Choiceless Ramsey Theory of Linear Orders

The study of partition relations for cardinal numbers introduced by Erdös and his school in the 1950's has, for the past several years, had a profound impact in logic. Unfortunately, quite early in their development, it was noticed by Rado [1] that the potentially most fruitful class of such relations, infinite exponent partition relations, were always in contradiction with the axiom of choice (AC). As a result, such relations were overlooked. This turned out to be a mistake; for, as has been noticed recently, a close study of infinite exponent partition relations is both interesting and rewarding. For example, there are weakened versions of such relations which are provable in ZF and which have valuable applications in recursion theory and set theory. In addition, the pure theory of these relations, like that of the axiom of determinateness, is fruitful as well as elegant. For more background here one should refer to [3].At any rate, with a more detailed look at infinite exponent partition relations came a more refined version of Rado's original theorem. Specifically, Rado used the full axiom of choice to carefully construct partitions to violate any desired relation—a more sophisticated look at the actual theory of such relations indicated how one could put together some desired partitions using only well-ordered choice [3].The distinction between well-ordered choice and full choice is by no means vacuous in this context. For Mathias has shown [4] that the simplest infinite exponent partition relation, ω → (ω)ω, is consistent with countable choice (well-ordered choice of length ℵ0) and, in fact, is consistent with dependent choice.

It is hard to find nontrivial positive partition relations which hold for many ordinals in ordinary set theory, or even ordinary set theory with the additional assumption of the Generalized Continuum Hypothesis.Erds, Hajnal and Milner have proved that limit ordinals a < u;"+ satisfy a positive partition relation that can be expressed in graph theoretic terms.In symbols one writes a - (a, infinite path)2 to mean that every graph on an ordinal a either has a subset order isomorphic to a in which no two points are joined by an edge or has an infinite path.This positive result generalizes to ordinals of cardinality Nm for m a natural number.However, the argument, based on a set mapping theorem, works only on the initial segment of the limit ordinals of cardinality Nm for which the set mapping theorem is true.In this paper, the Generalized Continuum Hypothesis is used to construct counterexamples for a cofinal set of ordinals of cardinality Nm, where m is a natural number at least two. Introduction.Erds, Hajnal and Milner [3] were the first to look at the partition relation a -> (a, infinite path)2 in their 1969 paper on set mappings.They were able to prove positive results for many ordinals, but their method using set mappings only worked for ordinals of relatively simple order structure.However, this particular relation looked most promising as a nontrivial one that many ordinals might satisfy.J. Larson [6] used Martin's Axiom to extend the positive result to limit ordinals less than the continuum.(It is not hard to construct counterexamples for sucessor ordinals.)However, under the assumption of the Generalized Continuum Hypothesis, above the continuum, counterexamples start to appear, as indicated in the statement of the main theorem below.THEOREM.Assume the Generalized Continuum Hypothesis.For every positive integer n > 2, there is a cofinal set of ordinals a < ojn so that a -*+ (a, infinite path)2.For ordinals of power continuum, the situation appears to be more delicate.In A diamond example of an ordinal graph with no infinite paths, J. Baumgartner and J. Larson [1] used Jensen's Diamond Principle to show that for all ordinals a with w"+2 < a < w2, the negative partition relation a -** (a, infinite path)2 holds.

This Chapter extends the ideas deal with description of individual opinions and social decisions that were briefly presented in Chapter 1. Section 2.2 investigates various types of binary relations and defines the notion of pair-dominant, or rationalizable by a binary relation, choice. In this Section main types of binary relations used in social choice theory are defined — linear orders, weak orders, strict partial orders, and acyclic relations. Section 2.3 gives another well-known concept of choice, based on the ‘extremization paradigm’ — that of a choice function rationalizable by some numerical (utility) function, studies the classical unicriterial and multicriterial choice models. We study three different types of multicriteral models — Paretian, weak Paretian, and joint-extremal ones. The equivalence of Paretian and weak Paretian models is shown. Section 2.4 lists Expansion-Contraction Axioms (rationality conditions) for choice functions, and studies their mutual relations. The axioms of Heredity, Concordance, Outcast, Arrow’s Choice Axiom in different versions, Choice Monotonicity, Choice Resolutness are studied. Section 2.5 establishes relations between the classes of rationalizable choice functions and the domains in the space of choice functions described in Section 2.4. The classical pair-dominant choice functions are shown to define a small, but important part of the models describing individual opinions and social decisions. Section 2.6 contains concluding remarks and gives a guide to the literature.

We investigate whether the existence of long linear orders can be proved without the Axiom of Choice. This question has two different answers depending on its formalization.

We classify definable linear orders in o-minimal structures expanding groups. For example, let ( P , ≺ ) (P,\prec ) be a linear order definable in the real field. Then ( P , ≺ ) (P,\prec ) embeds definably in ( R n + 1 , &gt; lex ) (\mathbb {R}^{n+1},&gt;_{\text {lex}}) , where &gt; lex &gt;_{\text {lex}} is the lexicographic order and n n is the o-minimal dimension of P P . This improves a result of Onshuus and Steinhorn in the o-minimal group context.

In this article, we will first discuss the completeness of real numbers in the context of an alternate definition of the straight line as a geometric continuum. According to this definition, points are not regarded as the basic constituents of a line segment and a line segment is considered to be a fundamental geometric object. This definition is in particular suitable to coordinatize different points on the straight line preserving the order properties of real numbers. Geometrically fundamental nature of line segments are required in physical theories like the string theory. We will construct a new topology suitable for this alternate definition of the straight line as a geometric continuum. We will discuss the cardinality of rational numbers in the later half of the article. We will first discuss what we do in an actual process of counting and define functions well-defined on the set of all positive integers. We will follow an alternate approach that depends on the Hausdorff topology of real numbers to demonstrate that the set of positive rationals can have a greater cardinality than the set of positive integers. This approach is more consistent with an actual act of counting. We will illustrate this aspect further using well-behaved functionals of convergent functions defined on the finite dimensional Cartezian products of the set of positive integers and non-negative integers. These are similar to the partition functions in statistical physics. This article indicates that the axiom of choice can be a better technique to prove theorems that use second-countability. This is important for the metrization theorems and physics of spacetime.

In the paper [S1] I introduced a game, denoted by MG(J) (where J is a free ideal on some infinite set S) and called “the meager nowhere dense game for J”. The special case when J is the collection of finite subsets of the set S is called the countable-finite game on S. It proceeds as follows.First player ONE picks a countable set C1, then player TWO picks a finite set F1. Then in the second inning ONE picks a countable set C2 with C1 ⊂ C2 (unless explicitly indicated otherwise, “⊂” means “is a proper subset of”) and TWO responds with a finite set F2, and so on. The players construct a sequence (C1,F1,C2,F2,…,Ck,Fk,…) where for each positive integer k(i) Ck denotes ONE's countable set picked during the kth inning,(ii) Fk denotes TWO's finite set picked during the kth inning, and(iii) Ck ⊂ Ck + 1.Such a sequence is a play of the countable-finite game on S, and TWO wins this play if is contained in . The notion of a winning perfect information strategy is defined as usual (see, for example, [S1]). Zermelo-Fraenkel set theory together with the axiom of choice (denoted by ZFC; for a statement of the axioms see pp. xv–xvi of [K]) is a strong enough theory to build a winning perfect information strategy for player TWO in this game.Does TWO have a winning strategy requiring less than perfect information? Fix a positive integer k. A strategy of TWO which requires knowledge of only at the most the k most recent moves of ONE is said to be a k-tactic. For the countable-finite game on an infinite set S the following facts about the existence of winning k-tactics for TWO are proved in [S1]:1) TWO does not have a winning 1-tactic (Theorem 1 of [S1]).2) If the cardinality of S is less than ℵ2 then TWO has a winning 2-tactic (Corollary 4 of [S1]).3) If TWO has a winning k-tactic in the countable-finite game on an infinite set S, then TWO has a winning 3-tactic (Proposition 15 of [SI]).

Set theory is a rich and beautiful subject whose fundamental concepts permeate virtually every branch of mathematics. One could say that set theory is a unifying theory for mathematics, since nearly all mathematical concepts and results can be formalized within set theory. This textbook is meant for an upper undergraduate course in set theory. In this text, the fundamentals of abstract sets, including relations, functions, the natural numbers, order, cardinality, transfinite recursion, the axiom of choice, ordinal numbers, and cardinal numbers, are developed within the framework of axiomatic set theory. The reader will need to be comfortable reading and writing mathematical proofs. The proofs in this textbook are rigorous, clear, and complete, while remaining accessible to undergraduates who are new to upper-level mathematics. Exercises are included at the end of each section in a chapter, with useful suggestions for the more challenging exercises.

The conjugation class of a special Sturmian morphism carries a natural linear order by virtue of the two elementary conjugations \(conj_a\) and \(conj_b\) with the single letters a and b, with the standard morphism of the class as the smallest element in this order. We show that a lexicographic order on the morphisms of the given conjugation class can be defined that matches the conjugation order.

Combinatorics with all its various aspects is a broad field of Mathematics which has many applications in areas like Topology, Group Theory and even Analysis. A reason for its wide range of applications might be that Combinatorics is rather a way of thinking than a homogeneous theory, and consequently Combinatorics is quite difficult to define. Nevertheless, let us start with a definition of Combinatorics which will be suitable for our purpose: Combinatorics is the branch of Mathematics which studies collections of objects that satisfy certain criteria, and is in particular concerned with deciding how large or how small such collections might be. Below we give a few examples which should illustrate some aspects of infinitary Combinatorics. At the same time, we present the main topics of this book, which are the Axiom of Choice, Ramsey Theory, cardinal characteristics of the continuum, and forcing.KeywordsRamsey TheoryAxiomsCardinal CharacteristicsInfinite Pigeonhole PrincipleRamsey UltrafilterThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This paper defends the claim that Habermas’s distinction between the moral and the ethical depends on some un-stated assumptions that he is not willing to accept. One of the assumptions concerns the presumption of the equality of the discourse participants; the other implicit assumption in his discourse theory of morality concerns the issue of reciprocity and symmetrical power relations. He takes each discourse participant as free to initiate a dialogue and to challenge the validity claims of discourse participants. Such assumptions undermine Habermas’s commitment for ethical neutrality and his rejection of the substantive notions of truth. He argues that although our everyday interactions are pervaded by distortions and asymmetrical power relations there are operative principles such as reciprocity and symmetrical power relations inherent in language. The question precisely is: why should we take the principles that are inherent in language for granted? It appears as though Habermas has a pre-given reason to endorse the principles in question. By disapproving the use of threat, coercion and intimidation in the process of argumentation, and instead by insisting for the employment of, among others, symmetrical relation, Habermas is affirming the claim that humans are beings that command each other’s respect. But the question precisely is, why do they command each other’s respect? It appears that we cannot commit ourselves to this claim without affirming a given ontology of the human and Habermas is exactly doing that.

Interval temporal logics formalize reasoning about interval structures over (usually) linearly ordered domains, where time intervals are the primitive ontological entities and truth of formulae is defined relative to time intervals, rather than time points. In this paper, we introduce and study Metric Propositional Neighborhood Logic (MPNL) over natural numbers. MPNL features two modalities referring, respectively, to an interval that is &amp;ldquo;met by&amp;rdquo; the current one and to an interval that &amp;ldquo;meets&amp;rdquo; the current one, plus an infinite set of length constraints, regarded as atomic propositions, to constrain the lengths of intervals. We argue that MPNL can be successfully used in different areas of artificial intelligence to combine qualitative and quantitative interval temporal reasoning, thus providing a viable alternative to well-established logical frameworks such as Duration Calculus. We show that MPNL is decidable in double exponential time and expressively complete with respect to a well-defined subfragment of the two-variable fragment FO2[N, =, &amp;lt;, s] of first-order logic for linear orders with successor function, interpreted over natural numbers. Moreover, we show that MPNL can be extended in a natural way to cover full FO2[N, =, &amp;lt;, s], but, unexpectedly, the latter (and hence the former) turns out to be undecidable.

We work in a Zermelo-Fraenkel set theory without the Axiom of Choice. In the appendix to his paper “Sur les ensembles finis”, Tarski proposed a finiteness criterion that we have called “C-finiteness”: a nonempty set is called “C-finite” if it cannot be partitioned into two blocks, each block being equivalent to the whole set. Despite the fact that this criterion can be shown to possess several features that are undesirable in a finiteness criterion, it has a fair amount of intrinsic interest. In Section 1 of this paper we look at a certain class of C-finite sets; in Section 2 we derive a few consequences from the negation of C-finiteness; and in Section 3 we show that not every C-infinite set necessarily possesses a linear ordering. Any unexplained notation is given in my paper, “Some definitions of finiteness”, Bull. Austral. Math. Soc. 5 (1971).

This chapter shows that Ludwig Wittgenstein's philosophy of mathematics has much less in common with intuitionism than is usually assumed, although some strategical moves are barely distinguishable. At least one such move cannot be overlooked because of its importance for the development of Wittgenstein's philosophy: it concerns quantification theory. This chapter shows that Wittgenstein's stance on quantification was, however, even more radical than that of intuitionists. This is a topic about which Wittgenstein changed his mind around 1929. The search for the source of Wittgenstein's new ideas on quantification leads directly to Hermann Weyl. He actually developed original ideas on topics such as quantification. According to him, propositions containing an existential quantifier ranging over the natural numbers do not possess the full status of judgement.

We write and for the cardinalities of the set of permutations with n non‐fixed points and the set of subsets with n elements, respectively, of a set which is of cardinality , where n is a natural number greater than 1. With the Axiom of Choice, and are equal for all infinite cardinals . We show, in ZF, that if is assumed, then for any infinite cardinal . Moreover, the assumption cannot be removed for and the superscript cannot be replaced by n. We also show under that for any infinite cardinal , implies is Dedekind‐infinite.