I present an outline of the Omega Point theory, which is a model for an omnipresent, omniscient, omnipotent, evolving, personal God who is both transcendent to spacetime and immanent in it, and who exists necessarily. The model is a falsifiable physical theory, deriving its key concepts not from any religious tradition but from modern physical cosmology and computer science; from scientific materialism rather than revelation. Four testable predictions of the model are given. The theory assumes that thinking is (...) a purely physical process of the brain, and that personality dies with the brain. Nevertheless, I show that the Omega Point theory suggests a future universal resurrection of the dead very similar to the one predicted in the Judeo‐Christian‐Islamic tradition. The notions of “grace” and the “beatific vision” appear naturally in the model. (shrink)

An attempt to discover what can be learned from the recent work of Frank Tipler on the Omega Point theory requires an analysis of his framework of understanding from scientific, philosophical, and theological perspectives. A critique of his crucial ideas, and of the salient points raised by some of his critics, can then be undertaken within the compass of his strengths. A critique of the critiques of Tipler's work allows one to evaluate the extent and limitations of his (...) contributions. (shrink)

An Omega Point Theory says that reality is making progress from some initial state to some final state. It moves from some Alpha Point (the initial state) to some Omega Point (the final state). The progress is an increase in some quality. For example, reality is making progress from the chaotic to the orderly; or it is making progress from the simple to the complex; or from the mindless to the mental; or from evil to (...) good. Here we focus on the Omega Point theory of Peirce. An Omega Point Theory is an evolutionary cosmology – it says that the universe is evolving from the Alpha Point to the Omega Point. Peirce refers to his evolutionary cosmology as a Cosmogonic Philosophy would suppose that in the beginning – infinitely remote – there was a chaos of unpersonalized feeling, which being without connection or regularity would properly be without existence. This feeling, sporting here and there in pure arbitrariness, would have started the germ of a generalizing tendency. Its other sportings would be evanescent, but this would have a growing virtue. Thus the tendency to habit would be started; and from this, with the other principles of evolution, all the regularities of the universe would be evolved. At any time, however, an element of pure chance survives and will remain until the world becomes an absolutely perfect, rational, and symmetrical system, in which mind is at last crystallized in the infinitely distant future. (Peirce, Collected Papers , 6.33). (shrink)

This article focuses on the Omega Point, an essential concept in Teilhard de Chardin’s evolutionary metaphysics. In certain passages about the Omega Point, Teilhard mentions the primeval atom hypothesis, a theory about the beginning of the universe proposed by Georges Lemaître, another contemporary Jesuit priest who was also a scientist. Although Teilhard and Lemaître are essentially evolutionists, besides being Jesuit priests, their evolutionary metaphysics and their philosophies of science are radically divergent, and two important differences are (...) presented here – about the eventual cosmic determinism and about the relationship between scientific knowledge and religious beliefs. This article also shows how Teilhard’s original concept was projected into two cosmological theories in which it reveals an essential transfiguration by losing its initial anthropic and religious matrix – the cosmological anthropic principle, proposed by physicists John Barrow and Frank Tipler, and the Omega Point Theory proposed later by Frank Tipler. (shrink)

Summary. The prerequisites of this study have three interwoven sources, the natural sciences and philosophical and socio-political ones. They are trends in the way of being of a modern, technogenic civilization. The COVID-19 pandemic caused significant damage to the image of the omnipotent techno-science that has developed in the mentality of this sociocultural type.Our goal was to study the co-evolutionary nature of this phenomenon as a natural consequence of the nature of the evolutionary strategy of our biological species. Technological civilization (...) as a cultural-civilizational type is one of the evolutionary options of Homo sapiens, the ontological basis of which is a stable evolutionary strategy (SESH). The latter is capable to spontaneous development and is consisting from biological, sociocultural, and techno-rationalistic modules. Evolutionary risks inextricably linked with it. At present, humankind has entered an era when its existence should be ensured by the meta-system engineering of its own cultural and ecological niche. By such artificially created elements, it implies the design and implementation of ecological systems of various levels of complexity, from the design of elements of such systems (which are biological individuals, populations and species) to the design of a global ecological system (biosphere). At the same time, there is a sharp jump in the value of anthropogenic and technological risk due to a sharp increase in the instability of the structure of ecological systems. We are witnessing a third wave of breaking and reconstruction of the existing relations between the elements of such systems after the Neolithic revolution and great geographical discoveries. Technological innovations led to sharp violations of normal social ecodynamics, leading to outbreaks of new infections, in particular. Theuncontested development of all sectors of controlled evolution technology is the way out of the last such crisis and the prevention of similar ones in the future. There are global nature of the organization of modern technological civilization and the transformation of the biosphere not so much into the noosphere (according to Vernadsky) but into the technosphere that determines the effect of a cascade reaction: any local co-adaptive conflict between elements of a cultural, socio-ecological niche tends to turnglobal and systemic problems. Humanity is constantly forced to make a choice between the stability and adaptability of the technosphere. (shrink)

Since the beginning of the scientific revolution, believers have had to reconcile their beliefs with science. This has always proved difficult. If an all-perfect God created the universe and its physical laws, why would he have to step in to perform miracles and answer prayers? If, as all the evidence indicates, the universe, including humans and their brains, is matter and nothing more, how can we possibly live forever? Theologians try hard, but never come up with satisfactory answers.

Josef Pieper (1904–1997) wrote his autobiography in three distinct volumes. Composing one volume of his complete works in his Gesammelte Werke (Felix Meiner Verlag, 1995) edited by Berthold Wald, he writes on his early (1904–1945), middle (1945–1964), and late years (1964–1985). Between Ignatius Press and St. Augustine’s Press, Pieper’s autobiographical writings have been translated into English over the last three decades. The first volume, Noch wusste es niemand: Autobiographische Aufzeichnungen 1904–1945 (Kosel-Verlag, 1979), was translated into English as No One Could (...) Have Known: An Autobiography: The Early Years 1904–1945 (Ignatius, 1987). Three decades later the second volume, Noch nicht aller Tage Abend: Autobiographische Aufzeichnungen 1945–1964 (Kosel-Verlag, 1979), was published in English as Not Yet the Twilight: An Autobiography 1945–1964 (St. Augustine’s Press, 2017). Finally, the third volume, Eine Geschichte wie ein Strahl: Autobiographische Aufzeichnungen seit 1964 (Kosel-Verlag, 1988), has been published under the title A Journey to Point Omega: Autobiography from 1964 (St. Augustine’s Press, 2020). (shrink)

In his book The Physics of Immortality. Frank Tipler has broken a longstanding intellectual taboo by dealing as a physicist with the theological themes of God and immortality, as well by arguing that theology can provide material for concept formation in the field of physics. His work on the anthropic principle convinced Tipler that, since the emergence of intelligent life is of the essence of the universe as a whole, the future of life is of fundamental significance. His Omega (...) Point theory takes theological theories of the future's significance seriously from a scientific point of view. Theories of computers play a central role in Tipler's theory of immortality, and even though many critics have misunderstood his thrust in these theories, they are worthy of further exploration. Perhaps Tipler's most important contribution is his insistence that the world as described by physics is more open to interaction with biblical and theological perspectives than is often believed. (shrink)

The main topics of this thesis are cardinal invariants, P -points and MAD families. Cardinal invariants of the continuum are cardinal numbers that are bigger than $\aleph _{0}$ and smaller or equal than $\mathfrak {c}.$ Of course, they are only interesting when they have some combinatorial or topological definition. An almost disjoint family is a family of infinite subsets of $\omega $ such that the intersection of any two of its elements is finite. A MAD family is a maximal (...) almost disjoint family. An ultrafilter $\mathcal {U}$ on $\omega $ is called a P-point if every countable $\mathcal {B\subseteq U}$ there is $X\in $ $\mathcal {U}$ such that $X\setminus B$ is finite for every $B\in \mathcal {B}.$ This kind of ultrafilters has been extensively studied, however there is still a large number of open questions about them.In the preliminaries we recall the principal properties of filters, ultrafilters, ideals, MAD families and cardinal invariants of the continuum. We present the construction of Shelah, Mildenberger, Raghavan, and Steprāns of a completely separable MAD family under $\mathfrak {s\leq a}.$ None of the results in this chapter are due to the author.The second chapter is dedicated to a principle of Sierpiński. The principle $\left $ of Sierpiński is the following statement: There is a family of functions $\left \{ \varphi _{n}:\omega _{1}\longrightarrow \omega _{1}\mid n\in \omega \right \} $ such that for every $I\in \left [ \omega _{1}\right ] ^{\omega _{1}}$ there is $n\in \omega $ for which $\varphi _{n}\left [ I\right ] =\omega _{1}.$ This principle was recently studied by Arnie Miller. He showed that this principle is equivalent to the following statement: There is a set $X=\left \{ f_{\alpha }\mid \alpha \alpha $ then $f_{\beta }\cap g$ is infinite. Miller showed that the principle of Sierpiński implies that non $\left =\omega _{1}.$ He asked if the converse was true, i.e., does non $\left =\omega _{1}$ imply the principle $\left $ of Sierpiński? We answer his question affirmatively. In other words, we show that non $\left =\omega _{1}$ is enough to construct an $\mathcal {IE}$ -Luzin set. It is not hard to see that the $\mathcal {IE}$ -Luzin set we constructed is meager. This is no coincidence, because with the aid of an inaccessible cardinal, we construct a model where non $\left =\omega _{1}$ and every $\mathcal {IE}$ -Luzin set is meager.The third chapter is dedicated to a conjecture of Hrušák. Michael Hrušák conjectured the following: Every Borel cardinal invariant is either at most non $\left $ or at least cov $\left $. Although the veracity of this conjecture is still an open problem, we were able to obtain some partial results: The conjecture is false for “Borel invariants of $\omega _{1}^{\omega }$ ” nevertheless, it is true for a large class of definable invariants. This is part of a joint work with Michael Hrušák and Jindřich Zapletal.In the fourth chapter we present a survey on destructibility of ideals and MAD families. We prove several classic theorems, but we also prove some new results. For example, we show that every almost disjoint family of size less than $\mathfrak {c}$ can be extended to a Cohen indestructible MAD family is equivalent to $\mathfrak {b=c}$. A MAD family $\mathcal {A}$ is Shelah–Steprāns if for every $X\subseteq \left [ \omega \right ] ^{<\omega }\setminus \left \{ \emptyset \right \} $ either there is $A\in \mathcal {I}\left $ such that $s\cap A\neq \emptyset $ for every $s\in X$ or there is $B\in \mathcal {I}\left $ that contains infinitely many elements of X $ denotes the ideal generated by $\mathcal {A}$ ). This concept was introduced by Raghavan which is connected to the notion of “strongly separable” introduced by Shelah and Steprāns. We prove that Shelah–Steprāns MAD families have very strong indestructibility properties: Shelah–Steprāns MAD families are indestructible for “many” definable forcings that does not add dominating reals. According to the author’s best knowledge, this is the strongest notion that has been considered in the literature so far. In spite of their strong indestructibility, Shelah–Steprāns MAD families can be destroyed by a ccc forcing that does not add unsplit or dominating reals. We also consider some strong combinatorial properties of MAD families and show the relationships between them.The fifth chapter is one of the most important chapters in the thesis. A MAD family $\mathcal {A}$ is called $+$ -Ramsey if every tree that branches into $\mathcal {I}\left $ -positive sets has an $\mathcal {I}\left $ -positive branch. Michael Hrušák’s first published question is the following: Is there a $+$ -Ramsey MAD family? It was previously known that such families can consistently exist. However, there was no construction of such families using only the axioms of ZFC. We solve this problem by constructing such a family without any extra assumptions.In the fourth and fifth chapters, we introduce several notions of MAD families, in the sixth chapter we prove several implications and non implications between them. We construct several MAD families with different properties.In the seventh chapter we build models without P -points. We show that there are no P -points after adding Silver reals either iteratively or by the side by side product. These results have some important consequences: The first one is that is its possible to get rid of P -points using only definable forcings. This answers a question of Michael Hrušák. We can also use our results to build models with no P -points and with arbitrarily large continuum, which was also an open question. These results were obtained with David Chodounský.prepared by Osvaldo Guzmán GonzálezE-mail : [email protected] : https://arxiv.org/abs/1810.09680. (shrink)

In this paper we establish the proof-theoretic equivalence of (i) $\hbox {\sf ATR}$ and $\widehat{\hbox{\sf ID}}_{\omega}$ , (ii) $\hbox{\sf ATR}_0+ (\Sigma^1_1-\hbox{\sf DC})$ and $\widehat{\hbox {\sf ID}}_{<\omega^\omega} , and (iii) $\hbox {\sf ATR}+(\Sigma^1_1-\hbox{\sf DC})$ and $\widehat{\hbox {\sf ID}}_{<\varepsilon_0} $.

In the first part of this paper we discuss some aspects of Detlefsen's attempt to save Hilbert's Program from the consequences of Godel's Second Incompleteness Theorem. His arguments are based on his interpretation of the long standing and well-known controversy on what, exactly, finitistic means are. In his paper [1] Detlefsen takes the position that there is a form of the ω-rule which is a finitistically valid means of proof, sufficient to prove the consistency of elementary number theory Z. On (...) the other hand, he claims that Z with its first-order logic is not strong enough to allow a formalization of such an ω-rule. This would explain why the unprovability of $\operatorname{Con}(Z)$ in Z does not imply that the consistency of Z cannot be proved by finitistic means. We show that Detlefsen's proposal is unacceptable as originally formulated in [1], but that a reasonable modification of the rule he suggest leads to a partial program already studied for many years. We investigate the scope of such a program in terms of proof-theoretic reducibilities. We also show that this partial program encompasses mathematically important theories studied in the "Reverse Mathematics" program. In order to investigate the provability with such a modified rule, we define new consistency and provability predicates which are weaker than the usual ones. We then investigate their properties, including a few that have no apparent philosophical significance but compare interestingly with the properties of the program based on the iteration of our ω-rule. We determine some of the limitations of such programs, pointing out that these limitations partly explain why partial programs that have been successfully carried out use quite different and substantially more radical extensions of finitistic methods with more general forms of restricted reasoning. (shrink)

The interplay between ultrafilters and unbounded subsets of \ with the order \ of strict eventual domination is studied. Among the tools are special kinds of non-principal ultrafilters on \. These include simple P-points; that is, ultrafilters with a base that is well-ordered with respect to the reverse of the order \ of almost inclusion. It is shown that the cofinality of such a base must be either \, the least cardinality of \-unbounded set, or \, the least cardinality of (...) a \-cofinal set. The small uncountable cardinal \ is introduced. Consequences of \ and of \ are explored; in particular, both imply \. Here \ is the reaping number, and is also the least cardinality of a \-base for a free ultrafilter. Both of these inequalities are shown to occur if there exist simple P-points of different cofinalities and there exist simple \-points and \-points), but this is a long-standing open problem. Six axioms on nonprincipal ultrafilters on \ and the relationships between them are discussed along with various models of set theory in which one or more are known to hold. The strongest of these, Axiom 1, is that for every free ultrafilter \ and for every \-unbounded \-chain C of increasing functions in \, C is also unbounded in the ultraproduct \. The other axioms replace one or both quantifiers with “there exists.” The negation of Axiom 3 in a model provides a family of normal sequentially compact spaces whose product is not countably compact. The question of whether such a family exists in ZFC, even with “normal” weakened to “regular”, is a famous unsolved problem of set-theoretic topology, known as the Scarborough–Stone problem. (shrink)

We give modest upper bounds for consistency strengths for two well-studied combinatorial principles. These bounds range at the level of subcompact cardinals, which is significantly below a κ+-supercompact cardinal. All previously known upper bounds on these principles ranged at the level of some degree of supercompactness. We show that by using any of the standard modified Prikry forcings it is possible to turn a measurable subcompact cardinal into ℵω and make the principle □ℵω,<ω fail in the generic extension. We also (...) show that by using Lévy collapse followed by standard iterated club shooting it is possible to turn a subcompact cardinal into ℵ2 and arrange in the generic extension that simultaneous reflection holds at ℵ2, and at the same time, every stationary subset of ℵ3 concentrating on points of cofinality ω has a reflection point of cofinality ω1. (shrink)

This critical review focuses on Smith's recent revision of his 1988 book: Teilhard and the New Religion: A Thorough Analysis of the Teachings of Pierre Teilhard de Chardin. Curiously, both the revised and original works have been largely ignored. Unfortunately, sometimes the best way to silence critics is to ignore them. This work will look into some of the primary concepts of Teilhard's "scientific theology" and assess Smith's evaluation, including: evolution, the law of complexity and consciousness, the Creative Union, and (...) the Omega Point. (shrink)

Paul Tillich and Pierre Teilhard de Chardin both have made contributions to a theology of evolution. In a 2002 essay John Haught expresses doubt that Tilllich's rather classical theology of “being” is radical enough to account for the “becoming” of evolution. Tillich's ontology of being includes the polarity of form and dynamics. Dynamics is the potentiality of being, that is, becoming. Tillich's dynamic dialectic of being and nonbeing is a more descriptive metaphor for the five mass extinctions of evolutionary history (...) than Teilhard's progress. This dialectic is also a more realistic description of cosmic evolution. Tillich's “Kingdom of God” within history as well as “the End of History,” in contrast to Teilhard's Omega Point, does not appear to contradict the Second Law of Thermodynamics, which predicts that the universe will ultimately disintegrate. Haught's contrast/contact modes of relating science and religion would regard Teilhard's Omega Point as an expression of spiritual hope and purpose rather than a scientifically verifiable principle. The contrast/ contact position is consonant with Tillich's description of religion as part of the vertical dimension of ultimate concern and science as part of the horizontal dimension of relationships between finite objects. Tillich did not share Teilhard's optimistic vision of the future. (shrink)

A P-point ultrafilter over \(\omega \) is called an interval P-point if for every function from \(\omega \) to \(\omega \) there exists a set _A_ in this ultrafilter such that the restriction of the function to _A_ is either a constant function or an interval-to-one function. In this paper we prove the following results. (1) Interval P-points are not isomorphism invariant under \(\textsf{CH}\) or \(\textsf{MA}\). (2) We identify a cardinal invariant \(\textbf{non}^{**}({\mathcal {I}}_{\tiny {\hbox {int}}})\) (...) such that every filter base of size less than continuum can be extended to an interval P-point if and only if \(\textbf{non}^{**}({\mathcal {I}}_{\tiny {\hbox {int}}})={\mathfrak {c}}\). (3) We prove the generic existence of slow/rapid non-interval P-points and slow/rapid interval P-points which are neither quasi-selective nor weakly Ramsey under the assumption \({\mathfrak {d}}={\mathfrak {c}}\) or \(\textbf{cov}({\mathcal {B}})={\mathfrak {c}}\). (shrink)

M. E. Rudin proved, under CH, that for each P-point p there exists a P-point q strictly RK-greater than p. This result was proved under ${\mathfrak {p}= \mathfrak {c}}$ by A. Blass, who also showed that each RK-increasing $ \omega $ -sequence of P-points is upper bounded by a P-point, and that there is an order embedding of the real line into the class of P-points with respect to the RK-ordering. In this paper, the results cited (...) above are proved under the assumption that $\mathfrak { b}=\mathfrak {c}$. A. Blass asked in 1973 which ordinals can be embedded in the set of P-points, and pointed out that such an ordinal cannot be greater than $ \mathfrak {c}^{+}$. In this paper it is proved, under $\mathfrak {b}=\mathfrak {c}$, that for each ordinal $\alpha < \mathfrak {c}^{+}$, there is an order embedding of $ \alpha $ into P-points. It is also proved, under $\mathfrak {b}=\mathfrak {c}$, that there is an embedding of the long line into P-points. (shrink)

A function is recursive (in given operations) if its values are computed explicitly and uniformly in terms of other "previously computed" values of itself and (perhaps) other "simultaneously computed" recursive functions. Here, "explicitly" includes definition by cases. We investigate those recursive functions on the structure $\mathbf{N} = \langle \omega, 0, \operatorname{succ,pred}\rangle$ that are computed in terms of themselves only, without other simultaneously computed recursive functions.

Frank Tipler's The Physics of Immortality provides abundant cause for intellectual offense—including challenges to physics, to theology, and, seemingly, to common sense. Few philosophical conundrums remain unaddressed. Still, the book is stimulating and well presented.

Fifty years after his death, the thought of the French scientist and Jesuit Pierre Teilhard de Chardin (1881–1955) continues to inspire new ways of understanding humanity’s future. Trained as a paleontologist and philosopher, Teilhard was an innovative synthesizer of science and religion, developing an idea of evolution as an unfolding of material and mental worlds into an integrated, holistic universe at what he called the Omega Point. His books, such as the bestselling The Phenomenon of Man, have influenced (...) generations of ecologists, environmentalists, planners, and others concerned with the fate of the earth.This book brings together original essays by leading experts who reflect on Teilhard’s legacy for today’s globalized world. They explore such topics as: the idea of God and the person; quantum reality and Teilhard’s vision; spiritual resources for the future; politics and economics; and a charter for co-evolution. (shrink)

This contribution addresses the anthropocenic challenge from a dialectical perspective, combining a diagnostics of the present with a prognostic of the emerging future. It builds on the oeuvres of two prominent dialectical thinkers, namely Georg Wilhelm Friedrich Hegel and Pierre Teilhard de Chardin. Hegel himself was a pre-anthropocenic thinker who did not yet thematise the anthropocenic challenge as such, but whose work allows us to emphasise the unprecedented newness of the current crisis. I will especially focus on his views on (...) Earth as a planetary process, emphasising that the “spirit” of technoscience is basically monitoring the impacts of its own activities on geochemistry and evolution. Subsequently, I will turn attention to Teilhard de Chardin, a palaeontologist and philosopher rightfully acknowledged as one of the first thinkers of the Anthropocene whose oeuvre provides a mediating middle term between Hegel’s conceptual groundwork and the anthropocenic present. Notably, I will discuss his views on self-directed evolution, on the on-going absorption of the biosphere by the noosphere, and on emerging options for “sublating” the current crisis into a synthetic convergence towards the Omega point. I will conclude that, after disclosing the biomolecular essence of life, biotechnology must now take a radical biomimetic turn ; that reflection itself must become distributed and collective; and, that the anthropocenic crisis must be sublated into the noocene. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:The Daring and Disappointing Dreams of Transhumanism's Secular EschatologyMichael Baggot L.C.IntroductionAlthough it is a largely secular movement, contemporary transhumanism borrows heavily from both Christian orthodoxy and heresies to construct a vision for human happiness. This article traces the roots of transhumanism's soteriology and eschatology and then examines the underlying anthropological problems that drive the hoped-for salvation through digital immortality. Unfortunately, the admirable desire to extend life sacrifices an appreciation (...) for the integral harmony of the human person's animal and spiritual dimensions. Since human actions manifest the person's intrinsic corporeality, attempts to detach the human personality from the body through digitalization would produce replicas of the dead without achieving true immortality. The surprising quasi-religious thrust of contemporary transhumanism's secular eschatology presents an opportunity to rediscover the Catholic patrimony's reflection on authentic divinization through the transhumanizing effects of divine grace. The article thus concludes with a Thomistic theosis-centered reorientation of secular transhumanism's eschatological aspirations for immanent immortality toward true human flourishing. [End Page 841]The Secular Eschatology of Contemporary TranshumanismThe leading transhumanist philosopher, Oxford University professor Nick Bostrom, thinks that his movement will provide real solutions to the perennial problems of humanity that traditional religions have tried and failed to address:While not a religion, transhumanism might serve a few of the same functions that people have traditionally sought in religion. It offers a sense of direction and purpose and suggests a vision that humans can achieve something greater than our present condition. Unlike most religious believers, however, transhumanists seek to make their dreams come true in this world, by relying not on supernatural powers or divine intervention but on rational thinking and empiricism, through continued scientific, technological, economic, and human development.1Yet, for all its protests against antiquated creeds, secular transhumanism frequently tends toward quasi-religious expressions. For instance, Meghan O'Gieblyn finds striking parallels between the evangelical and transhumanist worldviews she has embraced and abandoned during her life.2 Her writing explores her harrowing transition from Christian fundamentalism to trans-humanism and her eventual disillusionment with the techno-movement. Bible school had taught her about the various epochs of salvation history and her position within the "dispensation of grace" that marked the final preparation for the longed-for "millennial kingdom." As an adult, she spurned the false promises of organized religion and sought a more rational organization for her life. Later, a book recommendation plunged her into the thought of Google guru Ray Kurzweil and his narrative of evolutionary eras pointing toward an imminent "singularity" in which intelligent machines would emerge from an informational explosion. Such advanced robots would bring digital immortality through mind uploads and a nanotech new earth [End Page 842] that would put Eden to shame. Through his projects, Kurzweil purports to continue the Enlightenment tradition of empirical science and rational rigor. Similarly, many transhumanists tend to follow their Enlightenment predecessors in chiding traditional religion for obscurantist opposition to scientific progress. Nonetheless, O'Gieblyn discovered that the very movement that pretends to supplant the superstitions of the old-time religion is itself "a secular outgrowth of Christian eschatology."3The Christian-trained author could not help but detect familiar religious themes in the technocentric movement. Before it spread through the offices of Silicon Valley, the term "transhuman" first appeared in Henry Francis Carey's 1814 version of Dante's Paradiso, where the poet pilgrim recognizes that not even his own genius can describe the joy of risen life when he writes, "words may not tell of that transhuman change."4 Recent efforts to substitute supernatural resurrection with technological advancement find antecedents in the thirteenth-century friar Roger Bacon's quest for an elixir of life. Later, the nineteenth-century Russian Orthodox philosopher Nikolai Fedorov claims humans should use technology to control Darwin's unguided evolutionary process. Similarly, the twentieth-century Jesuit paleontologist Pierre Teilhard de Chardin predicted a global machine network that would merge consciousnesses worldwide and prompt the Omega Point of transphysical union with the divine.The Eugenicist Julian Huxley strips such notions of their religious trappings to present a secular transhumanist project that will "overcome limitations…to arrive at... (shrink)

As we know, the whole system of Hegel 's thought is based totally on the self-development process of the Geist. In other words, according to dialectical scheme of neoplatonism which Proclus systematized as a triad, μονή-πρόοδος-ἐπιστροφή, Hegel says that the Geist remains in himself, comes out from himself, and then returns to himself. With this process of self-development of the Geist, Hegel tries to explain the realities in general such as nature, history, art, religion, and philosophy. This process of the (...) development of the Geist makes a circle mode with the triadic way of an sich - für sich - an und für sich, but itself as the final destination of the Geist after finishing the whole process, is not a mere itself at the beginning, but the completed itself filled with the self-consciousness and freedom in full bloom. The last moment when the Geist returns to itself, Hegel says it is ‘Ruhe in Gott’. This is the final moment which the Geist as reality of the world and the history arrives and also the conclusion of his philosophy. Then, what happens to this final stage of the return of the Geist to itself, and what does it mean? Most of all, Hegel saw that, as the goal of world history is the achievement of self-consciousness of the Geist on its freedom and the realization of the freedom, this goal was achieved in the self-return of the Geist. And he has seen that the finite and the infinite, which showed constant opposition in the history of Western thought, are united here. According to Hegel, the infinite and finite in the bad concept that came out of the intellect(Verstand) are confronted with each other, and existence and non-existence are substantially separated. However, in the true concept of the infinite derived from reason(Vernunft), is formed the unified moment, in which the confrontations of both parties has been resolved and everything had been included in itself. The transcendence that he advocates is not a transmundane transcendence that is separated absolutely from the finite sensory world in existence and then unreachable absolutely, but a ‘right transcendence’ in which the finite and the infinite are really inseparable and remain as moments of the process of the Geist. In his view, the former which Platonism represents in Western philosophy, is an 'empty transcendence,' but the transcendence that he speaks is a 'full transcendence' that encompasses all moments of the finite in itself. The return of the absolute idea to itself means not only the sublation (Aufhebung) of the finite but also the the sublation of the reality, of the other, of the outwardness, of the abandonment of history, and secures the opposites to those such as ideality, self-identity, inwardness, and trans-historicity. Hegel's Geist as returning to itself has self-consciousness recognizing that it negates and rules the world and the history at the same time. In other words, it is honored and then the glorifying of God has been achieved in the consciousness of itself as the Absolute that it has been the only subject and the sole ruler of world history. According to Hegel, this glorifying of God is the most valuable object of the Geist and history. The Geist that has completed the process of self-development in the return to the itself enjoys the serenity, which is not a 'vacant serenity' but a ‘full serenity’ integrating the finite and the infinite into its inner simplicity. Hegel regarded the transition from the other to being self-internal as a transition from necessity to freedom, and in this serenity the Geist enjoys the freedom that it has in its own nature, as possible as it can. That is to say, in this serenity, the Geist becomes absolutely free and became itself, negating and sublating all other-being(Anderssein). In his view, the "Serenity in God," where the Geist becomes itself and its absolute freedom is obtained, is the ultimate goal of the Geist, and the omega point of the history. (shrink)

1. Introduction to a dilemma. 2. Returning to the beginning of time. 3. From a non-existent time in classical physics to an strange destiny of time in nowadays science. 4. Time of those who want understand the reality of history. 5. What will be, then, an existent time from which we can talk about the history of time (or the time of history) denoting and "omega point"?

Spiritual naturalists say that spirit is a natural force. Far from being novel or unconventional, spiritual naturalism spans the entire history of Western thought, from the Stoics, through leading modern thinkers, to the transhumanists. Spirit drives the self-organization of matter. The spirituality of any thing is just its degree of self-organization, which is its evolved complexity. But self-organization is self-regulation and self-control. Many spiritual naturalists, especially the transhumanists, have developed naturalistic phenomenologies of spirit. These are narratives which trace the ideal (...) evolution of complexity from the origin of our universe to its end. They are spiritually optimal possible histories of our universe. During the phases of any spiritually optimal history, humanity will perfect itself. But transhumanists say that the ideal evolution of complexity will greatly surpass humans. We will be surpassed by superhuman animals; but these will be surpassed by superhuman robots and godlike computers. The ideal end of the cosmos is an Omega Point with infinite spirituality. (shrink)

Teilhard is among the first to seriously explore the future of human evolution. He advocates both bio-technologies (e.g. genetic engineering) and intelligence technologies. He discusses the emergence of a global computation - communication system (and is said by some to have been the first to have envisioned the Internet). He advocates the development of a global society. He is almost surely the first to discuss the acceleration of technological progress to a Singularity in which human intelligence will become super-intelligence. He (...) discusses the spread of human intelligence into the universe and its amplification into a cosmic-intelligence. His work has been taken up by Barrow and Tipler; Tipler; Moravec; and Kurzweil. Of course, Teilhard's Omega Point Theory is deeply Christian. For secular transhumanists, this may be difficult. But transhumanism cannot avoid a fateful engagement with Christianity. Christian institutions may support or oppose transhumanism. Since Christianity is an extremely powerful cultural force in the West, it is imperative for transhumanism to engage it carefully. A serious study of Teilhard can help that engagement and will thus be rewarding to both communities. (shrink)

This study originally appeared in German in 1963. It was revised for the English edition and the translation is smooth. It is an introduction aimed at the layman. The language is simple and, except for the most important of Teilhard's terms, technical terms are scrupulously avoided. The book is organized around what Wildiers feels are Teilhard's major motivating concerns: God and the universe, or love of God vs. love of world. Wildiers explains how Teilhard sees the universe evolving from the (...) geosphere through the biosphere and noosphere and converging toward a specific end, the omega point, which Teilhard identifies with the parousia. Man has a unique position in this scheme. He represents the noosphere and is, in a sense, evolution in charge of itself. Wildiers shows how Teilhard's original concern was scientific and how he tenaciously made use of a phenomenological scientific methodology even in his most speculative theological reflections. Teilhard's christology, philosophy of history, and concept of evil are also discussed. The book does perhaps leave the reader "introduced," but there is no substitute for discovering Teilhard for oneself.--S. O. H. (shrink)

The Cosmos we live in consists of Invisible Prakriti and Visible World. In Visible World, we do live. All the galaxies, Milky Ways, nebulas and planets, stars, and physical bodies belong to this world are governed by the physical and mathematical laws of nature. Prakriti which is invisible spiritually governed and wave-formed existed even before the Big-Bang. Purush holds the Visible World and Prakriti around makes entire Cosmos in existence. Purush which is an absolutely positively charged and quality less with (...) no traces of negative portions always existed, neither being created nor destroyed. A fraction of a Purush known as Omega Point appears into the portion of Prakriti due to some causal effect for further expansion of the Visible World. Purush is situated in the middle of the cylindrical universe surrounded by the Prakriti and Visible World. Theories proposed by different scientists and philosophers to verify the Purush seem to be incomplete trial and error of the zigzag puzzle show. Today, our scientists, cosmologists, and mathematicians are trying to explain the function of the universe through only four physical dimensions of science, which seems to be incomplete without explaining the transcendental factors while dealing with the living bodies. (shrink)

A very felicitous translation of a work of major importance for science and philosophy. The book attempts to provide a coherent vision of the process of evolution starting from the formation of our planet through the emergence of life, and later, thought, to an imagined end state or Omega Point. The book is rich in imaginative theories about the various transitions of evolution but its greatest merit is in providing an overall pattern of high plausibility rendering the past (...) more intelligible and the future, in some measure, predictable. The central concept of the book seems to be that the energy present in each element of the universe is divided into two distinct components: a 'tangential' energy which links the element with all others of the same order of complexity and a 'radial' energy which draws the element towards ever greater complexity and centricity. The elaboration of the concept of 'radial' energy gives this book its great originality.--D. D. O. (shrink)

Death is an invariant of human existence, a kind of its “omega point”. At the same time, the attitude towards it is not a constant, but rather undergoes changes, being incorporated in a wide sociocultural context. The mail goal of the article is an analysis of the nature of these changes at the present stage specifically under the influence of rapid biotechnological progress and dilemmas generated by these changes. Firstly, there is a tendency of increasing optimism: some theorists (...) tend to perceive death not as an inevitable reality, but rather as a problem and a challenge. At the same time a radical direction arises, believing that death can be overcome only through the complete separation of man from its body substrate. The utopian nature of this project forces more moderate theorists to look for indirect way, which consists in a “war” with aging as the main predecessor of death, up to attempts of its complete abolition. Gradually, the formation of scientific immortology as a kind of opposition for thanatology takes place. At the same time, many theorists, and even more ordinary people remain skeptical position about the prospects for victory over death. It can be concluded that biomedical progress actualizes general human intention for self-preservation and death avoidance, but science requires an assessment of not only the objective possibilities for overcoming death but also both ontological and anthropological consequences of it. (shrink)

This article explores various issues with Pierre Teilhard de Chardin's theology including his view of God, his articulation of a "new religion," and his thoughts about transhumanism. This article published is a commemoration of Monsignor Vincent Foy's contributions to Catholic thought. -/- After having examined many of Teilhard's writings and commentaries that have been articulated by both supporters and detractors, I am persuaded that one should not take an all-or-nothing approach to Teilhard (or any other thinker for that matter). The (...) works of Teilhard are often written in poetic prose and remain obscure, in part because of his mysticism. He uses scientific language which at times is conflated with his evolutionary and spiritual understanding of the world. This article looks at some of the theological missteps of Teilhard's works. (shrink)

. Philip Hefner's focus on contingency and field as the guiding concepts in my thinking and his characterization of my theological enterprise as a Lakatosian research program are appropriate and helpful.I welcome Jeffrey Wicken's holistic approach to the emergence of life. Theology can appropriate the language of self‐organizing systems exploiting the thermodynamic flow of energy degradation for interpreting organic life as a creation of the Spirit of God.However, I cannot sympathize with Lindon Eaves's equation of “hard science” with a reductionism (...) which raises the double helix to the status of icon; the “meaning” of DNA derives from its place in the total phenomenon of life—not the reverse.Frank Tipler's cosmology raises the prospect of a rapprochement between physics and theology in the area of eschatology. A Christian cosmology, however, would require at least three modifications: contingency in the history of creation; the uniqueness of Jesus' resurrection; and the relation of these to the problem of evil. (shrink)

For X a separable metric space and $\alpha$ an infinite ordinal, consider the following three games of length $\alpha$ : In $G^{\alpha}_1$ ONE chooses in inning $\gamma$ an $\omega$ –cover $O_{\gamma}$ of X; TWO responds with a $T_{\gamma}\in O_{\gamma}$ . TWO wins if $\{T_{\gamma}:\gamma<\alpha\}$ is an $\omega$ –cover of X; ONE wins otherwise. In $G^{\alpha}_2$ ONE chooses in inning $\gamma$ a subset $O_{\gamma}$ of ${\sf C}_p(X)$ which has the zero function $\underline{0}$ in its closure, and TWO responds with (...) a function $T_{\gamma}\in O_{\gamma}$ . TWO wins if $\underline{0}$ is in the closure of $\{T_{\gamma}:\gamma<\alpha\}$ ; otherwise, ONE wins. In $G^{\alpha}_3$ ONE chooses in inning $\gamma$ a dense subset $O_{\gamma}$ of ${\sf C}_p(X)$ , and TWO responds with a $T_{\gamma}\in O_{\gamma}$ . TWO wins if $\{T_{\gamma}:\gamma<\alpha\}$ is dense in ${\sf C}_p(X)$ ; otherwise, ONE wins. After a brief survey we prove: 1. If $\alpha$ is minimal such that TWO has a winning strategy in $G^{\alpha}_1$ , then $\alpha$ is additively indecomposable (Theorem 4) 2. For $\alpha$ countable and minimal such that TWO has a winning strategy in $G^{\alpha}_1$ on X, the following statements are equivalent (Theorem 9): a) TWO has a winning strategy in $G^{\alpha}_2$ on ${\sf C}_p(X)$ . b) TWO has a winning strategy in $G^{\alpha}_3$ on ${\sf C}_p(X)$ . 3. The Continuum Hypothesis implies that there is an uncountable set X of real numbers such that TWO has a winning strategy in $G^{\omega^2}_1$ on X (Theorem 10). (shrink)

This book concerns the foundations of epistemic modality and hyperintensionality and their applications to the philosophy of mathematics. David Elohim examines the nature of epistemic modality, when the modal operator is interpreted as concerning both apriority and conceivability, as well as states of knowledge and belief. The book demonstrates how epistemic modality and hyperintensionality relate to the computational theory of mind; metaphysical modality and hyperintensionality; the types of mathematical modality and hyperintensionality; to the epistemic status of large cardinal axioms, undecidable (...) propositions, and abstraction principles in the philosophy of mathematics; to the modal and hyperintensional profiles of the logic of rational intuition; and to the types of intention, when the latter is interpreted as a hyperintensional mental state. Chapter \textbf{2} argues for a novel type of expressivism based on the duality between the categories of coalgebras and algebras, and argues that the duality permits of the reconciliation between modal and hyperintensional cognitivism and modal and hyperintensional expressivism. Elohim develops a novel, topic-sensitive truthmaker semantics for dynamic epistemic logic, and develops a novel, dynamic two-dimensional semantics grounded in two-dimensional hyperintensional Turing machines. Chapter \textbf{3} provides an abstraction principle for two-dimensional (hyper-)intensions. Chapter \textbf{4} advances a topic-sensitive two-dimensional truthmaker semantics, and provides three novel interpretations of the framework along with the epistemic and metasemantic. Chapter \textbf{5} applies the fixed points of the modal $\mu$-calculus in order to account for the iteration of epistemic states in a single agent, by contrast to availing of modal axiom 4 (i.e. the KK principle). The fixed point operators in the modal $\mu$-calculus are rendered hyperintensional, which yields the first hyperintensional construal of the modal $\mu$-calculus in the literature and the first application of the calculus to the iteration of epistemic states in a single agent instead of the common knowledge of a group of agents. Chapter \textbf{6} advances a solution to the Julius Caesar problem based on Fine's `criterial' identity conditions which incorporate conditions on essentiality and grounding. Chapter \textbf{7} provides a ground-theoretic regimentation of the proposals in the metaphysics of consciousness and examines its bearing on the two-dimensional conceivability argument against physicalism. The topic-sensitive epistemic two-dimensional truthmaker semantics developed in chapters \textbf{2} and \textbf{4} is availed of in order for epistemic states to be a guide to metaphysical states in the hyperintensional setting. -/- Chapters \textbf{8-12} provide cases demonstrating how the two-dimensional hyperintensions of hyperintensional, i.e. topic-sensitive epistemic two-dimensional truthmaker, semantics, solve the access problem in the epistemology of mathematics. Chapter \textbf{8} examines the interaction between Elohim's hyperintensional semantics and the axioms of epistemic set theory, large cardinal axioms, the Epistemic Church-Turing Thesis, the modal axioms governing the modal profile of $\Omega$-logic, Orey sentences such as the Generalized Continuum Hypothesis, and absolute decidability. These results yield inter alia the first hyperintensional Epistemic Church-Turing Thesis and hyperintensional epistemic set theories in the literature. Chapter \textbf{9} examines the modal and hyperintensional commitments of abstractionism, in particular necessitism, and epistemic hyperintensionality, epistemic utility theory, and the epistemology of abstraction. Elohim countenances a hyperintensional semantics for novel epistemic abstractionist modalities. Elohim suggests, too, that higher observational type theory can be applied to first-order abstraction principles in order to make first-order abstraction principles recursively enumerable, i.e. Turing machine computable, and that the truth of the first-order abstraction principle for two-dimensional hyperintensions is grounded in its being possibly recursively enumerable and the machine being physically implementable. Chapter \textbf{10} examines the philosophical significance of hyperintensional $\Omega$-logic in set theory and discusses the hyperintensionality of metamathematics. Chapter \textbf{11} provides a modal logic for rational intuition and provides a hyperintensional semantics. Chapter \textbf{12} avails of modal coalgebras to interpret the defining properties of indefinite extensibility, and avails of hyperintensional epistemic two-dimensional semantics in order to account for the interaction between interpretational and objective modalities and the truthmakers thereof. This yields the first hyperintensional category theory in the literature. Elohim invents a new mathematical trick in which first-order structures are treated as categories, and Vopenka's principle can be satisfied because of the elementary embeddings between the categories and generate Vopenka cardinals in the category of Set in category theory. Chapter \textbf{13} examines modal responses to the alethic paradoxes. Elohim provides a counter-example to epistemic closure for logical deduction. Chapter \textbf{14} examines, finally, the modal and hyperintensional semantics for the different types of intention and the relation of the latter to evidential decision theory. (shrink)

Paper presented in East-West Symposium on Science, Philosophy and Religion, Society for Asian and Comparative Philosophy Meeting with Australasian Association of Philosophy Annual Conference, The University of Melbourne, Melbourne, July 1999.

The starting point of this work is the existence of historical paradoxes in the set theory. These are: Russell's paradox, applied to the set W, Cantor's, for the set U, and Burali-Forti's, of the set Omega. A systematic analysis aimed at the simplification and the refining of such paradoxes showed that: (i) there exist at least eight contradictory expressions instead of three; (ii) another contradictory set is suggested by an extension of Burali-Forti's paradox; (iii) almost all of the (...) contradictions apply to more than one paradoxical set. (shrink)

In general, the ultimate end is the end beyond which there can be no further end. However, almost all the ultimate ends considered so far— “a man’s ultimate end”, “humanity’s ultimate end”, “the ultimate end of the universe”, and so on—are relative, in that they can in fact have a further end. Additionally, many of the ideas are based on dubious presuppositions such as teleology. Can there, then, be a meaningful idea of the absolute ultimate end without dubious presuppositions, beyond (...) which there can never be any further end? And if so, what will it be like? This paper, by raising such questions, suggests a possible study of the absolute ultimate end, which has never been attempted before, at least explicitly. At first, the classic ideas of the ultimate end and modern criticisms of them are surveyed; then, postulates for the concept of the absolute ultimate end are presented. Next, a definition of the absolute ultimate end is presented, and it is argued that the idea of AUE is not based on dubious presuppositions, and that AUE is the strongest meaningful ultimate end. Further, it is suggested that a study of AUE would be highly metaphysical—or even mystical—in character. Lastly, the possible relationships between us human and AUE are suggested. (shrink)

We show that Miller partition forcing preserves selective independent families and P-points, which implies the consistency of $\mbox {cof}(\mathcal {N})=\mathfrak {a}=\mathfrak {u}=\mathfrak {i}<\mathfrak {a}_T=\omega _2$. In addition, we show that Shelah’s poset for destroying the maximality of a given maximal ideal preserves tight mad families and so we establish the consistency of $\mbox {cof}(\mathcal {N})=\mathfrak {a}=\mathfrak {i}=\omega _1<\mathfrak {u}=\mathfrak {a}_T=\omega _2$.

The starting point of this work is the existence of historical paradoxes in the set theory. These are: Russell's paradox, applied to the set W, Cantor's, for the set U, and Burali-Forti's, of the set Omega. A systematic analysis aimed at the simplification and the refining of such paradoxes showed that: there exist at least eight contradictory expressions instead of three; another contradictory set is suggested by an extension of Burali-Forti's paradox; almost all of the contradictions apply to (...) more than one paradoxical set. (shrink)

A free ultrafilter $\mathcal{U}$ on $\omega$ is called a $T$-point if, for every countable group $G$ of permutations of $\omega$, there exists $U\in\mathcal{U}$ such that, for each $g\in G$, the set $\{x\in U:gx\ne x, gx\in U\}$ is finite. We show that each $P$-point and each $Q$-point in $\omega^*$ is a $T$-point, and, under CH, construct a $T$-point, which is neither a $P$-point, nor a $Q$-point. A question whether $T$-points exist in (...) ZFC is open. (shrink)

Certain instances of contraction are provable in Zardini’s system $\mathbf {IK}^\omega $ which causes triviality once a truth predicate and suitable fixed points are available.

Newtonian physics is based on Newtonian calculus applied to Newtonian dynamics. New paradigms such as ‘modified Newtonian dynamics’ change the dynamics, but do not alter the calculus. However, calculus is dependent on arithmetic, that is the ways we add and multiply numbers. For example, in special relativity we add and subtract velocities by means of addition β1⊕β2=tanh+tanh-1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _1\oplus \beta _2=\tanh \big +\tanh ^{-1}\big )$$\end{document}, although multiplication β1⊙β2=tanh·tanh-1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} (...) \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _1\odot \beta _2=\tanh \big \cdot \tanh ^{-1}\big )$$\end{document}, and division β1⊘β2=tanh/tanh-1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _1\oslash \beta _2=\tanh \big /\tanh ^{-1}\big )$$\end{document} do not seem to appear in the literature. The map fX=tanh-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_{\mathbb{X}}=\tanh ^{-1}$$\end{document} defines an isomorphism of the arithmetic in X=\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{X}}=$$\end{document} with the standard one in R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{R}}$$\end{document}. The new arithmetic is projective and non-Diophantine in the sense of Burgin, 1977), while ultrarelativistic velocities are super-large in the sense of Kolmogorov, 1961). Velocity of light plays a role of non-Diophantine infinity. The new arithmetic allows us to define the corresponding derivative and integral, and thus a new calculus which is non-Newtonian in the sense of Grossman and Katz. Treating the above example as a paradigm, we ask what can be said about the set X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{X}}$$\end{document} of ‘real numbers’, and the isomorphism fX:X→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_{{\mathbb{X}}}:{\mathbb{X}}\rightarrow {\mathbb{R}}$$\end{document}, if we assume the standard form of Newtonian mechanics and general relativity but demand agreement with astrophysical observations. It turns out that the observable accelerated expansion of the Universe can be reconstructed with zero cosmological constant if fX≈0.8sinh/\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_{\mathbb{X}}\approx 0.8\sinh /$$\end{document}. The resulting non-Newtonian model is exactly equivalent to the standard Newtonian one with ΩΛ=0.7\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _\Lambda =0.7$$\end{document}, ΩM=0.3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _M=0.3$$\end{document}. Asymptotically flat rotation curves are obtained if ‘zero’, the neutral element 0X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0_{{\mathbb{X}}}$$\end{document} of addition, is nonzero from the point of view of the standard arithmetic of R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{R}}$$\end{document}. This implies fX-1=0X>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f^{-1}_{{\mathbb{X}}}=0_{{\mathbb{X}}}>0$$\end{document}. The opposition Diophantine versus non-Diophantine, or Newtonian versus non-Newtonian, is an arithmetic analogue of Euclidean versus non-Euclidean in geometry. We do not yet know if the proposed generalization ultimately removes any need of dark matter, but it will certainly change estimates of its parameters. Physics of the dark universe seems to be both geometry and arithmetic. (shrink)

En este artículo, tengo dos objetivos distintos. En primer lugar, mostrar que no es una buena idea tener una teoría de la verdad que, aunque consistente, sea omega-inconsistente. Para discutir este punto, considero un caso particular: la teoría de Friedman-Sheard FS. Argumento que en los lenguajes de primer orden omega inconsistencia implica que la teoría de la verdad no tiene modelo estándar. Esto es, no hay un modelo cuyo dominio sea el conjunto de los números naturales en el (...) cual esta teoría de la verdad pueda tener una interpretación consistente. En ese sentido, la introducción del predicado veritativo no mantiene la ontología estándar. Además, cuando se considera un lenguaje de orden superior, la situación es aun peor. En teorías de segundo orden con semántica estándar, la misma introducción produce una teoría que no tiene modelo. Por eso, si la omega-inconsistencia es un mal síntoma, la insatisfacibilidad de una teoría es aun peor. En segundo lugar, propongo abandonar el principio de unión de teorías FSn y aceptar una extensibilidad indefinida de teorías FS0, FS1, FS2, FS3,.... De acuerdo a mi punto de vista, la secuencia de teorías tiene las mismas virtudes que FS sin sus desagradables consecuencias. In this paper, I have two different purposes. Firstly, I want to show that it's not a good idea to have a theory of truth that is consistent but omega-inconsistent. In order to bring out this point, it is useful to consider a particular case: FS. I argue that in First-order languages omega-inconsistency implies that a theory of truth has not standard model. Then, there is no model whose domain is the set of natural numbers in which this theory of truth could acquire a consistent interpretation. So, in theories of truth without standard models, the introduction of the truth-predicate to a first order theory does not maintain the standard ontology. I add that in Higher-order languages the situation is even worst. In second order theories with standard semantic the same introduction produces a theory that doesn't have a model. So, if an omega-inconsistent theory of truth is bad, an unsatisfiable theory is really bad. Secondly, I propose to give up the union principle of theories FSn and accept an indefinite extensibility of theories FS0, FS1, FS2, FS3,... According to my view, the sequence of theories has the same virtues of FS without its disgusting consequences. (shrink)

In this paper a logic for reasoning disquotationally about truth is presented and shown to have a standard model. This work improves on Hartry Field's recent results establishing consistency and omega-consistency of truth-theories with strong conditional logics. A novel method utilising the Banach fixed point theorem for contracting functions on complete metric spaces is invoked, and the resulting logic is shown to validate a number of principles which existing revision theoretic methods have heretofore failed to provide.

Henle, Mathias, and Woodin proved in [21] that, provided that${\omega }{\rightarrow }({\omega })^{{\omega }}$holds in a modelMof ZF, then forcing with$([{\omega }]^{{\omega }},{\subseteq }^*)$overMadds no new sets of ordinals, thus earning the name a “barren” extension. Moreover, under an additional assumption, they proved that this generic extension preserves all strong partition cardinals. This forcing thus produces a model$M[\mathcal {U}]$, where$\mathcal {U}$is a Ramsey ultrafilter, with many properties of the original modelM. This begged the question of (...) how important the Ramseyness of$\mathcal {U}$is for these results. In this paper, we show that several classes of$\sigma $-closed forcings which generate non-Ramsey ultrafilters have the same properties. Such ultrafilters include Milliken–Taylor ultrafilters, a class of rapid p-points of Laflamme,k-arrow p-points of Baumgartner and Taylor, and extensions to a class of ultrafilters constructed by Dobrinen, Mijares, and Trujillo. Furthermore, the class of Boolean algebras$\mathcal {P}({\omega }^{{\alpha }})/{\mathrm {Fin}}^{\otimes {\alpha }}$,$2\le {\alpha }<{\omega }_1$, forcing non-p-points also produce barren extensions. (shrink)