The Nothing from Infinity paradox arises when the combination of two infinitudes of point particles meet in a supertask and disappear. Corral-Villate claims that my arguments for disappearance fail and concedes that this failure also produces an extreme kind of indeterminism, which I have called plenitudinous. So my supertask at least poses a dilemma of extreme indeterminism within Newtonian point particle mechanics. Plenitudinous indeterminism might be trivial, although easy attempts to prove it so seem to fail in the (...) face of plausible continuity principles. However, the question of its triviality is here moot, since I show that, except in one case, Corral-Villate’s disproofs fail, and with a correction, the original arguments are unrefuted. Consequently, of the two contenders for the outcome of my supertask, the Nothing from Infinity paradox has won out. (shrink)

In this paper I present a novel supertask in a Newtonian universe that destroys and creates infinite masses and energies, showing thereby that we can have infinite indeterminism. Previous supertasks have managed only to destroy or create finite masses and energies, thereby giving cases of only finite indeterminism. In the Nothing from Infinity paradox we will see an infinitude of finite masses and an infinitude of energy disappear entirely, and do so despite the conservation of energy in all (...) collisions. I then show how this leads to the Infinity from Nothing paradox, in which we have the spontaneous eruption of infinite mass and energy out of nothing. I conclude by showing how our supertask models at least something of an old conundrum, the question of what happens when the immovable object meets the irresistible force. (shrink)

In this article, some classical paradoxes of infinity such as Galileo’s paradox, Hilbert’s paradox of the Grand Hotel, Thomson’s lamp paradox, and the rectangle paradox of Torricelli are considered. In addition, three paradoxes regarding divergent series and a new paradox dealing with multiplication of elements of an infinite set are also described. It is shown that the surprising counting system of an Amazonian tribe, Pirah ̃a, working with only three numerals (one, two, many) can (...) help us to change our perception of these paradoxes. A recently introduced methodology allowing one to work with finite, infinite, and infinitesimal numbers in a unique computational framework not only theoretically but also numerically is briefly described. This methodology is actively used nowadays in numerous applications in pure and applied mathematics and computer science as well as in teaching. It is shown in the article that this methodology also allows one to consider the paradoxes listed above in a new constructive light. (shrink)

Bowin begins with an apparent paradox about Aristotelian infinity: Aristotle clearly says that infinity exists only potentially and not actually. However, Aristotle appears to say two different things about the nature of that potential existence. On the one hand, he seems to say that the potentiality is like that of a process that might occur but isn't right now. Aristotle uses the Olympics as an example: they might be occurring, but they aren't just now. On the other (...) hand, Aristotle says that infinity "exists in actuality as a process that is now occurring" (234). Bowin makes clear that Aristotle doesn't explicitly solve this problem, so we are left to work out the best reading we can. His proposed solution is that "infinity must be...a per se accident...of number and magnitude" (250). (Bryn Mawr Classical Review 2008.07.47). (shrink)

Benardete paradoxes involve a beginningless set each member of which satisfies some predicate just in case no earlier member satisfies it. Such paradoxes have been wielded on behalf of arguments for the impossibility of an infinite past. These arguments often deploy patchwork principles in support of their key linking premise. Here I argue that patchwork principles fail to justify this key premise.

Benardete paradoxes involve infinite collections of Grim Reapers, assassins, demons, deafening peals, or even sentences. These paradoxes have recently been used in arguments for finitist metaphysical theses such as temporal finitism, causal finitism, and discrete views of time. Here we develop a new finite Benardete-like paradox. We then use this paradox to defend a companions in guilt argument that challenges recent applications of patchwork principles on behalf of the aforementioned finitist arguments. Finally, we develop another problem for those (...) applications by examining the notion of exact duplication. (shrink)

We examine two competing solutions to Benardete paradoxes: causal finitism, according to which nothing can have infinitely many causes, and the unsatisfiable pair diagnosis (UPD), according to which such paradoxes are logically impossible and no metaphysical thesis need be adopted to avoid them. We argue that the UPD enjoys notable theoretical advantages over causal finitism. Causal finitists, however, have levelled two main objections to the UPD. First, they urge that the UPD requires positing a ‘mysterious force’ that prevents paradoxes from (...) arising. Since such a force is implausible, the UPD is in trouble. Second, they employ recombination or patchwork principles to argue that paradoxical situations would be possible if causal finitism were false. Since such situations are not possible, causal finitism is true, and so a substantive metaphysical thesis is needed to avoid the paradoxes. We argue that the UPD proponent can successfully respond to these objections. (shrink)

(1st draft for review) -/- WARNING: -/- Reading this book comes with certain dangers to be mindful of; Please consider the following suggestions to avoid them: -/- 1: Do not try to perceive infinity; Any kind of success here leads to psychosis. 2: Do not try to resolve the paradoxes; To understand the greater truth of this book, paradoxes must be accepted as true. 3: Do not read this book if your faith is unstable and having it challenged could (...) potentially put you into existential crisis. (shrink)

This paper gives a definition of self-reference on the basis of the dependence relation given by Leitgeb (2005), and the dependence digraph by Beringer & Schindler (2015). Unlike the usual discussion about self-reference of paradoxes centering around Yablo's paradox and its variants, I focus on the paradoxes of finitary characteristic, which are given again by use of Leitgeb's dependence relation. They are called 'locally finite paradoxes', satisfying that any sentence in these paradoxes can depend on finitely many sentences. I (...) prove that all locally finite paradoxes are self-referential in the sense that there is a directed cycle in their dependence digraphs. This paper also studies the 'circularity dependence' of paradoxes, which was introduced by Hsiung (2014). I prove that the locally finite paradoxes have circularity dependence in the sense that they are paradoxical only in the digraph containing a proper cycle. The proofs of the two results are based directly on König's infinity lemma. In contrast, this paper also shows that Yablo's paradox and its nested variant are non-self-referential, and neither McGee's paradox nor the omega-cycle liar paradox has circularity dependence. (shrink)

An analysis of the counter-intuitive properties of infinity as understood differently in mathematics, classical physics and quantum physics allows the consideration of various paradoxes under a new light (e.g. Zeno’s dichotomy, Torricelli’s trumpet, and the weirdness of quantum physics). It provides strong support for the reality of abstractness and mathematical Platonism, and a plausible reason why there is something rather than nothing in the concrete universe. The conclusions are far reaching for science and philosophy.

This work deals with problems involving infinities and infinitesimals. It explores the ideas behind zero, its relationship to ontological nothingness, finititude (such as finite numbers and quantities), and the infinite. The idea of infinity and zero are closely related, despite what many perceive as an intuitive inverse relationship. The symbol 0 generally refers to nothingness, whereas the symbol infinity refers to ``so much'' that it cannot be quantified or captured. The notion of finititude rests somewhere between complete nothingness (...) and something having no end. -/- My concern is that many of the philosophers arguing for or against the ontology (or possibility) of an actual infinite set are unaware or unfamiliar with the mathematical literature attempting to clearly and rigorously define these terms. I believe it is a mistake to leave mathematicians out of this conversation, as analysts in particular have defined (and used) infinity in a way that is relevant to the ongoing debate between philosophers. -/- For this paper, I have selected examples from Principles of Analysis by Walter Rudin, which has become a standard text for Real Analysis classes taken by pure mathematicians and engineers. I will also restrict the scope of examples to Euclidean spaces for simplicity. Finally, I will be using addition, + and multiplication * in their usual way. (shrink)

We look at extensions (i.e., stronger logics in the same language) of the Belnap–Dunn four-valued logic. We prove the existence of a countable chain of logics that extend the Belnap–Dunn and do not coincide with any of the known extensions (Kleene’s logics, Priest’s logic of paradox). We characterise the reduced algebraic models of these new logics and prove a completeness result for the first and last element of the chain stating that both logics are determined by a single finite (...) logical matrix. We show that the last logic of the chain is not finitely axiomatisable. (shrink)

Our evidence can be about different subject matters. In fact, necessarily equivalent pieces of evidence can be about different subject matters. Does the hyperintensionality of ‘aboutness’ engender any hyperintensionality at the level of rational credence? In this paper, I present a case which seems to suggest that the answer is ‘yes’. In particular, I argue that our intuitive notions of independent evidence and inadmissible evidence are sensitive to aboutness in a hyperintensional way. We are thus left with a paradox. (...) While there is strong reason to think that rational credence cannot make such hyperintensional distinctions, our intuitive judgements about certain cases seem to demand that it does. (shrink)

Previous claims to have resolved the two-envelope paradox have been premature. The paradoxical argument has been exposed as manifestly fallacious if there is an upper limit to the amount of money that may be put in an envelope; but the paradoxical cases which can be described if this limitation is removed do not involve mathematical error, nor can they be explained away in terms of the strangeness of infinity. Only by taking account of the partial sums of the (...) infinite series of expected gains can the paradox be resolved. (shrink)

Infinity exists as a concept but has no existence in actuality. For infinity to have existence in actuality either time or space have to already be infinite. Unless something is already infinite, the only way to become infinite is by an 'infinity leap' in an infinitely small moment, and this is not possible. Neither does infinitely small have an existence since anything larger than zero is not infinitely small. Therefore infinity has no existence in actuality.

We look at extensions (i.e., stronger logics in the same language) of the Belnap–Dunn four-valued logic. We prove the existence of a countable chain of logics that extend the Belnap–Dunn and do not coincide with any of the known extensions (Kleene’s logics, Priest’s logic of paradox). We characterise the reduced algebraic models of these new log- ics and prove a completeness result for the first and last element of the chain stating that both logics are determined by a single (...) finite logical matrix. We show that the last logic of the chain is not finitely axiomatisable. (shrink)

Two strategies to infinity are equally relevant for it is as universal and thus complete as open and thus incomplete. Quantum mechanics is forced to introduce infinity implicitly by Hilbert space, on which is founded its formalism. One can demonstrate that essential properties of quantum information, entanglement, and quantum computer originate directly from infinity once it is involved in quantum mechanics. Thus, thеse phenomena can be elucidated as both complete and incomplete, after which choice is the border (...) between them. A special kind of invariance to the axiom of choice shared by quantum mechanics is discussed to be involved that border between the completeness and incompleteness of infinity in a consistent way. The so-called paradox of Albert Einstein, Boris Podolsky, and Nathan Rosen is interpreted entirely in the same terms only of set theory. Quantum computer can demonstrate especially clearly the privilege of the internal position, or “observer”, or “user” to infinity implied by Henkin’s proposition as the only consistent ones as to infinity. (shrink)

Georg Cantor's absolute infinity, the paradoxical Burali-Forti class Ω of all ordinals, is a monstrous non-entity for which being called a "class" is an undeserved dignity. This must be the ultimate vexation for mathematical philosophers who hold on to some residual sense of realism in set theory. By careful use of Ω, we can rescue Georg Cantor's 1899 "proof" sketch of the Well-Ordering Theorem––being generous, considering his declining health. We take the contrapositive of Cantor's suggestion and add Zermelo's choice (...) function. This results in a concise and uncomplicated proof of the Well-Ordering Theorem. (shrink)

The purpose of this paper is to dissolve paradoxes of the infinite by correctly identifying the infinite natural numbers.

The seventeenth century was an important period in the conceptual development of the notion of the infinite. In 1643, Evangelista Torricelli (1608-1647)—Galileo’s successor in the chair of mathematics in Florence—communicated his proof of a solid of infinite length but finite volume. Many of the leading metaphysicians of the time, notably Spinoza and Leibniz, came out in defense of actual infinity, rejecting the Aristotelian ban on it, which had been almost universally accepted for two millennia. Though it would be another (...) two centuries before the notion of the actually infinite was rehabilitated in mathematics by Dedekind and Cantor (Cauchy and Weierstrass still considered it mere paradox), their impenitent advocacy of the concept had significant reverberations in both philosophy and mathematics. In this essay, I will attempt to clarify one thread in the development of the notion of the infinite. In the first part, I study Spinoza’s discussion and endorsement, in the Letter on the Infinite (Ep. 12), of Hasdai Crescas’ (c. 1340-1410/11) crucial amendment to a traditional proof of the existence of God (“the cosmological proof” ), in which he insightfully points out that the proof does not require the Aristotelian ban on actual infinity. In the second and last part, I examine the claim, advanced by Crescas and Spinoza, that God has infinitely many attributes, and explore the reasoning that motivated both philosophers to make such a claim. Similarities between Spinoza and Crescas, which suggest the latter’s influence on the former, can be discerned in several other important issues, such as necessitarianism, the view that we are compelled to assert or reject a belief by its representational content, the enigmatic notion of amor Dei intellectualis, and the view of punishment as a natural consequent of sin. Here, I will restrict myself to the issue of the infinite, clearly a substantial topic in itself. (shrink)

David Builes presents a paradox concerning how confident you should be that any given member of an infinite collection of fair coins landed heads, conditional on the information that they were all flipped and only finitely many of them landed heads. We argue that if you should have any conditional credence at all, it should be 1/2.

I propose a revision of Cantor’s account of set size that understands comparisons of set size fundamentally in terms of surjections rather than injections. This revised account is equivalent to Cantor's account if the Axiom of Choice is true, but its consequences differ from those of Cantor’s if the Axiom of Choice is false. I argue that the revised account is an intuitive generalization of Cantor’s account, blocks paradoxes—most notably, that a set can be partitioned into a set that is (...) bigger than it—that can arise from Cantor’s account if the Axiom of Choice is false, illuminates the debate over whether the Axiom of Choice is true, is a mathematically fruitful alternative to Cantor’s account, and sheds philosophical light on one of the oldest unsolved problems in set theory. (shrink)

This paper is a response to David Oderberg's discussion of the Tristram Shandy paradox. I defend the claim that the Tristram Shandy paradox does not support the claim that it is impossible that the past is infinite.

This thesis attempts to present a new singular type of paradox called a “Monoletheia”, which itself creates a logical bridge to reconcile two subsequent paradoxes we call “Antinomies” and “Dialetheia”. The consequence follows therefrom, to state that the concept of infinity transcends its own regress of an existential construct to include a non-existent variable (representing void). This is done by applying the theoretical idea of co-existence between constructs and their own negation. -/- The working hypothesis can be summed (...) up in the following proposition: “Existence co-exists with non-existence“, and is expressed mathematically for convenience as “Ex + co-ex w/ X“. For the same purpose, the expression denoting an antinomy is “A*” and “Dt” is used for dialetheia. (shrink)

Russell’s initial project in philosophy (1898) was to make mathematics rigorous reducing it to logic. Before August 1900, however, Russell’s logic was nothing but mereology. First, his acquaintance with Peano’s ideas in August 1900 led him to discard the part-whole logic and accept a kind of intensional predicate logic instead. Among other things, the predicate logic helped Russell embrace a technique of treating the paradox of infinite numbers with the help of a singular concept, which he called ‘denoting phrase’. (...) Unfortunately, a new paradox emerged soon: that of classes. The main contention of this paper is that Russell’s new conception only transferred the paradox of infinity from the realm of infinite numbers to that of class-inclusion. Russell’s long-elaborated solution to his paradox developed between 1905 and 1908 was nothing but to set aside of some of the ideas he adopted with his turn of August 1900: (i) With the Theory of Descriptions, he reintroduced the complexes we are acquainted with in logic. In this way, he partly restored the pre-August 1900 mereology of complexes and simples. (ii) The elimination of classes, with the help of the ‘substitutional theory’, and of propositions, by means of the Multiple Relation Theory of Judgment, completed this process. (shrink)

When matter is falling into a black hole, the associated information becomes unavailable to the black hole's exterior. If the black hole disappears by Hawking evaporation, the information seems to be lost in the singularity, leading to Hawking's information paradox: the unitary evolution seems to be broken, because a pure separate quantum state can evolve into a mixed one.



This article proposes a new interpretation of the black hole singularities, which restores the information conservation. For the Schwarzschild black hole, it (...) presents new coordinates, which move the singularity at the future infinity (although it can still be reached in finite proper time). For the evaporating black holes, this article shows that we can still cure the apparently destructive effects of the singularity on the information conservation. For this, we propose to allow the metric to be degenerate at some points, and use the singular semiriemannian geometry. This view, which results naturally from Ashtekar's new variables formulation of Einstein's equation, repairs the incomplete geodesics.



The reinterpretation of singularities suggested here allows (in the context of standard General Relativity) the information conservation and unitary evolution to be restored, both for eternal and for evaporating black holes. (shrink)

When matter is falling into a black hole, the associated information becomes unavailable to the black hole's exterior. If the black hole disappears by Hawking evaporation, the information seems to be lost in the singularity, leading to Hawking's information paradox: the unitary evolution seems to be broken, because a pure separate quantum state can evolve into a mixed one.



This article proposes a new interpretation of the black hole singularities, which restores the information conservation. For the Schwarzschild black hole, it (...) presents new coordinates, which move the singularity at the future infinity (although it can still be reached in finite proper time). For the evaporating black holes, this article shows that we can still cure the apparently destructive effects of the singularity on the information conservation. For this, we propose to allow the metric to be degenerate at some points, and use the singular semiriemannian geometry. This view, which results naturally from the Cauchy problem, repairs the incomplete geodesics.



The reinterpretation of singularities suggested here allows (in the context of standard General Relativity) the information conservation and unitary evolution to be restored, both for eternal and for evaporating black holes.

. (shrink)

It is modernly debated whether application of the free will has potential to cause harm to nature. Power possessed to the discourse, sensory/perceptual, physical influences on life experience by the slow moving machinery of change is a viral element in the problems of civilization; failed resolution of historical paradox involving mind and matter is a recurring source of problems. Reference is taken from the writing of Euclid in which a oneness of nature as an indivisible point of thought is (...) made prerequisite in criteria of interpretation to demonstrate that contemporary scientific methodologies alternately ensue from the point of empirically centered induction. A qualification for conceptualizations is proposed that involves a physically describable form bound to energy in addition to contemporary notions of energy bound to form and a visually based mathematical-physical form is elaborated and discussed with respect to biological and natural processes. (shrink)

This article draws on several crucial and unpublished manuscripts from the Scholem Archive in exploration of Gershom Scholem's youthful statements on mathematics and its relation to extra-mathematical facts and, more broadly, to a concept of history that would prove to be consequential for Walter Benjamin's own thinking on "messianism" and a "futuristic politics." In context of critiquing the German Youth Movement's subsumption of active life to the nationalistic conditions of the "earth" during the First World War, Scholem turns to mathematics (...) for a genuine and self-consistent theory of action. In the concept of actual infinity (in Cantor and Bolzano) he finds an explanation of how mathematics relates to "the physical" without reducing the former to an "image" of the latter, and without relying on the concept of geometric intuition. This explanation, insofar as it relies on the notion of actual infinity, provides Scholem with a conception of mathematics (and the history of mathematics) that reconciles freedom and necessity—remarks on which he outlines in his diaries and communicates to Benjamin in early March 1916. (shrink)

We show that the statement "I only know that I know nothing," attributed to the Greek philosopher Socrates, contains, at its core, Zermelo's Axiom of Selection and the arithmetic of the infinite cardinal aleph-0.

I defend a new argument for causal finitism, the view that nothing can have an infinite causal history. I begin by defending a number of plausible metaphysical principles, after which I explore a host of novel variants of the Littlewood-Ross and Thomson’s Lamp paradoxes that violate such principles. I argue that causal finitism is the best solution to the paradoxes.

It is commonly assumed that Aristotle denies any real existence to infinity. Nothing is actually infinite. If, in order to resolve Zeno’s paradoxes, Aristotle must talk of infinity, it is only in the sense of a potentiality that can never be actualized. Aristotle’s solution has been both praised for its subtlety and blamed for entailing a limitation of mathematic. His understanding of the infinite as simply indefinite (the “bad infinite” that fails to reach its accomplishment), his conception of (...) the cosmos and even its prime mover as finite (in the sense of autarchic/self-contained) have been contrasted with the subsequent claim of God as ens infinitum (understood as a “positive” infinity). The goal of this essay is to reexamine the major texts (notably De caelo) and to demonstrate that (1) Aristotle’s claim according to which there is no actual infinite concerns only substances, not processes. (2) That Aristotle does not deny an actual infinite as such. (3) That when considering time and God (qua eternal) Aristotle acknowledges an actual infinite. (shrink)

Several interesting themes emerge from G. E. Moore’s previously unpub­lished review of _The Principles of Mathematics_. These include a worry concerning whether mathematical notions are identical to purely logical ones, even if coextensive logical ones exist. Another involves a conception of infinity based on endless series neglected in the Principles but arguably involved in Zeno’s paradox of Achilles and the Tortoise. Moore also questions the scope of Russell’s notion of material implication, and other aspects of Russell’s claim that (...) mathematics reduces to logic. (shrink)

Kierkegaard’s ideal supports a radical form of “deep diversity,” to use Charles Taylor’s expression. It is radical because it embraces not only irreducible conceptions of the good but also incompatible ones. This is due to its paradoxical nature, which arises from its affirmation of both monism and pluralism, the One and the Many, together. It does so in at least three ways. First, in terms of the structure of the self, Kierkegaard describes his ideal as both unified (the “positive third”) (...) and plural (a “negative unity”). Second, he affirms a process which brings together unity, as implied by the linear notion of “stages”, with plurality, in the form of “spheres of existence” (aesthetic, ethical, and religious). And third, the culmination of the process implies that we should embrace both a unified dialectic (“Religiousness A”) alongside the plural remnants of the ethical/aesthetic, that is, both the infinity of the former and the finitudes of the latter. Unsurprisingly, while Kierkegaard describes those who are able to exemplify his ideal in practice as “always joyful,” he also considers the ideal to be “extremely hard, the hardest task of all.” This is why those such as Hubert L. Dreyfus are wrong to claim that it provides an experience of bliss; on the contrary, those who realize it “are always in danger.” As I shall show, one form this danger takes is that it threatens to dirty the hands of those who manage to uphold Kierkegaard’s ideal. Moreover, it does so in ways that, I claim, tend to be missed by Kierkegaard himself. Nevertheless, the danger is also essential to the creativity of his approach, and I conclude by pointing out how this creativity makes it capable of tackling one of the profoundest challenges in contemporary ethics: that arising from what just war theorists call a “supreme emergency.” . (shrink)

This work studies some problems connected to the role of negation in logic, treating the positive fragments of propositional calculus in order to deal with two main questions: the proof of the completeness theorems in systems lacking negation, and the puzzle raised by positive paradoxes like the well-known argument of Haskel Curry. We study the constructive com- pleteness method proposed by Leon Henkin for classical fragments endowed with implication, and advance some reasons explaining what makes difficult to extend this constructive (...) method to non-classical fragments equipped with weaker implications (that avoid Curry's objection). This is the case, for example, of Jan Lukasiewicz's n-valued logics and Wilhelm Ackermann's logic of restricted implication. Besides such problems, both Henkin's method and the triviality phenomenon enable us to propose a new positive tableau proof system which uses only positive meta-linguistic resources, and to mo- tivate a new discussion concerning the role of negation in logic proposing the concept of paratriviality. In this way, some relations between positive reasoning and infinity, the possibilities to obtain a ¯first-order positive logic as well as the philosophical connection between truth and meaning are dis- cussed from a conceptual point of view. (shrink)

In 1922, Thoralf Skolem introduced the term of «relativity» as to infinity от set theory. Не demonstrated Ьу Zermelo 's axiomatics of set theory (incl. the axiom of choice) that there exists unintended interpretations of anу infinite set. Тhus, the notion of set was also «relative». We сan apply his argurnentation to Gödel's incompleteness theorems (1931) as well as to his completeness theorem (1930). Then, both the incompleteness of Реапо arithmetic and the completeness of first-order logic tum out to (...) bе also «relative» in Skolem 's sense. Skolem 's «relativity» argumentation of that kind сan bе applied to а vету wide range of problems and one сan spoke of the relativity of discreteness and continuity or, of finiiteness and infinity, or, of Cantor 's kinds of infinities, etc. The relativity of Skolemian type helps us for generaIizing Einstein 's principle of relativity from the invariance of the physical laws toward diffeomorphisms to their invariance toward anу morphisms (including and especiaIly the discrete ones). Such а kind of generalization from diffeomorphisms (then, the notion of velocity always makes sense) to anу kind of morphism (when 'velocity' mау оr maу not make sense) is an extension of the general Skolemian type оГ relativity between discreteness and continuity от between finiteness and infinity. Particularly, the Lorentz invariance is not valid in general because the notion of velocity is limited to diffeomorphisms. [п the case of entanglement, the physical interaction is discrete0. 'Velocity" and consequently, the 'Lorentz invariance'"do not make sense. Тhat is the simplest explanation ofthe argurnent EPR, which tums into а paradox оnJу if the universal validity of 'velocity' and 'Lогелtz invariance' is implicitly accepted. (shrink)

Peano arithmetic cannot serve as the ground of mathematics for it is inconsistent to infinity, and infinity is necessary for its foundation. Though Peano arithmetic cannot be complemented by any axiom of infinity, there exists at least one (logical) axiomatics consistent to infinity. That is nothing else than a new reading at issue and comparative interpretation of Gödel’s papers (1930; 1931) meant here. Peano arithmetic admits anyway generalizations consistent to infinity and thus to some addable (...) axiom(s) of infinity. The most utilized example of those generalizations is the complex Hilbert space. Any generalization of Peano arithmetic consistent to infinity, e.g. the complex Hilbert space, can serve as a foundation for mathematics to found itself and by itself. (shrink)

This paper responds to commentaries by Kaiserman and Magidor, and Hawthorne. The case of the infinite clowns can teach us several things.

Kierkegaard’s ideal supports a radical form of “deep diversity,” to use Charles Taylor’s expression. It is radical because it embraces not only irreducible conceptions of the good but also incompatible ones. This is due to its paradoxical nature, which arises from its affirmation of both monism and pluralism, the One and the Many, together. It does so in at least three ways. First, in terms of the structure of the self, Kierkegaard describes his ideal as both unified (the “positive third”) (...) and plural (a “negative unity”). Second, he affirms a process which brings together unity, as implied by the linear notion of “stages”, with plurality, in the form of “spheres of existence” (aesthetic, ethical, and religious). And third, the culmination of the process implies that we should embrace both a unified dialectic (“Religiousness A”) alongside the plural remnants of the ethical/aesthetic, that is, both the infinity of the former and the finitudes of the latter. Unsurprisingly, while Kierkegaard describes those who are able to exemplify his ideal in practice as “always joyful,” he also considers the ideal to be “extremely hard, the hardest task of all.” This is why those such as Hubert L. Dreyfus are wrong to claim that it provides an experience of bliss; on the contrary, those who realize it “are always in danger.” As I shall show, one form this danger takes is that it threatens to dirty the hands of those who manage to uphold Kierkegaard’s ideal. Moreover, it does so in ways that, I claim, tend to be missed by Kierkegaard himself. Nevertheless, the danger is also essential to the creativity of his approach, and I conclude by pointing out how this creativity makes it capable of tackling one of the profoundest challenges in contemporary ethics: that arising from what just war theorists call a “supreme emergency.”. (shrink)

The paper continues the consideration of Hilbert mathematics to mathematics itself as an additional “dimension” allowing for the most difficult and fundamental problems to be attacked in a new general and universal way shareable between all of them. That dimension consists in the parameter of the “distance between finiteness and infinity”, particularly able to interpret standard mathematics as a particular case, the basis of which are arithmetic, set theory and propositional logic: that is as a special “flat” case of (...) Hilbert mathematics. The following four essential problems are considered for the idea to be elucidated: Fermat’s last theorem proved by Andrew Wiles; Poincaré’s conjecture proved by Grigori Perelman and the only resolved from the seven Millennium problems offered by the Clay Mathematics Institute (CMI); the four-color theorem proved “machine-likely” by enumerating all cases and the crucial software assistance; the Yang-Mills existence and mass gap problem also suggested by CMI and yet unresolved. They are intentionally chosen to belong to quite different mathematical areas (number theory, topology, mathematical physics) just to demonstrate the power of the approach able to unite and even unify them from the viewpoint of Hilbert mathematics. Also, specific ideas relevant to each of them are considered. Fermat’s last theorem is shown as a Gödel insoluble statement by means of Yablo’s paradox. Thus, Wiles’s proof as a corollary from the modularity theorem and thus needing both arithmetic and set theory involves necessarily an inverse Grothendieck universe. On the contrary, its proof in “Fermat arithmetic” introduced by “epoché to infinity” (following the pattern of Husserl’s original “epoché to reality”) can be suggested by Hilbert arithmetic relevant to Hilbert mathematics, the mediation of which can be removed in the final analysis as a “Wittgenstein ladder”. Poincaré’s conjecture can be reinterpreted physically by Minkowski space and thus reduced to the “nonstandard homeomorphism” of a bit of information mathematically. Perelman’s proof can be accordingly reinterpreted. However, it is valid in Gödel (or Gödelian) mathematics, but not in Hilbert mathematics in general, where the question of whether it holds remains open. The four-color theorem can be also deduced from the nonstandard homeomorphism at issue, but the available proof by enumerating a finite set of all possible cases is more general and relevant to Hilbert mathematics as well, therefore being an indirect argument in favor of the validity of Poincaré’s conjecture in Hilbert mathematics. The Yang-Mills existence and mass gap problem furthermore suggests the most general viewpoint to the relation of Hilbert and Gödel mathematics justifying the qubit Hilbert space as the dual counterpart of Hilbert arithmetic in a narrow sense, in turn being inferable from Hilbert arithmetic in a wide sense. The conjecture that many if not almost all great problems in contemporary mathematics rely on (or at least relate to) the Gödel incompleteness is suggested. It implies that Hilbert mathematics is the natural medium for their discussion or eventual solutions. (shrink)

As with so many mystics, Alexis karpouzos intuitively know the oneness of cosmic creation and historic humanity as part of all that is and all there isn't. So, the originality of Alexis Karpouzos thought is that it crosses the most diverse fields, the most opposing philosophies, to unite them into an often contradictory and broken whole. Marx and Heidegger, Nietzsche, Freud and Heraclitus, poets and political theorists all come together in the same distance and the same unusual proximity. Alexis karpouzos (...) use Pre-Socratics philosophy and generally the ancient Greek philosophy, as well as the pre-philosophical thinking of The Upanishads, the Vedas and Buddhism in India, of Lao Tzu, of Zen Buddhism and the Taoist tradition in China, of the Arab mystics and poets, with their metaphysical religiosity as the metaphysical basis for the interpretation and understanding of the world and existence. At the same time the ancient metaphysics is connect with Hegel’s dialectical ontology and with the modern thinking of Marx, Nietzsche, Freud, Heidegger, and others as the interpretive context for understanding the central problems of the technical and scientific world during his time. Novalis, Hölderlin, Rimbaud, Whitman, Eliot and others show the dreamy nature of existence, the transcendence of the Cosmos, welcome the infinity. For Alexis karpouzos the combination of ancient and modern thought creates the holistic experience of universal space-time and the consciousness of the unity. The poetic thought of Alexis karpouzos is a expressions of soul's inner experiences, expression of universality. The inspiring visual images and the symbolic use of language offer a description of elevating experiences of consciousness, a glimpse of higher worlds. The philosophy of Alexis karpouzos speak to the human experience from a universal perspective, transcending all religions, cultural and national boundaries. Using vivid images and a direct language that speaks to the heart, his philosophy evokes a sense of deep communication with the collective unconscious, a sense of connection to all the creatures of the world, compassion for others, admiration for the beauty of nature, reverence for all life, and an abiding faith in the invisible touch of world. Alexis karpouzos thoughts are often terse and paradoxical, challenging us to to break out of the box of limiting beliefs and see things from a new perspective. Above all, Alexis karpouzos continually calls to us to wake up and explore the mysteries within our own selves, i.e the mysteries of universe. (shrink)

The essay aims to analyse the gradual historical process of the partial overlap, replacement and expansion of the theoretical paradigm of depression with respect to that of melancholy. The first part is devoted to analysing some of the central features of the multivalent thematizations of melancholy drawn up during modernity, also with relation to the spirit of capitalism (in its Weberian acceptation). This is followed by an overview of the birth of the modern category of depression, and the process that (...) during the twentieth century led to its transformation into a theoretical and nosological paradigm which in the end replaced that of melancholy. The second part is the attempt to draw up a Zeitdiagnose: the aim is to assess the ‘depressive’ psychosocial impact exercised by particular socio-economic mechanisms and cultural trends, with particular reference to what has been defined as the ‘new spirit of capitalism’, as well as the new forms of individualism and their paradoxical dynamics. In conclusion, I will form a bridge between the history of the two aforesaid paradigms and the contemporary psychosocial dynamics correlated to the phenomenon today interpreted as a ‘depression epidemic’. Il saggio mira ad analizzare il graduale processo storico di parziale sovrapposizione, sostituzione ed ampliamento del paradigma teoretico della depressione rispetto a quello della melanconia. La prima parte è dedicata ad analizzare alcuni dei tratti nevralgici delle polivalenti tematizzazioni della malinconia avvenuti nel corso della modernità, anche in relazione allo spirito del capitalismo (nella sua accezione weberiana). Segue una panoramica sulla nascita della categoria moderna di depressione, e del processo che nel corso del XX secolo l’ha condotta alla sua trasformazione in un paradigma teoretico e nosologico che ha infine sostituito quello della melanconia. La seconda parte rappresenta il tentativo di elaborare una Zeitdiagnose: l’obiettivo è valutare l’impatto psicosociale di natura ‘depressiva’ esercitato da particolari meccanismi socioeconomici e tendenze culturali, con particolare riferimento a ciò che è stato definito il «nuovo spirito del capitalismo», nonché alle nuove forme dell’individualismo e alle loro dinamiche paradossali. In conclusione viene gettato un ponte tra la storia dei due suddetti paradigmi e le dinamiche psicosociali contemporanee correlate al fenomeno oggi interpretato quale «epidemia depressiva». (shrink)

This wide ranging discourse covers many disciplines of science and the human condition in an attempt to fully understand the manifestation of time. Time's Paradigm is, at its inception, a philosophical debate between the theories of 'Presentism' and 'The Block Model', beginning with a pronounced psychological analysis of 'free will' in an environment where the past and the future already exist. It lays the foundation for the argument that time is a cyclical, contained progression, rather than a meandering voyage into (...) infinity, bringing into question the validity of a commensurate 'Big Bang'. Following, the proposal widens to encompass physics. It tackles clock rates and time dilation, acausality and the nuisance of a universal clock, and demonstrates that conscious consideration creates the present moment - time's flow - separating the solid state past and future whose reality is devoid of space. Arguments relating to Quantum Physics theory, including the Uncertainty Principle and a Superposition of States, lend credibility to key areas involving cognitive awareness. It is posited that defined points in time and space prohibit progress in linear models for progression. Thus motile paradoxes can be resolved with the absence of infinities; temporal perception, it is concluded, being the result of uncertainty. Time's Paradigm takes the bold step of asking us to consider a tangible dimension of time, representing an intimate extension of our three, known spatial dimensions. Chaos theory is briefly introduced leading to the configuration of a fractal fourth dimension of time whose assumption demands only one direction of flow. Further, it asks whether our universe is expanding or contracting. It considers the simple physics of bodies contracting in a fourth dimension of time (UC), and how that marries comfortably with standard scientific models such as Special Relativity. The rate at which matter is contracting in the universe is illustrated in a reduction factor of 1.618... coinciding with Fibonacci's Ratio and countering Time Dilation. Lastly the more complex aspects of relativistic velocities are tackled together with the conundrum of Zero Velocity and The Speed of Light being attributes of the same event in Cyclical Space-Time, and ultimately, the prospect of superluminal velocities by interaction with parallel time-zones in a multi-layered block universe. (shrink)

The paper addresses Leon Hen.kin's proposition as a " lighthouse", which can elucidate a vast territory of knowledge uniformly: logic, set theory, information theory, and quantum mechanics: Two strategies to infinity are equally relevant for it is as universal and t hus complete as open and thus incomplete. Henkin's, Godel's, Robert Jeroslow's, and Hartley Rogers' proposition are reformulated so that both completeness and incompleteness to be unified and thus reduced as a joint property of infinity and of all (...) infinite sets. However, only Henkin's proposition equivalent to an internal position to infinity is consistent . This can be retraced back to set theory and its axioms, where that of choice is a key. Quantum mechanics is forced to introduce infinity implicitly by Hilbert space, on which is founded its formalism. One can demonstrate that some essential properties of quantum information, entanglement, and quantum computer originate directly from infinity once it is involved in quantum mechanics. Thus, these phenomena can be elucidated as both complete and incomplete, after which choice is the border between them. A special kind of invariance to the axiom of choice shared by quantum mechanics is discussed to be involved that border between the completeness and incompleteness of infinity in a consistent way. The so-called paradox of Albert Einstein, Boris Podolsky, and Nathan Rosen is interpreted entirely in the same terms only of set theory. Quantum computer can demonstrate especially clearly the privilege of the internal position, or " observer'' , or "user" to infinity implied by Henkin's proposition as the only consistent ones as to infinity. An essential area of contemporary knowledge may be synthesized from a single viewpoint. (shrink)

Quantum computer is considered as a generalization of Turing machine. The bits are substituted by qubits. In turn, a "qubit" is the generalization of "bit" referring to infinite sets or series. It extends the consept of calculation from finite processes and algorithms to infinite ones, impossible as to any Turing machines (such as our computers). However, the concept of quantum computer mets all paradoxes of infinity such as Gödel's incompletness theorems (1931), etc. A philosophical reflection on how quantum computer (...) might implement the idea of "infinite calculation" is the main subject. (shrink)

Causal finitism, the view that the causal history of any event must be finite, has garnered much philosophical interest recently—especially because of its applicability to the Kalām cosmological argument. The most prominent argument for causal finitism is the Grim Reaper argument, which attempts to show that, if infinite causal histories are possible, then other paradoxical states of affairs must also be possible. However, this style of argument has been criticized on the grounds of (i) relying on controversial modal principles, and (...) (ii) providing a false diagnosis of the paradoxes involved. In this paper, I develop a new kind of Grim Reaper argument immune to these criticisms. I show that, by using insights from the literature on time travel, causal finitists should instead argue that infinite causal histories are problematically inexplicable, as they entail the possibility of unexplained foiling mechanisms. The fruits of this paper are that (i) a novel supporting argument for the Kalām is developed, and (ii) along the way of building this argument, it is shown that the literatures on time travel and causal finitism are deeply and intimately connected. (shrink)

What is so special and mysterious about the Continuum, this ancient, always topical, and alongside the concept of integers, most intuitively transparent and omnipresent conceptual and formal medium for mathematical constructions and the battle field of mathematical inquiries ? And why it resists the century long siege by best mathematical minds of all times committed to penetrate once and for all its set-theoretical enigma ? -/- The double-edged purpose of the present study is to save from the transfinite deadlock of (...) higher set theory the jewel of mathematical Continuum -- this genuine, even if mostly forgotten today raison d'etre of all set-theoretical enterprises to Infinity and beyond, from Georg Cantor to W. Hugh Woodin to Buzz Lightyear, by simultaneously exhibiting the limits and pitfalls of all old and new reductionist foundational approaches to mathematical truth: be it Cantor's or post-Cantorian Idealism, Brouwer's or post-Brouwerian Constructivism, Hilbert's or post-Hilbertian Formalism, Goedel's or post-Goedelian Platonism. -/- In the spirit of Zeno's paradoxes, but with the enormous historical advantage of hindsight, we claim that Cantor's set-theoretical methodology, powerful and reach in proof-theoretic and similar applications as it might be, is inherently limited by its epistemological framework of transfinite local causality, and neither can be held accountable for the properties of the Continuum already acquired through geometrical, analytical, and arithmetical studies, nor can it be used for an adequate, conceptually sensible, operationally workable, and axiomatically sustainable re-creation of the Continuum. -/- From a strictly mathematical point of view, this intrinsic limitation of the constative and explicative power of higher set theory finds its explanation in the identified in this study ultimate phenomenological obstacle to Cantor's transfinite construction, similar to topological obstacles in homotopy theory and theoretical physics: the entanglement capacity of the mathematical Continuum. (shrink)

I criticise a paper by Peter Forrest in which he argues that a principle of unrestricted countable fusion has paradoxical consequences. I argue that the paradoxical consequences that he exhibits may be due to his Whiteheadean assumptions about the nature of spacetime rather than to the principle of unrestricted countable fusion.

This book is my gift to Albert Einstein on the occasion of his 142nd birthday - and is also a gift to everybody in the world he helped to shape! -/- My book adopts the view that the universe is infinite and eternal - but scientifically created. This paradox of creating eternity depends on the advanced electronics developed by future humanity. Those humans will develop time travel, plus programs that use "imaginary" time and infinite numbers like pi. They'll also (...) become the El or Elohim (names used by various religions to mean "God" or "the gods"). As astronomer Carl Sagan wrote in "Pale Blue Dot", "Many religions teach that it is the goal of humans to become gods." (I think that Elohim would be termed supernatural today, though their infinite abilities are actually natural outcomes of progress.) -/- A look through the book will tell you that some ideas are frequently repeated. This is because each article is meant to be understood without reading the others … so the same ideas show up in more than one. I’ve tried to stay away from jargon and equations unless they’re necessary (I find that they often make a subject harder to understand, not easier). All objects and events on Earth, in space, and in time (including the inevitability of world peace and immortality) are just one thing – strings of electronics' binary digits 1 and 0. -/- -/- TABLE OF CONTENTS -/- CHAPTER 1 - HYPOTHESIS OF QUANTUM GRAVITY -/- CHAPTER 2 - THE PHYSICIST AND THE PHILOSOPHER -/- CHAPTER 3 - ATTENTION MEDICAL DOCTORS! PHYSICS THEORY IMPLIES ALL PATIENTS CAN BE NORMALISED USING GRAVITY -/- CHAPTER 4 - SETI, EVOLUTION, AND TIME -/- CHAPTER 5 - ANYONS - ONE KEY TO UNLOCK GRAVITATIONAL-ELECTROMAGNETIC UNION, THE TOPOLOGICAL UNIVERSE, SPACE-TIME TRAVEL, FUTURE COMPUTERS, DARK MATTER/DARK ENERGY -/- CHAPTER 6 - WHAT CAUSES THE PLANETS TO SPEED UP AND SLOW DOWN WHILE THEY ARE ORBITING THE SUN -/- CHAPTER 7 - A BRIEF OUTLINE OF THE POSSIBLE BASICS OF COSMOLOGY IN THE 22nd CENTURY, AND WHAT IT MEANS FOR RELIGION -/- CHAPTER 8 - THE WORLD'S REAL-LIFE EXPERIMENTS WITH COVID-19 AND PHYSICS ALTER SOCIETY AND GLOBAL ECONOMICS -/- CHAPTER 9 - UNITING TIME -/- CHAPTER 10 - EXTRACTING ENERGY FROM BLACK HOLES -/- CHAPTER 11 - TIME TREK: THE 25TH CENTURY'S ANSWER TO STAR TREK. (shrink)