Some axioms for constructive analysis

This note explores the common core of constructive, intuitionistic, recursive and classical analysis from an axiomatic standpoint. In addition to clarifying the relation between Kleene’s and Troelstra’s minimal formal theories of numbers and number-theoretic sequences, we propose some modified choice principles and other function existence axioms which may be of use in reverse constructive analysis. Specifically, we consider the function comprehension principles assumed by the two minimal theories EL and M, introduce an axiom schema CF_d asserting that every decidable property of numbers has a characteristic function, and use it to describe a precise relationship between the minimal theories. We show that the axiom schema AC₀₀ of countable choice can be decomposed into a monotone choice schema AC ₀₀ ^m (which guarantees that every Cauchy sequence has a modulus) and a bounded choice schema BC₀₀. We relate various (classically correct) axiom schemas of continuous choice to versions of the bar and fan theorems, suggest a constructive choice schema AC_1/2,0 (which incidentally guarantees that every continuous function has a modulus of continuity), and observe a constructive equivalence between restricted versions of the fan theorem and correspondingly restricted bounding axioms ${AB_{1/2,0}^{2^{\mathbb{N}}}}$ . We also introduce a version WKL!! of Weak K?nig’s Lemma with uniqueness which is intermediate in strength between WKL and the decidable fan theorem FT_d.     

Finite axiomatizability of local set theory

The need for modifying axiomatic set theories was caused, in particular, by the development of category theory. The ZF and NBG axiomatic theories turned out to be unsuitable for defining the notion of a model of category theory. The point is that there are constructions such as the category of categories in naïve category theory, while constructions like the set of sets are strongly restricted in the ZF and NBG axiomatic theories. Thus, it was required, on the one hand, to restrict constructions similar to the category of categories and, on the other hand, adapt axiomatic set theory in order to give a definition of a category which survives restricted construction similar to the category of categories. This task was accomplished by promptly inventing the axiom of universality (AU) asserting that each set is an element of a universal set closed under all NBG constructions. Unfortunately, in the theories ZF + AU and NBG + AU, there are toomany universal sets (as many as the number of all ordinals), whereas to solve the problem stated above, a countable collection of universal sets would suffice. For this reason, in 2005, the first-named author introduced local-minimal set theory, which preserves the axiom AU of universality and has an at most countable collection of universal sets. This was achieved at the expense of rejecting the global replacement axiom and using the local replacement axiom for each universal class instead. Local-minimal set theory has 14 axioms and one axiom scheme (of comprehension). It is shown that this axiom scheme can be replaced by finitely many axioms that are special cases of the comprehension scheme. The proof follows Bernays’ scheme with significant modifications required by the presence of the restricted predicativity condition on the formula in the comprehension axiom scheme.     

Eine ErweiterungT(V′) des Ordinalzahlensystems 58-0158-0158-01(Λ0) von G. Jäger

Summary This paper gives a recursive generalization of a strong notation system of ordinals, which was devellopped by Jäger [3]. The generalized systemT(V) is based on a hierarchy of Veblen-functions for inaccessible ordinals. The definition ofT(V) assumes the existence of a weak Mahlo-ordinal. The wellordering ofT(V) is provable in a formal system of second order arithmetic with the axiom schema of ₂ ¹ -comprehension in a similar way, as it is proved in [6] for the weaker notation systemT(V).     

An axiom system for the weak monadic second order theory of two successors

A compelte axiom system for the weak monadic second order theory of two successor functions, W2S, is presented. The axiom system consists, roughly, of the generalized Peano axioms and of an inductive definition of the finite sets. For the proof, methods of J. R. Buchi and J. Doner are used to obtain a new decision procedure for W2S, whose proofs are easily formalized. Different finiteness axioms are discussed. This paper was written while the author was visiting at Purdue University, and appeared first as Report CSD TR-56, Purdue University, 1971.     

On the Consistency of a Positive Theory

In positive theories, we have an axiom scheme of comprehension for positive formulas. We study here the “generalized positive” theory GPK_∞⁺. Natural models of this theory are hyperuniverses. The author has shown in [2] that GPK_∞⁺ interprets the Kelley Morse class theory. Here we prove that GPK_∞⁺ + AC_WF (AC_WF being a form of the axiom of choice allowing to choose elements in well-founded sets) and the Kelley-Morse class theory with the axiom of global choice and the axiom “On is ramifiable” are mutually interpretable. This shows that GPK_∞⁺ + AC_WF is a “strong” theory since “On is ramifiable” implies the existence of a proper class of inaccessible cardinals.     

The non-planarity of K 5 and K 3,3 as axioms for plane ordered geometry

The Pasch axiom is shown to be equivalent, given the linear order axioms, to the conjunction of its outer form with the statement that K ₅ (or K _3,3) is not planar.     

Localizing the axioms

We examine what happens if we replace ZFC with a localistic/relativistic system, LZFC, whose central new axiom, denoted by Loc(ZFC), says that every set belongs to a transitive model of ZFC. LZFC consists of Loc(ZFC) plus some elementary axioms forming Basic Set Theory (BST). Some theoretical reasons for this shift of view are given. All ${\Pi_2}We examine what happens if we replace ZFC with a localistic/relativistic system, LZFC, whose central new axiom, denoted by Loc(ZFC), says that every set belongs to a transitive model of ZFC. LZFC consists of Loc(ZFC) plus some elementary axioms forming Basic Set Theory (BST). Some theoretical reasons for this shift of view are given. All P₂{\Pi_2} consequences of ZFC are provable in LZFC. LZFC strongly extends Kripke-Platek (KP) set theory minus Δ₀-Collection and minus ?{\in} -induction scheme. ZFC+ “there is an inaccessible cardinal” proves the consistency of LZFC. In LZFC we focus on models rather than cardinals, a transitive model being considered as the analogue of an inaccessible cardinal. Pushing this analogy further we define α-Mahlo models and P₁¹{\Pi_1^1} -indescribable models, the latter being the analogues of weakly compact cardinals. Also localization axioms of the form Loc(ZFC+f){Loc({\rm ZFC}+\phi)} are considered and their global consequences are examined. Finally we introduce the concept of standard compact cardinal (in ZFC) and some standard compactness results are proved.     

On the predecessor relation in abstract algebras

In this paper we generalize the Dedekind theory of order for the natural numbers N to abstract algebras with arbitrarily many finitary or infinitary operations. For any algebra ??, we introduce an algebraic predecessor relation P?? and its transitive hull P*?? coinciding in N with the unary injective successor function' resp. the >-relation. For some important classes of algebras ??, including Peano algebras (absolutely free algebras, word algebras), the algebraic predecessor relation is well-founded. Hence, its transitive hull, the natural ordering >?? of ??, is a well-founded partial order, which turns out to be a convenient device for classifying Peano algebras with respect to the number of operations and their arities. Moreover, the property of well-foundedness is an efficient tool for giving simple proofs of structure theorems as, e. g., that the class of all Peano algebras is closed under subalgebras and non-void direct products. - Finally, we will show how in the case of a formal language ??, i. e., the Peano algebra ?? of expressions (= terms & formulas), relations P??, resp. P*_?? can be used to define basic syntactical notions as occurences of free and bound variables etc. without any reference to a particular representation (“coding”) of the formal language. MSC: 03B22, 03E30, 03E75, 03F35, 08A55, 08B20.     

Starkes und überstarkes Auswahlaxiom

It is well known that equivalence holds between the weak axiom of choice (AC) and the well ordering principle (WOP) for sets, resp. between strong AC and WOP for classes. It will be shown that in a theory PC^* with inpredicative classes (i. e. with no restriction of quantification in the defining formula) the super-strong AC used by the informally working mathematician is equivalent to a superstrong WOP. The equivalence between strong AC and super-strong AC is implied by a conditionC refutable in PC^* but provable in PC which is PC^* with predicative classes only and with the general ordered pair axiom. PC^* [super-strong AC] is inconsistent because the super-strong AC impliesC. Therefore the application of choice functions to non-empty classes generally makes a predicative definition of these classes necessary. Connected with these problems is a statement equivalent to the conjunction of the axioms of power set and foundation based on a function which coincides with the von Neumann-function under the assumption of one of the mentioned axioms.     

On replacement axioms for the Jacobi identity for vertex algebras and their modules

We discuss the axioms for vertex algebras and their modules, using formal calculus. Following certain standard treatments, we take the Jacobi identity as our main axiom and we recall weak commutativity and weak associativity. We derive a third companion property that we call “weak skew-associativity”. This third property in some sense completes an S₃-symmetry of the axioms, which is related to the known S₃-symmetry of the Jacobi identity. We do not initially require a vacuum vector, which is analogous to not requiring an identity element in ring theory. In this more general setting, one still has a property, occasionally used in standard treatments, which is closely related to skew-symmetry, which we call “vacuum-free skew-symmetry”. We show how certain combinations of these properties are equivalent to the Jacobi identity for both vacuum-free vertex algebras and their modules. We then specialize to the case with a vacuum vector and obtain further replacement axioms. In particular, in the final section we derive our main result, which says that, in the presence of certain minor axioms, the Jacobi identity for a module is equivalent to either weak associativity or weak skew-associativity. The first part of this result has appeared previously and has been used to show the (nontrivial) equivalence of representations of and modules for a vertex algebra. Many but not all of our results appear in standard treatments; some of our arguments are different from the usual ones.     

Ein Wohlordnungsbeweis für das OrdinalzahlensystemT(J)

Summary A recursive notation system of a strong segment of ordinals was developped by Jäger [3]. An unessential modified versionT(J) of this notation system was described in [4]. In the following, the well-ordering ofT(J) is proved in a formal system of second order arithmetic with the axiom schema of ₂ ¹ -comprehension. It follows, that the proof theoretical ordinal of ₂ ¹ -analysis is greater than the order type ofT(J).     

Monotonicity and independence axioms

Given σ, a family ofchoice problems, subsets ofR ⁿ representing the payoff vectors (measured in von Neumann-Morgenstern utility scales) attainable by a group ofn players, asolution f on σ associates to everyS in σ a unique elementf (S) ofS. Amonotonicity axiom specifies how the solution outcome should change when the choice problem is subjected to certain geometric transformations while anindependence axiom requires, in similar circumstances, the invariance of the solution outcome. A number of such axioms are here formulated and the logical relationships among them are established.Strong monotonicity is shown to be the strongest axiom and strongly monotonic solutions are characterized.     

拓扑分子格的分离公理

在[1]中我们建立了拓扑分子格的理论,它既是古典的点集拓扑学的推广,又是晚近发展起来的Fuzzy拓扑学的推广,对于某些Fuzzy格L(如L是线性序集或L是分子格等),它也是L—Fuzzy拓扑学的推广。因此,凡在拓扑分子格中得到的结果自然都是上述各种拓扑学中相应定理的一般化形式。在本文中我们将讨论拓扑分子格的分离公理。 我们熟知点集拓扑学中的分离公理有多种不同的等价形式。以正则性为例,设X是拓扑空间,X叫正则的,当且仅当对每个点a∈X以及a的每个开邻域U,a有开邻域V满足条件V~-U。这一分离公理又可表述为:设a∈X,F是X中不包含a的闭集,则有开集P     

The strength of extensionality II — Weak weak set theories without infinity

By obtaining several new results on Cook-style two-sorted bounded arithmetic, this paper measures the strengths of the axiom of extensionality and of other weak fundamental set-theoretic axioms in the absence of the axiom of infinity, following the author’s previous work [K. Sato, The strength of extensionality I — weak weak set theories with infinity, Annals of Pure and Applied Logic 157 (2009) 234-268] which measures them in the presence. These investigations provide a uniform framework in which three different kinds of reverse mathematics-Friedman-Simpson’s “orthodox” reverse mathematics, Cook’s bounded reverse mathematics and large cardinal theory-can be reformulated within one language so that we can compare them more directly.     

Implications of large-cardinal principles in homotopical localization

The existence of arbitrary cohomological localizations on the homotopy category of spaces has remained unproved since Bousfield settled the same problem for homology theories in the decade of 1970. This is related with another open question, namely whether or not every homotopy idempotent functor on spaces is an f-localization for some map f. We prove that both questions have an affirmative answer assuming the validity of a suitable large-cardinal axiom from set theory (Vopěnka's principle). We also show that it is impossible to prove that all homotopy idempotent functors are f-localizations using the ordinary ZFC axioms of set theory (Zermelo-Fraenkel axioms with the axiom of choice), since a counterexample can be displayed under the assumption that all cardinals are nonmeasurable, which is consistent with ZFC.     

Mazur intersection property for Asplund spaces

The main result of the present note states that it is consistent with the ZFC axioms of set theory (relying on Martin's Maximum MM axiom), that every Asplund space of density character ω₁ has a renorming with the Mazur intersection property. Combined with the previous result of Jiménez and Moreno (based upon the work of Kunen under the continuum hypothesis) we obtain that the MIP renormability of Asplund spaces of density ω₁ is undecidable in ZFC.     

Interpretations of the alternative set theory

Summary We show an axiom A such that there is no nontrivial interpretation of the alternative set theory (AST) inAST+A keeping , sets and the class of all standard natural numbers. Furthermore, there is no interpretation ofAST inAST without the prolongation axiom, but there is an interpretation ofAST in the theory having the prolongation axiom and the basic set-theoretical axioms only.     

Δ2 degrees without Σ1 induction

In this paper, we study the structure of Turing degrees below 0′ in the theory that is a fragment of Peano arithmetic without Σ₁ induction, with special focus on proper d-r.e. degrees and non-r.e. degrees. We prove:

1. P ^? + BΣ₁ + Exp ? There is a proper d-r.e. degree.
2. P ^? +BΣ₁+ Exp ? IΣ₁ ? There is a proper d-r.e. degree below 0′.
3. P ^? + BΣ₁ + Exp ? There is a non-r.e. degree below 0′.