situation” by means of “real numbers” (iii) Klein’s vision of having “a fusion of arithmetic and geometry” where “a point is replaced by a small spot” (iv) Goedel’s assertion about “non-standard analysis, in some version” being “the analysis of the future”. (shrink)

In 1893, Schatunowsky showed that 30 is the largest number all of whose totatives are primes; we show that this result cannot be proved, in any form, in Chebyshev's theorem (Bertrand's postulate), even if all irreducibles are primes. Bonse's inequality is shown to be indeed weaker than Chebyshev's theorem. Schatunowsky's theorem holds in together with Bonse's inequality, the existence of the greatest prime dividing certain types of numbers, and the condition that all irreducibles be prime.

In mathematical logic, Craig’s Theorem states that any recursively enumerable theory is recursively axiomatizable. Its epistemological interest concerns its possible use as a method of eliminating “theoretical content” from scientific theories.

I demonstrate that Bell's theorem is based on circular reasoning and thus a fundamentally flawed argument. It unjustifiably assumes the additivity of expectation values for dispersion-free states of contextual hidden variable theories for non-commuting observables involved in Bell-test experiments, which is tautologous to assuming the bounds of ±2 on the Bell-CHSH sum of expectation values. Its premises thus assume in a different guise the bounds of ±2 it sets out to prove. Once this oversight is ameliorated from Bell's argument (...) by identifying the impediment that leads to it and local realism is implemented correctly, the bounds on the Bell-CHSH sum of expectation values work out to be ±2√2 instead of ±2, thereby mitigating the conclusion of Bell's theorem. Consequently, what is ruled out by any of the Bell-test experiments is not local realism but the linear additivity of expectation values, which does not hold for non-commuting observables in any hidden variable theories to begin with. I also identify similar oversight in the GHZ variant of Bell's theorem, invalidating its claim of having found an inconsistency in the premisses of the argument by EPR for completing quantum mechanics. Conceptually, the oversight in both Bell's theorem and its GHZ variant traces back to the oversight in von Neumann's theorem against hidden variable theories identified by Grete Hermann in the 1930s. (shrink)

In this short survey article, I discuss Bell’s theorem and some strategies that attempt to avoid the conclusion of non-locality. I focus on two that intersect with the philosophy of probability: (1) quantum probabilities and (2) superdeterminism. The issues they raised not only apply to a wide class of no-go theorems about quantum mechanics but are also of general philosophical interest.

The equivalence of (classical) categorial grammars and context-free grammars, proved by Gaifman [4], is a very basic result of the theory of formal grammars (an essentially equivalent result is known as the Greibach normal form theorem [1], [14]). We analyse the contents of Gaifman's theorem within the framework of structure and type transformations. We give a new proof of this theorem which relies on the algebra of phrase structures and exhibit a possibility to justify the key construction (...) used in Gaifman's proof by means of the Lambek calculus of syntactic types [15]. (shrink)

The paper considers the claim that quantum theories with a deterministic dynamics of objects in ordinary space-time, such as Bohmian mechanics, contradict the assumption that the measurement settings can be freely chosen in the EPR experiment. That assumption is one of the premises of Bell’s theorem. I first argue that only a premise to the effect that what determines the choice of the measurement settings is independent of what determines the past state of the measured system is needed for (...) the derivation of Bell’s theorem. Determinism as such does not undermine that independence . Only entanglement could do so. However, generic entanglement without collapse on the level of the universal wave-function can go together with effective wave-functions for subsystems of the universe, as in Bohmian mechanics. The paper argues that such effective wave-functions are sufficient for the mentioned independence premise to hold. (shrink)

The Jewish religious tradition summons its adherents to save life. For religious Jews preservation of life is the ultimate religious commandment. At the same time Jewish law recognizes that the agony of a moribund person may not be stretched. When the time to die has come this has to be respected. The process of dying should not needlessly be prolonged. We discuss the position of two prominent Orthodox Jewish authorities – the late Rabbi Moshe Feinstein and Rabbi J David Bleich (...) – towards the role of life-sustaining treatment in end-of-life care. From the review, the characteristic halachic and heterogeneous character of Jewish ethical reasoning appears. The specificity of Jewish dealing with ethical dilemmas in health care indicates the importance for contemporary healthcare professionals of providing care which is sensitive to a patient’s culture and worldview. (shrink)

Bell’s theorem admits several interpretations or ‘solutions’, the standard interpretation being ‘indeterminism’, a next one ‘nonlocality’. In this article two further solutions are investigated, termed here ‘superdeterminism’ and ‘supercorrelation’. The former is especially interesting for philosophical reasons, if only because it is always rejected on the basis of extra-physical arguments. The latter, supercorrelation, will be studied here by investigating model systems that can mimic it, namely spin lattices. It is shown that in these systems the Bell inequality can be (...) violated, even if they are local according to usual definitions. Violation of the Bell inequality is retraced to violation of ‘measurement independence’. These results emphasize the importance of studying the premises of the Bell inequality in realistic systems. (shrink)

We consider the relation between versions of Ramsey’s Theorem and König’s Infinity Lemma, in the absence of the axiom of choice.

This paper addresses the strength of Ramsey's theorem for pairs ($RT^2_2$) over a weak base theory from the perspective of 'proof mining'. Let $RT^{2-}_2$ denote Ramsey's theorem for pairs where the coloring is given by an explicit term involving only numeric variables. We add this principle to a weak base theory that includes weak König's Lemma and a substantial amount of $\Sigma^0_1$-induction (enough to prove the totality of all primitive recursive functions but not of all primitive recursive functionals). (...) In the resulting theory we show the extractability of primitive recursive programs and uniform bounds from proofs of $\forall\exists$-theorems. There are two components of this work. The first component is a general proof-theoretic result, due to the second author, that establishes conservation results for restricted principles of choice and comprehension over primitive recursive arithmetic PRA as well as a method for the extraction of primitive recursive bounds from proofs based on such principles. The second component is the main novelty of the paper: it is shown that a proof of Ramsey's theorem due to Erdős and Rado can be formalized using these restricted principles. So from the perspective of proof unwinding the computational content of concrete proofs based on $RT^2_2$ the computational complexity will, in most practical cases, not go beyond primitive recursive complexity. This even is the case when the theorem to be proved has function parameters f and the proof uses instances of $RT^2_2$ that are primitive recursive in f. (shrink)

We argue that Löb's Theorem implies a limitation on mechanism. Specifically, we argue, via an application of a generalized version of Löb's Theorem, that any particular device known by an observer to be mechanical cannot be used as an epistemic authority (of a particular type) by that observer: either the belief-set of such an authority is not mechanizable or, if it is, there is no identifiable formal system of which the observer can know (or truly believe) it to (...) be the theorem-set. This gives, we believe, an important and hitherto unnoticed connection between mechanism and the use of authorities by human-like epistemic agents. (shrink)

This paper initiates the reverse mathematics of social choice theory, studying Arrow's impossibility theorem and related results including Fishburn's possibility theorem and the Kirman–Sondermann theorem within the framework of reverse mathematics. We formalise fundamental notions of social choice theory in second-order arithmetic, yielding a definition of countable society which is tractable in RCA0. We then show that the Kirman–Sondermann analysis of social welfare functions can be carried out in RCA0. This approach yields a proof of Arrow's (...) class='Hi'>theorem in RCA0, and thus in PRA, since Arrow's theorem can be formalised as a Π01 sentence. Finally we show that Fishburn's possibility theorem for countable societies is equivalent to ACA0 over RCA0. (shrink)

Bayes's theorem is a tool for assessing how probable evidence makes some hypothesis. The papers in this volume consider the worth and applicability of the theorem. Richard Swinburne sets out the philosophical issues. Elliott Sober argues that there are other criteria for assessing hypotheses. Colin Howson, Philip Dawid and John Earman consider how the theorem can be used in statistical science, in weighing evidence in criminal trials, and in assessing evidence for the occurrence of miracles. David Miller (...) argues for the worth of the probability calculus as a tool for measuring propensities in nature rather than the strength of evidence. The volume ends with the original paper containing the theorem, presented to the Royal Society in 1763. (shrink)

Applying Weglorz' mode s of set theory without the axiom of choice, we investigate Arrow-type social we fare functions for infinite societies with restricted coalition algebras. We show that there is a reasonable, nondictatorial social welfare function satisfying “finite discrimination”, if and only if in Weglorz' mode there is a free ultrafilter on a set representing the individuals.

G¨odel’s a theorem concerns an arithmetical statement and the truth of this statement does not depend on self-reference; nevertheless its interpretation is of tremendous interest. G¨odel’s theorem allows one to conclude that formal arithmetic is not axiomatizable. But there is another very interesting logico-philosophical result: the possibility of a statement to exist such that it is improvable in the object-theory and at the same time its truth is provable in the metatheory. It seems that in the real history (...) G¨odel’s theorem was absolutely unexpected, and even in retrospect, now it is difficult to point out logico-philosophical reasons for the above possibility. In my opinion though it was still conceivable at that time, but unfortunately the philosophers and logicians missed to notice it. In the terms of recursive arithmetic Hilbert’s programme can be formulated as follows: Build a decidable, complete, consistent logical system such that the class of derivable statements coincides with the class of the mathematical statements which are intuitively true; prove this equivalence by means of an arithmetization of the theory in the framework of the recursive arithmetic! Now, if the system is so strong as to admit the construction of the classical mathematics, then in particular it will contain the classical arithmetic. But Hilbert’s requirements reduce to a proof with the means of the recursive arithmetic, by arithmetization. So, on one hand the system will contain the arithmetic and, on the other hand it will be expressible in it, in order words the system will include its own syntax, as well as its own semantics. Now it is clear that Hilbert’s programme contains a possibility of a situation similar to some well known semantic antinomies, situations of self-reference. (shrink)