This is a commentary on Andrea Lavazza and Murilo Vilaça's article "Human Extinction and AI: What We Can Learn from the Ultimate Threat" (Lavazza & Vilaça, 2024). I discuss the potential concern that their proposal to create artificial successors to "insure" against the tragedy of human extinction might mean being too quick to accept that catastrophic prospect as inevitable, rather than single-mindedly focusing on avoiding it. I also consider the question of the value that we might reasonably assign to such (...) a future of artificial successors under the assumption that they do not enjoy phenomenal consciousness. (shrink)

In this paper, we investigate the extent to which techniques used in [10], [2], and [3]—developed to prove coloring theorems at successors of singular cardinals of uncountable cofinality—can be extended to cover the countable cofinality case.

Abstract.We investigate the existence of strong colorings on successors of singular cardinals. This work continues Section 2 of [1], but now our emphasis is on finding colorings of pairs of ordinals, rather than colorings of finite sets of ordinals.

We consider successor-invariant first-order logic (FO + succ)inv, consisting of sentences Φ involving an “auxiliary” binary relation S such that (.

We extend BDDs for plain propositional logic to the fragment of first order logic, consisting of quantifier free logic with zero, successor and equality. We allow equations with zero and successor in the nodes of a BDD, and call such objects -BDDs. We extend the notion of Ordered BDDs in the presence of zero, successor and equality. -BDDs can be transformed to equivalent Ordered -BDDs by applying a number of rewrite rules until a normal form is reached. (...) All paths in these ordered -BDDs represent satisfiable conjunctions. The major advantage of transforming a formula to an equivalent Ordered -BDD is that on the latter it can be observed in constant time whether the formula is a tautology, a contradiction, or just satisfiable. (shrink)

We show that, like singular cardinals, and weakly compact cardinals, Jensen's core model K for measures of order zero [4] calculates correctly the successors of Jónsson cardinals, assuming $O^{Sword}$ does not exist. Namely, if $\kappa$ is a Jónsson cardinal then $\kappa^+ = \kappa^{+K}$ , provided that there is no non-trivial elementary embedding $j:K \longrightarrow K$ . There are a number of related results in ZFC concerning $\cal{P}(\kappa)$ in V and inner models, for $\kappa$ a Jónsson or singular cardinal.

Humans possess two nonverbal systems capable of representing numbers, both limited in their representational power: the first one represents numbers in an approximate fashion, and the second one conveys information about small numbers only. Conception of exact large numbers has therefore been thought to arise from the manipulation of exact numerical symbols. Here, we focus on two fundamental properties of the exact numbers as prerequisites to the concept of EXACT NUMBERS : the fact that all numbers can be generated by (...) a successor function and the fact that equality between numbers can be defined in an exact fashion. We discuss some recent findings assessing how speakers of Munduruc (an Amazonian language), and young Western children (3-4 years old) understand these fundamental properties of numbers. (shrink)

The realism/antirealism controversy has gone on for centuries, and gives every indication that it will continue to go on for centuries. Dismayed, I take a closer look at it. I find that the question it poses--very roughly, whether scientific knowledge is true (approximately true, put forward as true, etc.) or only useful (empirically adequate, a convenient method of representation, etc.)--actually suppresses socially critical thought and discussion about science (e.g., concerning whether scientific knowledge is sexist or racist or socially harmful in (...) other ways, or whether scientific knowledge is useful for achieving the goals we have or the goals we ought to have). I find, as well, that two of the most important responses to the realism/antirealism controversy--that which construes it as concerned with science's aims and that which construes it as concerned with science's results--fail to make sufficient empirical or normative contact with science. As a consequence, they provide representations of science that either do not help scientists (or nonscientists) make informed decisions about science, or actually hinder these individuals from doing so. I conclude that we should either stop engaging in the realism/antirealism controversy entirely, or else engage in it in a more socially responsible way--by gathering the right kinds of empirical and normative data, and framing more helpful versions of the questions we want to ask. I end by responding to the strong objections sure to be voiced by my colleagues in philosophy of science. (shrink)

I prove that there is a recursive function T that does the following: Let X be transitive and rudimentarily closed, and let X ′ be the closure of X ∪ {X } under rudimentary functions. Given a Σ0-formula φ and a code c for a rudimentary function f, T is a Σω-formula such that for any equation image ∈ X, X ′ ⊧ φ [f ] iff X ⊧ T [equation image]. I make this precise and show relativized versions of (...) this. As an application, I prove that under certain conditions, if Y is the Σω extender ultrapower of X with respect to some extender F that also is an extender on X ′, then the closure of Y ∪ {Y } under rudimentary functions is the Σ0 extender ultrapower of X′ with respect to F, and the ultrapower embeddings agree on X. (shrink)

Hegel remains widely known but largely unread in Anglo-American philosophy. Although the earlier hostility to his thought in these circles has begun to fade, Hegel still remains for many philosophers a more or less peripheral figure, somebody to be taught once other subjects in the philosophy department have been covered. This is partly because of his obscure style and mostly because of the standard picture of Hegel that remains in the psychic geography of many academic philosophers. Hegel is conceived as (...) the last great thinker who tried to fashion a unified systematic picture of God, man and the world through something called dialectic. On this standard view, Hegel is seen as arguing for a kind of grand Spirit who is gradually coming to self-consciousness by struggling through the contradictions He has created, using people as instruments for His coming to self-consciousness, until finally He succeeds somewhere in Berlin. Spirit-God comes to full self-consciousness and as parts of this grand Spirit-God, we too come to a full awareness of what we really are. This kind of grand metaphysical cosmology and theodicy does not fit the more skeptical temperaments of many twentieth-century academic thinkers. (shrink)

My interest in China was rekindled several years ago by an invitation to a conference, "Modernization, Globalization and China's Path to Economic Development," to he held in Hangzhou, July, 2002. The conference was organized by Cao Tian Yu, a philosopher of science at Boston University and his wife Lin Chun of the London School of Economics--both deeply concerned about the future of China. It was attended by a number of Western Leftists (Samir Amin, Perry Anderson, Robin Blackburn and myself), by (...) China specialist Joseph Fewsmith, by representatives from Singapore, Taiwan and India, by representatives from China's developing "New Left," (among them Wang Hui, whose book China's New Order was recently published by Harvard University Press1), by the president of Hangzhou College of Commerce (where the event was held) and by three retired, once prominent government officials, among them Du Runsheng, a principal architect of China's agricultural reform of the late 70s, early 80s. (shrink)

If [Formula: see text] is regular and [Formula: see text], then the existence of a weakly presaturated ideal on [Formula: see text] implies [Formula: see text]. This partially answers a question of Foreman and Magidor about the approachability ideal on [Formula: see text]. As a corollary, we show that if there is a presaturated ideal [Formula: see text] on [Formula: see text] such that [Formula: see text] is semiproper, then CH holds. We also show some barriers to getting the tree (...) property and a saturated ideal simultaneously on a successor cardinal from conventional forcing methods. (shrink)

L’articolo passa in rassegna le diverse tradizioni che trattano del passaggio, nell’Accademia del 339 a. C., dello scolarcato da Speusippo all’homo novus Senocrate. La nostra tradizione in proposito è alquanto tardiva: essa prende la sua origine da Filodemo, che dipenderebbe da fonti relativamente vicine a Platone ed ai suoi immediati discepoli, del III o II secolo. Pur essendo autore di età augustea, Filodemo non sembra conoscere certe vicissitudini subite dalla tradizione platonico-accademica. La tradizione seguente è notevolmente più tardiva: scendiamo all’età (...) dei medioplatonici, e troviamo documenti rappresentati, oltre che da alcune scarne notizie, dalle Epistole socratiche cui non potremmo conferire altra antichità che quella della nuova sofistica, in età di Adriano. Infine, esiste una tradizione conciliatoria che ci viene indicata dalle Vite di Aristotele nella tarda antichità e che trova eco nello scritto di alcuni commentatori neoplatonici; sulle fonti di ciò possiamo avanzare puramente delle ipotesi. (shrink)

Most accounts of our knowledge of the successor axiom claim that this is based on the procedure of adding one. While they usually don’t claim to provide an account of how children actually acquire this knowledge, one may well think that this is how they get that knowledge. I argue that when we look at children’s responses in interviews, the time when they learn the successor axiom and the intermediate learning stages they find themselves in, that there is (...) an empirically viable alternative. I argue that they could also learn it on the basis of a method that has to do with the structure of the numeral system. Specifically, that they (1) use the syntactic structure of the numeral system and (2) attend to the leftmost digits, the one with the highest place-value. Children can learn that this is a reliable method of forming larger numbers by combining two elements. First, a grasp of the syntactic structure of the numeral system. That way they know that the leftmost digit receives the highest value. Second, an interpretation of numerals as designating cardinal values, so that they also realise that increasing or adding digits on the lefthand side of a numeral produces a larger number. There are thus two, currently equally well-supported, ways in which children might learn that there are infinitely many natural numbers. (shrink)

In his “Reply to Williams” (2013), a response to my “David-Hillel Ruben’s ‘Traditions and True Successors’: A Critical Reply.” (2013), David Ruben reports that there is much that we disagree about concerning the nature of true succession. I am not entirely persuaded by what he says of these disagreements.

The paper is concerned with the existence of a universal graph at the successor of a strong limit singular μ of cofinality ℵ0. Starting from the assumption of the existence of a supercompact cardinal, a model is built in which for some such μ there are $\mu^{++}$ graphs on μ+ that taken jointly are universal for the graphs on μ+, while $2^{\mu^+} \gg \mu^{++}$ . The paper also addresses the general problem of obtaining a framework for consistency results at (...) the successor of a singular strong limit starting from the assumption that a supercompact cardinal κ exists. The result on the existence of universal graphs is obtained as a specific application of a more general method. (shrink)

A successor relation between theories T und T₁ express that T₁, the successor of T, has justifiably superseded T. In physics, for instance, Newton's theory of gravitation has superseded Kepler's laws and, in turn, Einstein's theory has become the successor of Newton's. By now there is no agreement on how a general concept of successor relation would have to be construed. In the present paper attention is drawn to two types of such relations, one deductive the (...) other confirmatory. It seems that what most people are looking for is the deductive type of relation. However, a deductive successor relation has to be justified by showing that if it holds then another, confirmatory relation holds, saying roughly that the actual empirical success of the deducing theory is at least as good as the corresponding success of the deduced theory. Under certain restrictions such a justification is given in the paper. (shrink)

This essay considers the implications for the powers metaphysic of the no-successor problem: As there are no successors in the set of real numbers, one state cannot occur just after another in continuous time without there being a gap between the two. I show how the no-successor problem sets challenges for various accounts of the manifestation of powers. For powers that give rise to a manifestation that is a new state, the challenge of no-successors is similar to that (...) faced on Bertrand Russell's analysis by causal relations. Powers whose manifestation is a processes and powers that manifest through time are challenged differently. To avoid powers appearing enigmatic, these challenges should be addressed, and I point to some possible ways this might be achieved. A prerequisite for addressing these challenges is a careful focus on the nature and timing of the manifesting and manifestation of powers. (shrink)

Assuming some large cardinals, a model of ZFC is obtained in which $\aleph_{\omega+1}$ carries no Aronszajn trees. It is also shown that if $\lambda$ is a singular limit of strongly compact cardinals, then $\lambda^+$ carries no Aronszajn trees.

Suppose that λ is the successor of a singular cardinal μ whose cofinality is an uncountable cardinal κ. We give a sufficient condition that the club filter of λ concentrating on the points of cofinality κ is not λ+-saturated.1 The condition is phrased in terms of a notion that we call weak reflection. We discuss various properties of weak reflection. We introduce a weak version of the ♣-principle, which we call ♣*−, and show that if it holds on a (...) stationary subset S of λ, then no normal filter on S is λ+-saturated. Under the above assumptions, ♣*− is true for any stationary subset S of λ which does not contain points of cofinality κ. For stationary sets S which concentrate on points of cofinality κ, we show that ♣*− holds modulo an ideal obtained through the weak reflection. (shrink)

We prove that if μ is a regular cardinal and ℙ is a μ-centered forcing poset, then ℙ forces that $(I[\mu ^{ + + } ])^V $ generates I[µ⁺⁺] modulo clubs. Using this result, we construct models in which the approachability property fails at the successor of a singular cardinal. We also construct models in which the properties of being internally club and internally approachable are distinct for sets of size the successor of a singular cardinal.

This paper is devoted to the proof of the following upward categoricity theorem: Let K be a tame abstract elementary class with amalgamation, arbitrarily large models, and countable Löwenheim-Skolem number. If K is categorical in ‮א‬₁ then K is categorical in every uncountable cardinal. More generally, we prove that if K is categorical in a successor cardinal λ⁺ then K is categorical everywhere above λ⁺.

From large cardinals we obtain the consistency of the existence of a singular cardinal κ of cofinality ω at which the Singular Cardinals Hypothesis fails, there is a bad scale at κ and κ ++ has the tree property. In particular this model has no special κ +-trees.

We establish that for every computably enumerable (c.e.) Turing degree b the upper cone of c.e. Turing degrees determined by b is the degree spectrum of the successor relation of some computable linear ordering. This follows from our main result, that for a large class of linear orderings the degree spectrum of the successor relation is closed upward in the c.e. Turing degrees.

We prove that from categoricity in λ+ we can get categoricity in all cardinals ≥ λ+ in a χ-tame abstract elementary classe [Formula: see text] which has arbitrarily large models and satisfies the amalgamation and joint embedding properties, provided [Formula: see text] and λ ≥ χ. For the missing case when [Formula: see text], we prove that [Formula: see text] is totally categorical provided that [Formula: see text] is categorical in [Formula: see text] and [Formula: see text].

What constitutes numerically one and the same tradition diachronically, at different times? This question is the focus of often violent dispute in societies. Is it capable of a rational resolution? Many accounts attempt that resolution with a diagnosis of ambiguity of the disputed concept-Islam, Marxism, or democracy for example. The diagnosis offered is in terms of vagueness, namely the vague criteria for sameness or similarity of central beliefs and practices.

REF is the statement that every stationary subset of a cardinal reflects, unless it fails to do so for a trivial reason. The main theorem, presented in Sect. 0, is that under suitable assumptions it is consistent that REF and there is a κ which is κ+n -supercompact. The main concepts defined in Sect. 1 are PT, which is a certain statement about the existence of transversals, and the “bad” stationary set. It is shown that supercompactness (and even the failure (...) of PT) implies the existence of non-reflecting stationary sets. E.g., if REF then for manyλ ⌝ PT(λ, ℵ1). In Sect. 2 it is shown that Easton-support iteration of suitable Levy collapses yield a universe with REF if for every singular λ which is a limit of supercompacts the bad stationary set concentrates on the “right” cofinalities. In Sect. 3 the use of oracle c.c. (and oracle proper—see [Sh-b, Chap. IV] and [Sh 100, Sect. 4]) is adapted to replacing the diamond by the Laver diamond. Using this, a universe as needed in Sect. 2 is forced, where one starts, and ends, with a universe with a proper class of supercompacts. In Sect. 4 bad sets are handled in ZFC. For a regular λ {δ<+ : cfδ<λ} is good. It is proved in ZFC that ifλ=cfλ>ℵ1 then {α<+ : cfα<λ} is the union of λ sets on which there are squares. (shrink)

(1996). Democratization in the successor states to socialist Yugoslavia. The European Legacy: Vol. 1, Fourth International Conference of the International Society for the study of European Ideas, pp. 459-473.

All three books reviewed here are turning over again for us the pages of perennially irresistible thinkers whose ideas never cease to hold us transfixed; all three are inviting us to notice that the material that we thought we knew has got more to do with what Nehamas calls 'the art of living' than we might have realised; and all three are making space for attitudes, responses and areas of self-understanding that are, by traditional classifications, irrational and hence sometimes inadequately (...) acknowledged by philosophy as we usually understand it. And, of course, all three are juxtaposing thinkers from the ancient world with major figures from recent and early modern times. (shrink)

The term "public lover" has been a staple term in the Chinese language. Yet another term—"public concern" —has only lately become part of our language. This is largely because China lacks any sense of pluralism and any sense of limitations imposed on rulers by the common people. At present, if we consider both magnitude and timeliness, the "public concern" of both the Chinese people and foreigners is as follows: Will Jiang Zemin, whom Deng Xiaoping appointed, come to the same end (...) as Hua Guofeng, whom Mao Zedong appointed? (shrink)