Cantor Paradox

2014

International Journal of Open Problems in Computer Science and Mathematics, 2012

The author introduces the concept of intrinsic set property, by means of which the well-known Cantor's Theorem can be deduced. As a natural consequence of this fact, it is proved that Cantor's Theorem need not imply the existence of a tower of different-size infinities, because the impossibility of defining a bijection between any infinite countable set and its power can be a consequence of the existence of any intrinsic property which does not depend on size.

Mathematics, 2020

Cantor thought of the principles of set theory or intuitive principles as universal forms that can apply to any actual or possible totality. This is something, however, which need not be accepted if there are totalities which have a fundamental ontological value and do not conform to these principles. The difficulties involved are not related to ontological problems but with certain peculiar sets, including the set of all sets that are not members of themselves, the set of all sets, and the ordinal of all ordinals. These problematic totalities for intuitive theory can be treated satisfactorily with the Zermelo and Fraenkel (ZF) axioms or the von Neumann, Bernays, and Gödel (NBG) axioms, and the iterative conceptions expressed in them.

Scientific American, 1967

T he abstract t h e o~ of sets i5 currently in a state of change that ill meral wa!"~ IS ana10gou~ to the 19th-centup m~olution m g t w n e q . 145; m any revolution, phticd or scientific, it i s d B d t for those parhcipthp in the revolution or w5hlessi1lp it to foretell its ultimate mnsequences, except perhaps that they \\*ill be profound. One thing that can b done i s to to use the past as a guide to the future. It is an unreliable guide, to be sure. but ktter than none.

The actual infinity Aristotle-Cantor , potential infinity . The origins of Cantor’s infinity, aleph null, the diagonal argument The natural infinity , continuum The mathematical infinity A first classification of sets Three notable examples of countable sets The 1-1 correspondence, equivalent sets, cardinality . The theory of transfinite numbers Existence and construction, existence proofs

Notre Dame Journal of Formal Logic, 2015

This paper expands upon a way in which we might rationally doubt that there are multiple sizes of infinity. The argument draws its inspiration from recent work in the philosophy of truth and philosophy of set theory. More specifically, elements of contextualist theories of truth and multiverse accounts of set theory are brought together in an effort make sense of Cantor's troubling theorem. The resultant theory provides an alternative philosophical perspective on the transfinite, but has limited impact on everyday mathematical practice. This relativity of cardinalities is very striking evidence of how far abstract formalistic set theory is removed from all that is intuitive. One can indeed construct systems that faithfully represent set theory down to the last detail. But as soon as one applies the finer instruments of investigation all this fades away to nothing. Of all the cardinalities only the finite ones and the denumerable one remain. Only these have real meaning; every thing else is formalistic fiction. (von Neumann, 1967) The aim of this paper is to take seriously the idea that we have, in some sense, misunderstood the message of Cantor's theorem; or at the least, that in hindsight we have driven headlong into the transfinite when we could have paused a moment longer to consider an alternative. My goal is to demonstrate that Cantor's theorem can be understood more like the liar paradox, as a kind of fork in the road. The crucial idea is that, in admitting there are multiple sizes of infinity, we have done irreparable damage to our naïve conception of the infinite. My goal in this paper is to demonstrate that we may coherently reject the multiplicity of infinite cardinalities and to illustrate the value of this perspective. 1

Mathematics, 2019

In this paper, we illustrate the paradox concerning maximally consistent sets of propositions, which is contrary to set theory. It has been shown that Cantor paradoxes do not offer particular advantages for any modal theories. The paradox is therefore not a specific difficulty for modal concepts, and it also neither grants advantages nor disadvantages for any modal theory. The underlying problem is quite general, and affects anyone who intends to use the notion of "world" in its ontology.

In this article, as implied by the title, I intend to argue for the unattainability of Cantor’s Absolute at least in terms of the proof-theoretical means of set-theory and of the theory of large cardinals. For this reason a significant part of the article is a critical review of the progress of set-theory and of mathematical foundations toward resolving problems which to the one or the other degree are associated with the concept of infinity especially the one beyond that of the natural intuition of natural numbers.

European Journal for Philosophy of Religion

It is often alleged that Cantor’s views about how the set theoretic universe as a whole should be considered are fundamentally unclear. In this article we argue that Cantor’s views on this subject, at least up until around 1896, are relatively clear, coherent, and interesting. We then go on to argue that Cantor’s views about the set theoretic universe as a whole have implications for theology that have hitherto not been sufficiently recognised. However, the theological implications in question, at least as articulated here, would not have satisfied Cantor himself.

Annals of Pure and Applied Logic, 1995