A Poor Man's introduction to the Art of Counting Infinities

2012

For hundreds of years, we've accepted the 'uncountability' of the irrational numbers, a proper subset of the reals-our common 'everyday' numbers. We've 'proven' this and accepted it as fact. For math students, you won't find out until graduate school how 'different' the irrationals are from the rationals-the concept of Lebesgue measure is encountered in advanced undergraduate math courses for the first time (again, considering math students). Math is the most precise science. We speak in precise terms, we define things precisely, and we state the exact conditions when a theorem is applicable. It's the reason why all other sciences use math. In a very real sense, using math automatically gives 'credibility' to ideas. (Abuses of this relationship abound with cranks/crackpots and frauds.) To make the 'uncountable' countable, make this 'new' knowledge more accessible to younger students thereby encouraging young scientists and engineers to understand our common everyday numbers more deeply (and thereby encouraging more and better scientists and engineers), and make the unsatisfactory imprecision of 'uncountability' precise, we propose the following concepts. Let's attempt to appropriately modify the notion of infinite countability. We can label infinite countability any way we want.. You can call it IC or modify the symbol for infinity, ∞. It almost doesn't matter what you label it as long as you're consistent and explicit about what it means. Let's go with ∞ c for simplicity. Now that we've labeled infinite countability, what are the allowable ways, within mathematics, that we can modify that symbol? Immediately obvious are multiplication and exponentiation: 2∞ c and ∞ c 2 , but right away, we must decide what that means precisely! Let's ignore multiplication and focus on exponentiation. There seems to be more 'power' (to explain) and relevancy in that concept. If we define ∞ c 2 to mean the number of elements in Z 2 , the set of all ordered pairs of integers, we've made 'infinite exponentiation' precise. Right away, we have applicability: the rationals, Q. One famous proof of the countability of the rationals actually ties these concepts together. From this proof alone, we can inspect the order of the rationals: 2. So we've expanded the notion of 'size' (of a set) to include countably infinite sets and orders of magnitude of those. This is actually fundamental and relevant as we shall soon see. The size of a set no longer is limited to finite sets alone. We can meaningfully speak about countably infinite sets as well.. In this 'new framework' (it's not really new; it's 'been there' waiting for us to discover it), we can precisely identify the number of elements in countably infinite sets as well. The 'question' from mathematicians (rephrasing line one above) becomes: are the irrational numbers countable in the 'new' sense above? To assume "no" is not math/science. We must attempt to answer this question rationally-based on facts / logical arguments. The following six brief articles, together, do exactly that. Please forgive any 'sloppiness in notation'.. When appropriate, definitions are precise and explicit. And, we find we give way too much 'authority' (associate 'deep meaning') to the transcendentals.. It's my sincere hopes these essays do not cause resentment or divisiveness within the math (and math education) community.. It is my hope we can work together formally developing these ideas, integrating them appropriately into mathematics, math education, and above all else-keeping the knowledge accessible to 'laypeople' and students..

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

This article sets out to analyse some of the most basic elements of our number concept – of our awareness of the one and the many in their coherence with multiplicity, succession and equinumerosity. On the basis of the definition given by Cantor and the set theoretical definition of cardinal numbers and ordinal numbers provided by Ebbinghaus, a critical appraisal is given of Frege's objection that abstraction and noticing (or disregarding) differences between entities do not produce the concept of number. By introducing the notion of subject functions, an account is advanced of the (nominalistic) reason why Frege accepted physical, kinematic and spatial properties (subject functions) of entities, but denied the ontic status of their quantitative properties (their quantitative subject function). With reference to intuitionistic mathematics (Brouwer, Weyl, Troelstra, Kreisel, Van Dalen) the primitive meaning of succession is acknowledged and connected to an analysis of what is entailed in the term 'Gleichzahligkeit' ('equinumerosity'). This expression enables an analysis of the connections between ordinality and cardinality on the one hand and succession and wholeness (totality) on the other. The conceptions of mathematicians such as Frege, Cantor, Dedekind, Zermelo, Brou-wer, Skolem, Fraenkel, Von Neumann, Hilbert, Bernays and Weyl, as well as the views of the philosopher Cassirer, are discussed in order to arrive at an assessment of the relation between ordinality and cardinality (also taking into account the relation between logic and arithmetic) – and on the basis of this evaluation, attention is briefly given to the definition of an ordered pair in axiomatic set theory (with reference to the set theory of Zermelo-Fraenkel) and to the definition of an ordered pair advanced by Wiener and Kuratowski. Introduction We start our discussion with reference to the problem of the one and the many (multi-plicity, plurality) and then proceed to incorporate the nature of succession in our reflections. Considering numerical relations within non-arithmetical contexts makes it

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.

Problems with infinities and infinitesimals For the last hundred years or so, mathematicians have finally 'understood the infinite'-or so they think. Despite several thousand years of previous uncertainty, and very stiff opposition from notable mathematicians (Kronecker, Poincaré, Weyl, Brouwer and numerous others), the theory of infinite point sets and hierarchies of cardinal and ordinal numbers introduced by Cantor is now the established orthodoxy, and dominates modern logic, topology and analysis. Most mathematicians believe that there are infinite sets, but can't demonstrate this claim, and have difficulty resolving the paradoxes that beset the subject a hundred years ago, except to assert that there are dubious 'axioms' that supposedly extract us from the quagmire. If you have read my recent diatribe Set Theory: Should You Believe? you will know that I no longer share this religion, and I tried to convince you that you shouldn't either. Several readers of that paper have asked me: so what should we believe? What are the alternatives to the modern set theoretical paradigm? To begin the slow redevelopment of mathematics as a subject that actually makes complete sense, I now propose tell you what a number is, what an infinity is, and what an infinitesimal is. You'll see how to understand these notions without unnecessary philosophy. I am not going to define 'infinite sets'-for the simple reason that they don't really exist. But I will explain how some of the modern theory of 'ordinals' may be recast as a purely finite theory which makes complete and precise sense, requires no assumptions or 'axioms'-and reveals interesting natural connections with computer science. Then you'll see how the inverses of these infinities are naturally infinitesimals, giving a concrete, specifiable approach to nonstandard analysis. After an initial (skeptical) look at the usual 'set-theoretical' approach to 'ordinals', we will start by defining natural numbers in a simplified way, without use of set theory. The fact is that sets are just not the all-important data structures that many believe-in computer science, for example, the notions of list and multiset are arguably more important. James Franklin told me about

Resumen. El infinito como método en la teoría de conjuntos y la matemática Este artículo da cuenta de la aparición histórica de lo infinito en la teoría de conjuntos, y de cómo lo tratamos dentro y fuera de las matemáticas. La primera sección analiza el surgimiento de lo infinito como una cuestión de método en la teoría de conjuntos. La segunda sección analiza el infinito dentro y fuera de las matemáticas, y cómo deben adoptarse.

Synthese, 2004

I follow standard mathematical practice and theory to argue that the natural numbers are the finite von Neumann ordinals. I present the reasons standardly given for identifying the natural numbers with the finite von Neumann's (e.g., recursiveness; well-ordering principles; continuity at transfinite limits; minimality; and identification of n with the set of all numbers less than n). I give a detailed mathematical demonstration that 0 is { } and for every natural number n, n is the set of all natural numbers less than n. Natural numbers are sets. They are the finite von Neumann ordinals.

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