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In this paper I develop a philosophical account of actual mathematical infinity that does not demand ontologically or epistemologically problematic assumptions. The account is based on a simple metaphor in which we think of indefinitely continuing processes as defining objects. It is shown that such a metaphor is valid in terms of mathematical practice, as well as in line with empirical data on arithmetical cognition.

We provide a positive solution for Post’s Problem for ordinal register machines, and also prove that these machines and ordinal Turing machines compute precisely the same partial functions on ordinals. To do so, we construct ordinal register machine programs which compute the necessary functions. In addition, we show that any set of ordinals solving Post’s Problem must be unbounded in the writable ordinals.

We prove Los conjecture = Morley theorem in ZF. with the same characterization, i.e., of first order countable theories categorical in ℵα for some (eqiuvalently for every ordinal) α > 0. Another central result here in this context is: the number of models of a countable first order T of cardinality ℵα is either ≥ ∣α∣ for every α or it has a small upper bound (independent of α close to ⊐2).

We provide a short proof that if κ is a regular cardinal with κ < c, thenfor any ordinal α < min{, κ}. In particular,for any ordinal α < . This generalizes an unpublished result of E. Szemerédi that Martin's axiom implies thatfor any cardinal κ with κ < c.

We prove that, e.g., if mu >cf(mu)= aleph_0 and mu>2^{aleph_0} and every stationary family of countable subsets of mu^+ reflect in some subset of mu^+ of cardinality aleph_1, then the SCH for mu^+ (moreover, for mu^+, any scale for mu^+ has a bad stationary set of cofinality aleph_1). This answers a question of Foreman and Todorcevic who got such conclusion from the simultaneous reflection of four stationary sets.

Allow me to make a personal preliminary remark on how this article arose. Some time ago I got caught up in a dispute with a colleague, of whom I think very highly, that was conducted via e-mail and lasted two to three months. I believed that I was only stating something rather obvious and not at all original, but I was unable to convince my discussion partner.

The Bachmann-Howard structure, that is the segment of ordinal numbers below the proof theoretic ordinal of Kripke-Platek set theory with infinity, is fully characterized in terms of CARLSON’s approach to ordinal notation systems based on the notion of Σ1-elementarity.

We consider the cardinal sequences of compact scattered spaces in models where CH is false. We describe a number of models where the continuum is aleph_2 in which no such space can have aleph_2 countable levels., 6 pages

This paper is the second in a series of three culminating in an ordinal analysis of Π12-comprehension. Its objective is to present an ordinal analysis for the subsystem of second order arithmetic with Δ12-comprehension, bar induction and Π12-comprehension for formulae without set parameters. Couched in terms of Kripke-Platek set theory, KP, the latter system corresponds to KPi augmented by the assertion that there exists a stable ordinal, where KPi is KP with an additional axiom stating that every set is contained in an admissible set.

We show that it is consistent for the almost-disjointness numbera\mathfrak {a}to have countable cofinality. For example, it may be equal toℵω\aleph _\omega.

The notion of a function from ℕ to ℕ defined by recursion on ordinal notations is fundamental in proof theory. Here this notion is generalized to functions on the universe of sets, using notations for well orderings longer than the class of ordinals. The generalization is used to bound the rate of growth of any function on the universe of sets that is Σ1-definable in Kripke–Platek admissible set theory with an axiom of infinity. Formalizing the argument provides an ordinal analysis.

We describe a model-theoretic approach to ordinal analysis via the finite combinatorial notion of an α-large set of natural numbers. In contrast to syntactic approaches that use cut elimination, this approach involves constructing finite sets of numbers with combinatorial properties that, in nonstandard instances, give rise to models of the theory being analyzed. This method is applied to obtain ordinal analyses of a number of interesting subsystems of first- and second-order arithmetic.

This paper deals mainly with generalizations of results in finitary combinatorics to infinite ordinals. It is well-known that for finite ordinals ∑bT

We give generalizations of extended Poincar\'e supergravity with {\it arbitrarily many} supersymmetries in the absence of central charges in three-dimensions by gauging its intrinsic global $~SO(N)$~ symmetry. We call these \alephnull (Aleph-Null) supergravity theories. We further couple a non-Abelian supersymmetric Chern-Simons theory and an Abelian topological BF theory to \alephnull supergravity. Our result overcomes the previous difficulty for supersymmetrization of Chern-Simons theories beyond $~N=4$. This feature is peculiar to the Chern-Simons and BF theories including supergravity in three-dimensions. We also show that dimensional reduction schemes for four-dimensional theories such as $~N=1$~ self-dual supersymmetric Yang-Mills theory or $~N=1$~ supergravity theory that can generate \alephnull globally and locally supersymmetric theories in three-dimensions. As an interesting application, we present \alephnull supergravity Liouville theory in two-dimensions after appropriate dimensional reduction from three-dimensions.

A number is the number of a class which is an objective, nonactual, mathematical object. The concept of class is analyzed and it is concluded that a number is the number of a pure founded class. A tempting strategy of explaining numbers away is rejected. Some well‐known definitions of numbers are analyzed and it is concluded that this analysis purports the thesis that the unique notion of number does not exist. Numbers are conventional. Nevertheless, an argument is offered purporting the thesis that von Neumann's ordinal numbers are the ordinal numbers. Accordingly, the corresponding von Neumann's cardinal numbers are the numbers.

( S , (⩽ n ) nϵN ) is called an ordinal structure if S is a set and (⩽ n ) nϵN a family of quasi-orders on S . Since ordinal structures are used as models for ordinal data, we assign to each ordinal structure a canonical conceptual structure, a so-called concept lattice. The ordinal dimension of an ordinal structure S ≔ (S, (⩽ n ) nϵN ) is the smallest number of quasi-orders on S which determines the same conceptual structure as the concept lattice of S . It turns out that the ordinal dimension of S equals the chromatic number of a certain hypergraph. We show how to compute this hypergraph and analyse how the ordinal dimension behaves under several constructions. Furthermore, we discuss the linear case where all quasi-orders on S are assumed to be linear and the convex-ordinal case where, with every ⩽ i in the family (⩽ n ) nϵN , its dual ⩾ i is also in (⩽ n ) nϵN .

For each infinite ordinal less than or equal to ω · 2, we give a ZFC example of a topological space which has point-open type that ordinal.

The theorem that the arithmetic mean is greater than or equal to the geometric mean is investigated for cardinal and ordinal numbers. It is shown that whereas the theorem of the means can be proved for n pairwise comparable cardinal numbers without the axiom of choice, the inequality a2 + b2 ≥ 2ab is equivalent to the axiom of choice. For ordinal numbers, the inequality α2 + β2 ≥ 2αβ is established and the conditions for equality are derived; stronger inequalities are obtained for finite and infinite sequences of ordinals under suitable monotonicity hypotheses. MSC: 03E10, 04A10, 03E25, 04A25.

Recall from Chapter 13 that we write card A = card B if there is a bijection A → B. Then we say that A and B have the same cardinal number. We write card A ≤ card B if there is an injection A → B. In this latter case we say that the cardinal number of A is less than or equal to the cardinal number of B. Recall that card N ≤ card R but card N ≠ card R.

In the first part we show why ordinals and ordinal notations are naturally connected with proof theoretical research. We introduce the program of ordinal analysis. The second part gives examples of applications of ordinal analysis.

I catalog several concepts associated with finite cardinals, and then invoke them to interpret and evaluate several passages in Rips et al.'s target article. Like the literature it discusses, the article seems overly quick to ascribe the possession of certain concepts to children (and of set-theoretic concepts to non-mathematicians).

In Chapter 10 a non-empty set X was defined to have cardinality n , where n is a positive integer, when there is a bijection ℕ n → X from the standard set ℕ n = {1,2,…, n }. Such sets, together with the empty set, were defined as finite and all others as infinite . We saw (Exercise 10.1) that, given a finite set X , another set Y is also finite of the same cardinality if and only if there is a bijection X → Y . In this case we say that X and Y are equipotent . A bijection X → Y guarantees that X and Y have the same cardinality. These ideas may be extended to infinite sets. We can think of any two sets as having the same cardinality if they are equipotent so that there is a bijection between them. We can then describe this cardinality by introducing additional standard sets. For example, when we embark on counting a set X we assign a distinct element of the set to each positive integer in turn. If we exhaust the elements on reaching the integer n then we have constructed a bijection ℕ n → X and so the set is finite with cardinality n . If we never exhaust the elements we may nevertheless reach each element of the set eventually so that we have a bijection ℤ + → X . In this case the set is equipotent to the standard set ℤ + and we say that the set, although infinite, is denumerable .

In this paper, four cardinal functions are defined on the class of k-spaces. Some of the relationships between these cardinal functions are studied. Characterizations of various k-spaces are presented in terms of the existence of these cardinal functions. A bound for the ordinal invariant κ \kappa of Arhangelskii and Franklin is established in terms of the tightness of the space. Examples are presented which exhibit the interaction between these cardinal invariants and the ordinal invariants of Arhangelskii and Franklin.

This paper presents a definition of a natural number and the corresponding development of arithmetic which does not use the axiom of foundation and does not use the axiom of infinity, or in any other way use an infinite (reflexive) class, to an audience with a modest background in set theory. Both the cardinal and ordinal properties of a natural number are apparent in this definition and the ensuing development indicates clearly how these two properties plus the fact that the sets used in constructing a natural number are well founded combine to give the natural numbers their properties. The importance of using well founded sets at the most elementary level of the foundations of mathematics is thereby made clear and the groundwork laid for the introduction of the axioms of foundation and infinity.

We prove that the following are consistent with ZFC:1. 2ω = ℵω1 + #x039A;c = ℵω1 + ΚΒ = ΚU = ω2 (for measure and category simultaneously).2. .This concludes the discussion about the cofinality of Κc.

Partition properties are perhaps the most fruitful of the various methods for defining and discussing large cardinals in set theory. In this paper we weaken in a natural way the most well known of these partition properties and examine the extent to which the cardinals defined remain “large."

The proposition that a decreasing sequence of ordinals necessarily terminates has been given a new, and perhaps unexpected, importance by the rôle which it plays in Gentzen's proof of the freedom from contradiction of the “reine Zahlentheorie.” Gödel's construction of non-demonstrable propositions and the establishment of the impossibility of a proof of freedom from contradiction, within the framework of a certain type of formal system, showed that a proof of freedom from contradiction could be found only by transcending the axioms and proof processes of that formal system. Gentzen's proof succeeds by utilising transfinite induction to prove that certain sequences of reduction processes, enumerated by ordinals less than ε (the first ordinal to satisfy ε = ὡ) are finite. Were it possible to prove the restricted ordinal theorem, that a descending sequence of ordinals, less than ε, is finite, in Gentzen's “reine Zahlentheorie,” then it would be possible to determine a contradiction in that number system. In his paper, Gentzen proves the theorem of transfinite induction, which he requires, by an intuitive argument. There is also a method of reducing transfinite induction, for ordinals less than ε, to a number-theoretic principle given by Hilbert and Bernays, and a similar method by Ackermann. None of these proofs of transfinite induction is finitist.

A finite set, B, of ordinals will be called a sum-set if there are nonzero ordinals α 1 , α 2 , ⋯ , α n {\alpha _1},{\alpha _2}, \cdots ,{\alpha _n} such that the set of sums of α 1 , α 2 , ⋯ , α n {\alpha _1},{\alpha _2}, \cdots ,{\alpha _n} , in all n! permutations of the summands, is B. Let B k {B_k} denote an arbitrary k-element sum-set; we consider various matters related to the set of numbers n for which there are n summands for B k {B_k} .

The creation-of-matter hypothesis of the Bondi-Gold-Hoyle steady-state cosmology requires that in an infinite time to which the first transfinite number may be assigned the number of atoms of matter produced would be equal to the cardinal number of the set of mathematical points in the continuum. The existence of a set of finite atoms with that cardinal number is physically unacceptable. The argument for the production of a non-denumerable set of atoms, in infinite time, is given in terms of a model which is shown to be isomorphic with the original Cantor “diagonal” proof for the existence of a non-denumerable infinity. An alternative model which meets the requirements of the steady-state theory is presented; in this model, the number of atoms is explicitly no greater than countably infinite, and remains countably infinite as long as the past time of the universe is restricted to the unlimited set of finite unit-time intervals. If the origin of the steady-state universe is taken as being within that infinite set, expressed by the negative natural numbers, the contradiction of an atom at every mathematical point does not arise. The contradiction does arise if the origin is not within the set of finite numbers, and accordingly there is a restriction as to which concept of infinite past may properly be maintained in the steady-state theory.

A new property of pseudo-open functions is presented in this paper. Pseudo-open functions are monotonic decreasing with respect to ordinal and cardinal invariants defined by compact and sequential closures. A weak form of continuity, called k -continuous, is defined, characterized and used in the proof of the monotonicity properties of pseudo-open mappings. The relationships between the classes of k -continuous and sequentially continuous mappings in the category of all topological spaces and the continuous mappings in the subcategories of k -spaces and sequential spaces are presented.

Cantor defined the cardinal number of a set M to be “the general concept which, by means of our active faculty of thought, arises from the set M when we make abstraction of the nature of its various elements m and of the order in which they are given.” He denoted this cardinal number by \( \overline{\overline M} \). The two bars indicate the two levels of abstraction needed to produce the cardinal number from M. With only one level of abstraction, that is, by only abstracting of the nature of its various elements, we obtain the ordinal number \( \overline M \). Cantor’s definition of cardinal number is clearly not an operational one. Indeed Cantor’s words suggest that cardinal numbers are psychological entities rather than mathematical objects.

Publisher Summary This chapter discusses cardinal functions. Cardinal functions extend important topological properties as countable base, separable, and first countable to higher cardinality. Cardinal functions allow one to formulate, generalize, and prove results of a particular type in a systematic and elegant manner. The cardinal functions also allow one to make precise quantitative comparisons between certain topological properties. Prerequisite for work in cardinal functions is knowledge of cardinal and ordinal numbers and transfinite constructions. The chapter also describes the cardinal functions on the two most important classes of abstract topological spaces, namely, compact spaces and metrizable spaces. It also describes cardinal functions that are used to obtain bounds on the cardinality of a space. An infinite cardinal that is not regular is said to be singular. Theorems from combinatorial set theory play an important role in cardinal functions.