Size and Function

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

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.

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

Laerte Ferreira Morgado, 2024

In view of the paradox of Euclid's principle, that the part is smaller than the whole, in comparison with Cantor's one-to-one relationship, allowing proper subsets to be placed in a one-to-one relationship with the larger set, I present a refutation of this one-to-one relationship and, preserving, therefore, subject to further consideration, the principle of Euclid, I propose a construction of the natural numbers and an algorithmic definition of the size of finite sets of natural numbers, in addition to definitions of sizes of infinite sets of natural numbers and of sets of non-negative real numbers, bounded or not, which are based on the notion of "asymptotic density" and make use of the notion of limit and the Lebesgue measure regarding subsets of the non-negative real line.

Research Gate, 2024

Galileo found the idea of larger or smaller infinities impossible to comprehend, then unintentionally made them equal. Over 200 years later, Cantor insisted that his formulation of transfinite numbers was not arbitrary, but then unintentionally inherited Galileo's methodological error in the way one-to-one correspondence was used. Genuine one-to-one correspondence produces exact agreement between set size, cardinality, natural density, and probability measures. The error is subtle, and literally not visible, enabling it to evade detection for more than 380 years. It takes more than a few pages to unpick the detail, since every living mathematician has had to prove that some extremely low density (and some extremely high density) sets are countably infinite as a minor part of their undergraduate studies.

2003

In 1891 Cantor presented two proofs with the purpose to establish a general theorem that any set can be replaced by a set of greater power. Cantor's power set theorem can be considered to be an extension of Cantor's 1891 second proof and its argument makes use of a well-known self-referring statement. In this article it is shown that, defining the relative complement of the self-referring statement, Cantor's power set theorem cannot be derived. Moreover, it is given a refutation of the first proof, the so-called Cantor's diagonal argument.

Research Gate, 2024

Historically, Galilean 1:1 correspondence has been used to prove that almost all infinite sets have the same cardinality ('size') as the naturals. This is a fundamentally flawed procedure which results in Galileo's Paradox, the claim that two infinite sets are equal in size, despite one set having almost zero density compared to the other. Here Galileo's Paradox is re-categorised as a fallacy. Infinite set sizes can be correctly measured using genuine n:m correspondence where possible, recognising that some infinite sets cannot fit to such a simple pattern. Galileo's Gaff is thinking that infinite sets are in 1:1 correspondence with the naturals, when some cannot even be put into n:m correspondence. Here we use 'Galilean' as being synonymous with false, bogus, and invalid.

The Annals of Mathematics, 1998