DOCSLIB.ORG

Löwenheim-Skolem Theorem, the Axiom of Dependent Choice and A

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Löwenheim-Skolem Theorem, the Axiom of Dependent Choice and A L¨owenheim-Skolem theorem, the Axiom of Dependent Choice and a categorical extension principle Christian Esp´ındola∗ Abstract In this paper we prove the equivalence, over ZF , of L¨owenheim-Skolem theorem (LS), the Axiom of Dependent Choice (DC), and a certain functor extension property we introduce. We also consider variations of that property equivalent to other well known choice principles. 1 Introduction It has been known since 1920 ([6], [3]) that the the axiom of choice played a crucial r^olein the proof of the following form of L¨owenheim-Skolem theorem: Theorem 1.1. LS. Every model M of a first order theory T with countable signature has an elementary submodel N which is (at most)1countable. As noted by Skolem himself, the condition that the countable model for T be an elemen- tary submodel of M made an apparently essential use of the axiom of choice, by relying on the so called Skolem functions, which can be found in the modern standard proofs of the theorem. A careful examination of the proof shows that DC is sufficient already (see [4]), although the exact strength of LS was not known. On the other hand, a categorical proof of LS was given in [1], following a construction that Joyal had used in his unpublished proof of G¨odel'scompleteness theorem, and which allows to consider a certain functor extension property related to LS. The main theorem of this paper will show that the three statements are in fact equivalent over ZF . 2 The main theorem To state the functor extension principle we introduce some definitions: Definition 2.1. A Boolean category is a category having the following three properties: 1) It has all finite limits. 2) Every arrow f : A ! B can be factored as f : A C B, where C, called the image of f, is the least subobject of B through which f can factor. The arrow f : A C not factoring through any proper subobject of C is called a cover. 3) Images are stable under base change (i.e., pullbacks preserve covers). 4) The poset Sub(X) of subobjects of a given object X has finite unions and these are stable under pullbacks. 5) Every subobject A in the poset Sub(X) has a complement, i.e., there exists a subobject B such that the intersection A ^ B is initial in Sub(X) and A _ B = X (in particular, Sub(X) is a Boolean algebra, and we denote B = :A). Definition 2.2. A functor between Boolean categories is said to be Boolean provided it preserves finite limits and images factorizations, unions and complements. ∗This research was financed by the project \Constructive and category-theoretic foundations of mathematics (dnr 2008-5076)" from the Swedish Research Council (VR). 1From now on, we shall use the word countable to refer to an either finite or countably infinite set. 1 Consider the category of small Boolean categories with Boolean functors. Given a small Boolean category C0, [1] describes the construction of succesive conservative embeddings C0 !C1 ! ::: !C!, where each Cn+1 has sections for every cover from an object in Cn to the terminal object. It also provides a way to extend Boolean functors with domain C to Boolean functors with domain C!. Although the construction of C! can be performed in ZF by using Grothendieck fibrations and the construction of the bicolimit through the category of fractions (see [2]), assuming the axiom of choice it can be proved, in fact, that C! is the free completion of C by adding sections of covers over the terminal object. In other words, given a Boolean category D such that every cover over the terminal object has a section, any Boolean functor F : C!D has a Boolean extension F : C! !D which sends sections into sections and that is, moreover, the unique (up to natural isomorphism) Boolean functor with that property. Our functor extension principle concerns the particular case where C is countably small (i.e., it has a countable set of objects and morphisms) and D = Set is the (large) category of sets (note that Boolean functors F : C!Set can still be properly defined in ZF , but we use this notation for convenience). That this extension property is related to some choice principle should not be a surprise, since if we do not impose any restriction on the size of C, the existence of such an extension is trivially equivalent to the axiom of global choice: an extension of the identity functor in Set provides sections for each cover Si ! {∗}, i.e., provides a choice for each set. However, the interest of this principle lies in the fact that restricting the cardinality to a countable set makes it equivalent to DC. Theorem 2.3. In Zermelo-Fraenkel set theory, the following are equivalent: 1. The Axiom of Dependent Choice. 2. Every Boolean functor F : C0 !Set from a countably small Boolean category C0 has a Boolean extension F : C! !Set. 3. The L¨owenheim-Skolemtheorem. Proof. (1 ) 2) This implication is essentially contained in [1], lemma 4:2. The construction of F avoids choice because the pseudofunctor considered can be defined in ZF by using the countability of the set of objects and arrows when considering the pullback functors involved (and one can consider Grothendieck's construction of the bicolimit, as explained in [2], Ex:6). The first step of the extension from C to C1 only uses countable choice (derivable from DC), as it requires to consider choice functions for a countable family of sets. The iteration of this first step uses DC. (2 ) 3) This implication is also contained in the categorical proof of LS given in [1], but its essential set-theoretic content can be extracted as follows. Given a first order theory T with countable signature, its associated syntactic category CT is a small Boolean category with a countable set of objects. The process of embedding it into the finite coproduct completion C0 and constructing now the free completion of C0 by sections of covers over the terminal object gives a Boolean conservative embedding CT !C!, where C! is a small Boolean category with the property that every cover over a subobject of the terminal object has a section. 0 0 Let T be the Henkinization of T . We shall see that C! is a model for T , by finding suitable interpretations for the added constants. Suppose that φ(x) is a formula with one free variable containing only constants of level at most n − 1; we may also assume that these have already been interpreted in C!. Then the constant c associated to φ(x) can be interpreted in the following way: 2 m n [[x; φ(x)]] / / [x; >] o o P W W X q s p c r t m0 n0 [[fg; 9xφ(x)]] / / [fg; >] o o :[[fg; 9xφ(x)]] Given the interpretation m of [x; φ(x)], the resulting interpretation m0 for [fg; 9xφ(x)] is given by the image factorization of pm. Let s be a section2 of the cover q; consider the complement of the subobject [[fg; 9xφ(x)]] and take its pullback P along p. Let t be the section of the resulting cover r; then ms and nt determine a section from the terminal object which we can interpret as the constant c. With this interpretation, it is clear that the formula 9xφ(x) ! φ(c) is satisfied, since the interpretation of [fg; φ(c)] is the pullback of m along c, which is precisely the same as [[fg; 9xφ(x)]] with the projections s and m0. 0 If we consider the syntactic category CT 0 , the fact that C! is a model for T implies that 3 there is a Boolean functor J : CT 0 !C! through which the embedding I : CT !C! factors . By 2, any model M for T , interpreted as a functor F : CT !Set, has a Boolean extension 0 0 F : C! !Set, and the composition FJ provides a model M for the Henkinized theory T which extends M. Finally, a countable submodel for M can be constructed taking as the underlying universe the set N of the interpretations of all constants of T 0 and restricting to N the interpretations of all symbols of M. (3 ) 1) Let S be a set with a binary relation R such that for every x 2 S, the set fy 2 S = xRyg is nonempty. Consider the theory T over the language L = fRg that contains a binary relation symbol, and whose only non logical axiom is 8x9yR(x; y). Since S is a model for T which has a countable signature, by 3 it contains a countable submodel N . Hence, there is a bijection between the underlying set N of N and either the set of natural numbers or some finite ordinal. If f is such a bijection, we can now recursively define a sequence by −1 −1 putting x0 = f (minff(n) = n 2 Ng) and xn = f (minff(n) = n 2 N ^ xn−1Rng). Remark 2.4. The proof above implicitely shows a direct set-theoretic deduction of LS from DC that makes no use of Skolem functions, and instead replaces Skolemization with Henk- inization. Indeed, every model of a theory can be extended to a model of its Henkinization by just interpreting the constants (this requires an application of DC since the constants are added in levels), and then we can restrict the interpretations of the symbols to the countable set of constants of the Henkinized theory.
Load more

Recommended publications
  • Chapter 1 Preliminaries
    Chapter 1 Preliminaries 1.1 First-order theories In this section we recall some basic definitions and facts about first-order theories. Most proofs are postponed to the end of the chapter, as exercises for the reader. 1.1.1 Definitions Definition 1.1 (First-order theory) —A first-order theory T is defined by: A first-order language L , whose terms (notation: t, u, etc.) and formulas (notation: φ, • ψ, etc.) are constructed from a fixed set of function symbols (notation: f , g, h, etc.) and of predicate symbols (notation: P, Q, R, etc.) using the following grammar: Terms t ::= x f (t1,..., tk) | Formulas φ, ψ ::= t1 = t2 P(t1,..., tk) φ | | > | ⊥ | ¬ φ ψ φ ψ φ ψ x φ x φ | ⇒ | ∧ | ∨ | ∀ | ∃ (assuming that each use of a function symbol f or of a predicate symbol P matches its arity). As usual, we call a constant symbol any function symbol of arity 0. A set of closed formulas of the language L , written Ax(T ), whose elements are called • the axioms of the theory T . Given a closed formula φ of the language of T , we say that φ is derivable in T —or that φ is a theorem of T —and write T φ when there are finitely many axioms φ , . , φ Ax(T ) such ` 1 n ∈ that the sequent φ , . , φ φ (or the formula φ φ φ) is derivable in a deduction 1 n ` 1 ∧ · · · ∧ n ⇒ system for classical logic. The set of all theorems of T is written Th(T ). Conventions 1.2 (1) In this course, we only work with first-order theories with equality.
    [Show full text]
  • Automated ZFC Theorem Proving with E
    Automated ZFC Theorem Proving with E John Hester Department of Mathematics, University of Florida, Gainesville, Florida, USA https://people.clas.ufl.edu/hesterj/ [email protected] Abstract. I introduce an approach for automated reasoning in first or- der set theories that are not finitely axiomatizable, such as ZFC, and describe its implementation alongside the automated theorem proving software E. I then compare the results of proof search in the class based set theory NBG with those of ZFC. Keywords: ZFC · ATP · E. 1 Introduction Historically, automated reasoning in first order set theories has faced a fun- damental problem in the axiomatizations. Some theories such as ZFC widely considered as candidates for the foundations of mathematics are not finitely axiomatizable. Axiom schemas such as the schema of comprehension and the schema of replacement in ZFC are infinite, and so cannot be entirely incorpo- rated in to the prover at the beginning of a proof search. Indeed, there is no finite axiomatization of ZFC [1]. As a result, when reasoning about sufficiently strong set theories that could no longer be considered naive, some have taken the alternative approach of using extensions of ZFC that admit objects such as proper classes as first order objects, but this is not without its problems for the individual interested in proving set-theoretic propositions. As an alternative, I have programmed an extension to the automated the- orem prover E [2] that generates instances of parameter free replacement and comprehension from well formuled formulas of ZFC that are passed to it, when eligible, and adds them to the proof state while the prover is running.
    [Show full text]
  • Cantor-Von Neumann Set-Theory Fa Muller
    Logique & Analyse 213 (2011), x–x CANTOR-VON NEUMANN SET-THEORY F.A. MULLER Abstract In this elementary paper we establish a few novel results in set the- ory; their interest is wholly foundational-philosophical in motiva- tion. We show that in Cantor-Von Neumann Set-Theory, which is a reformulation of Von Neumann's original theory of functions and things that does not introduce `classes' (let alone `proper classes'), developed in the 1920ies, both the Pairing Axiom and `half' the Axiom of Limitation are redundant — the last result is novel. Fur- ther we show, in contrast to how things are usually done, that some theorems, notably the Pairing Axiom, can be proved without invok- ing the Replacement Schema (F) and the Power-Set Axiom. Also the Axiom of Choice is redundant in CVN, because it a theorem of CVN. The philosophical interest of Cantor-Von Neumann Set- Theory, which is very succinctly indicated, lies in the fact that it is far better suited than Zermelo-Fraenkel Set-Theory as an axioma- tisation of what Hilbert famously called Cantor's Paradise. From Cantor one needs to jump to Von Neumann, over the heads of Zer- melo and Fraenkel, and then reformulate. 0. Introduction In 1928, Von Neumann published his grand axiomatisation of Cantorian Set- Theory [1925; 1928]. Although Von Neumann's motivation was thoroughly Cantorian, he did not take the concept of a set and the membership-relation as primitive notions, but the concepts of a thing and a function — for rea- sons we do not go into here. This, and Von Neumann's cumbersome nota- tion and terminology (II-things, II.I-things) are the main reasons why ini- tially his theory remained comparatively obscure.
    [Show full text]
  • (Aka “Geometric”) Iff It Is Axiomatised by “Coherent Implications”
    GEOMETRISATION OF FIRST-ORDER LOGIC ROY DYCKHOFF AND SARA NEGRI Abstract. That every first-order theory has a coherent conservative extension is regarded by some as obvious, even trivial, and by others as not at all obvious, but instead remarkable and valuable; the result is in any case neither sufficiently well-known nor easily found in the literature. Various approaches to the result are presented and discussed in detail, including one inspired by a problem in the proof theory of intermediate logics that led us to the proof of the present paper. It can be seen as a modification of Skolem's argument from 1920 for his \Normal Form" theorem. \Geometric" being the infinitary version of \coherent", it is further shown that every infinitary first-order theory, suitably restricted, has a geometric conservative extension, hence the title. The results are applied to simplify methods used in reasoning in and about modal and intermediate logics. We include also a new algorithm to generate special coherent implications from an axiom, designed to preserve the structure of formulae with relatively little use of normal forms. x1. Introduction. A theory is \coherent" (aka \geometric") iff it is axiomatised by \coherent implications", i.e. formulae of a certain simple syntactic form (given in Defini- tion 2.4). That every first-order theory has a coherent conservative extension is regarded by some as obvious (and trivial) and by others (including ourselves) as non- obvious, remarkable and valuable; it is neither well-enough known nor easily found in the literature. We came upon the result (and our first proof) while clarifying an argument from our paper [17].
    [Show full text]
  • 1.4 Theories
    1.4 Theories Before we start to work with one of the most central notions of mathematical logic – that of a theory – it will be useful to make some observations about countable sets. Lemma 1.3. 1. A set X is countable iff there is a surjection ψ : N X. 2. If X is countable and Y X then Y is countable. 3. If X and Y are countable, then X Y is countable. 4. If X and Y are countable, then X ¢ Y is countable. k 5. If X is countable and k 1 then X is countable. ä 6. If Xi is countable for each i È N, then Xi is countable. iÈN 7. If X is countable, then ä Xk is countable. k ÈN Ø Ù Proof. 1. If X is countably infinite note that every bijection is a surjection. If X x0,...,xn Õ Ô Õ is finite, let ψ Ôi xi for 0 i n and ψ j xn for j n. For the other direction, Ô Õ Ô Õ Ô Õ Ù let ψ be a surjection, then X is Øψ 0 , ψ 1 , ψ 2 ,... and removing duplicates from the Ô ÕÕ Ô N sequence ψ i i¥0 yields a bijective ϕ : X. È 2. Obvious if Y is finite. Otherwise, let ϕ : N X be a bijection and let y0 Y . Define " Õ Ô Õ È ϕÔn if ϕ n Y ψ : N Y,n y0 otherwise which is a surjection. 3. Let ϕ : N X and ψ : N Y be bijections.
    [Show full text]
  • Lipics-FSCD-2021-29.Pdf (0.9
    On the Logical Strength of Confluence and Normalisation for Cyclic Proofs Anupam Das ! Ï University of Birmingham, UK Abstract In this work we address the logical strength of confluence and normalisation for non-wellfounded typing derivations, in the tradition of “cyclic proof theory” . We present a circular version CT of Gödel’ s system T, with the aim of comparing the relative expressivity of the theories CT and T. We approach this problem by formalising rewriting-theoretic results such as confluence and normalisation for the underlying “coterm” rewriting system of CT within fragments of second-order arithmetic. We establish confluence of CT within the theory RCA0, a conservative extension of primitive recursive arithmetic and IΣ1. This allows us to recast type structures of hereditarily recursive operations as “coterm” models of T. We show that these also form models of CT, by formalising a totality argument for circular typing derivations within suitable fragments of second-order arithmetic. Relying on well-known proof mining results, we thus obtain an interpretation of CT into T; in fact, more precisely, we interpret level-n-CT into level-(n + 1)-T, an optimum result in terms of abstraction complexity. A direct consequence of these model-theoretic results is weak normalisation for CT. As further results, we also show strong normalisation for CT and continuity of functionals computed by its type 2 coterms. 2012 ACM Subject Classification Theory of computation → Equational logic and rewriting; Theory of computation → Proof theory; Theory of computation → Higher order logic; Theory of computation → Lambda calculus Keywords and phrases confluence, normalisation, system T, circular proofs, reverse mathematics, type structures Digital Object Identifier 10.4230/LIPIcs.FSCD.2021.29 Related Version This work is based on part of the following preprint, where related results, proofs and examples may be found.
    [Show full text]
  • Inseparability and Conservative Extensions of Description Logic Ontologies: a Survey
    Inseparability and Conservative Extensions of Description Logic Ontologies: A Survey Elena Botoeva1, Boris Konev2, Carsten Lutz3, Vladislav Ryzhikov1, Frank Wolter2, and Michael Zakharyaschev4 1 Free University of Bozen-Bolzano, Italy fbotoeva,[email protected] 2 University of Liverpool, UK fkonev,[email protected] 3 University of Bremen, Germany [email protected] 4 Birkbeck, University of London, UK [email protected] Abstract. The question whether an ontology can safely be replaced by another, possibly simpler, one is fundamental for many ontology engi- neering and maintenance tasks. It underpins, for example, ontology ver- sioning, ontology modularization, forgetting, and knowledge exchange. What `safe replacement' means depends on the intended application of the ontology. If, for example, it is used to query data, then the answers to any relevant ontology-mediated query should be the same over any relevant data set; if, in contrast, the ontology is used for conceptual reasoning, then the entailed subsumptions between concept expressions should coincide. This gives rise to different notions of ontology insep- arability such as query inseparability and concept inseparability, which generalize corresponding notions of conservative extensions. We survey results on various notions of inseparability in the context of descrip- tion logic ontologies, discussing their applications, useful model-theoretic characterizations, algorithms for determining whether two ontologies are inseparable (and, sometimes, for computing the difference between them if they are not), and the computational complexity of this problem. 1 Introduction Description logic (DL) ontologies provide a common vocabulary for a domain of interest together with a formal modeling of the semantics of the vocabulary items (concept names and role names).
    [Show full text]
  • The Strength of Mac Lane Set Theory
    The Strength of Mac Lane Set Theory A. R. D. MATHIAS D´epartement de Math´ematiques et Informatique Universit´e de la R´eunion To Saunders Mac Lane on his ninetieth birthday Abstract AUNDERS MAC LANE has drawn attention many times, particularly in his book Mathematics: Form and S Function, to the system ZBQC of set theory of which the axioms are Extensionality, Null Set, Pairing, Union, Infinity, Power Set, Restricted Separation, Foundation, and Choice, to which system, afforced by the principle, TCo, of Transitive Containment, we shall refer as MAC. His system is naturally related to systems derived from topos-theoretic notions concerning the category of sets, and is, as Mac Lane emphasizes, one that is adequate for much of mathematics. In this paper we show that the consistency strength of Mac Lane's system is not increased by adding the axioms of Kripke{Platek set theory and even the Axiom of Constructibility to Mac Lane's axioms; our method requires a close study of Axiom H, which was proposed by Mitchell; we digress to apply these methods to subsystems of Zermelo set theory Z, and obtain an apparently new proof that Z is not finitely axiomatisable; we study Friedman's strengthening KPP + AC of KP + MAC, and the Forster{Kaye subsystem KF of MAC, and use forcing over ill-founded models and forcing to establish independence results concerning MAC and KPP ; we show, again using ill-founded models, that KPP + V = L proves the consistency of KPP ; turning to systems that are type-theoretic in spirit or in fact, we show by arguments of Coret
    [Show full text]
  • More Model Theory Notes Miscellaneous Information, Loosely Organized
    More Model Theory Notes Miscellaneous information, loosely organized. 1. Kinds of Models A countable homogeneous model M is one such that, for any partial elementary map f : A ! M with A ⊆ M finite, and any a 2 M, f extends to a partial elementary map A [ fag ! M. As a consequence, any partial elementary map to M is extendible to an automorphism of M. Atomic models (see below) are homogeneous. A prime model of T is one that elementarily embeds into every other model of T of the same cardinality. Any theory with fewer than continuum-many types has a prime model, and if a theory has a prime model, it is unique up to isomorphism. Prime models are homogeneous. On the other end, a model is universal if every other model of its size elementarily embeds into it. Recall a type is a set of formulas with the same tuple of free variables; generally to be called a type we require consistency. The type of an element or tuple from a model is all the formulas it satisfies. One might think of the type of an element as a sort of identity card for automorphisms: automorphisms of a model preserve types. A complete type is the analogue of a complete theory, one where every formula of the appropriate free variables or its negation appears. Types of elements and tuples are always complete. A type is principal if there is one formula in the type that implies all the rest; principal complete types are often called isolated. A trivial example of an isolated type is that generated by the formula x = c where c is any constant in the language, or x = t(¯c) where t is any composition of appropriate-arity functions andc ¯ is a tuple of constants.
    [Show full text]
  • Paradox and Foundation Zach Weber Submitted in Total Fulfilment of The
    Paradox and Foundation Zach Weber Submitted in total fulfilment of the requirements of the degree of Doctor of Philosophy May 2009 School of Philosophy, Anthropology and Social Inquiry The University of Melbourne This is to certify that - the thesis comprises only my original work towards the PhD, - due acknowledgement has been made in the text to all other material used, - the thesis is less than 100,000 words in length. Preface Dialethic paraconsistency is an approach to formal and philosophical theories in which some but not all contradictions are true. Advancing that program, this thesis is about paradoxes and the foundations of mathematics, and is divided accordingly into two main parts. The first part concerns the history and philosophy of set theory from Cantor through the independence proofs, focusing on the set concept. A set is any col- lection of objects that is itself an object, with identity completely determined by membership. The set concept is called naive because it is inconsistent. I argue that the set concept is inherently and rightly paradoxical, because sets are both intensional and extensional objects: Sets are predicates in extension. All consistent characterizations of sets are either not explanatory or not coherent. To understand sets, we need to reason about them with an appropriate logic; paraconsistent naive set theory is situated as a continuation of the original foundational project. The second part produces a set theory deduced from an unrestricted compre- hension principle using the weak relevant logic DLQ, dialethic logic with quantifiers. I discuss some of the problems involved with embedding in DLQ, especially related to identity and substitution.
    [Show full text]
  • Warren Goldfarb, Notes on Metamathematics
    Notes on Metamathematics Warren Goldfarb W.B. Pearson Professor of Modern Mathematics and Mathematical Logic Department of Philosophy Harvard University DRAFT: January 1, 2018 In Memory of Burton Dreben (1927{1999), whose spirited teaching on G¨odeliantopics provided the original inspiration for these Notes. Contents 1 Axiomatics 1 1.1 Formal languages . 1 1.2 Axioms and rules of inference . 5 1.3 Natural numbers: the successor function . 9 1.4 General notions . 13 1.5 Peano Arithmetic. 15 1.6 Basic laws of arithmetic . 18 2 G¨odel'sProof 23 2.1 G¨odelnumbering . 23 2.2 Primitive recursive functions and relations . 25 2.3 Arithmetization of syntax . 30 2.4 Numeralwise representability . 35 2.5 Proof of incompleteness . 37 2.6 `I am not derivable' . 40 3 Formalized Metamathematics 43 3.1 The Fixed Point Lemma . 43 3.2 G¨odel'sSecond Incompleteness Theorem . 47 3.3 The First Incompleteness Theorem Sharpened . 52 3.4 L¨ob'sTheorem . 55 4 Formalizing Primitive Recursion 59 4.1 ∆0,Σ1, and Π1 formulas . 59 4.2 Σ1-completeness and Σ1-soundness . 61 4.3 Proof of Representability . 63 3 5 Formalized Semantics 69 5.1 Tarski's Theorem . 69 5.2 Defining truth for LPA .......................... 72 5.3 Uses of the truth-definition . 74 5.4 Second-order Arithmetic . 76 5.5 Partial truth predicates . 79 5.6 Truth for other languages . 81 6 Computability 85 6.1 Computability . 85 6.2 Recursive and partial recursive functions . 87 6.3 The Normal Form Theorem and the Halting Problem . 91 6.4 Turing Machines .
    [Show full text]
  • Choice, Extension, Conservation. from Transfinite to finite Methods in Abstract Algebra
    Choice, extension, conservation. From transfinite to finite methods in abstract algebra Daniel Wessel Universita` degli Studi di Trento Universita` degli Studi di Verona December 3, 2017 Doctoral thesis in Mathematics Joint doctoral programme in Mathematics, 30th cycle Department of Mathematics, University of Trento Department of Computer Science, University of Verona Academic year 2017/18 Supervisor: Peter Schuster, University of Verona Trento, Italy December 3, 2017 Abstract Maximality principles such as the ones going back to Kuratowski and Zorn ensure the existence of higher type ideal objects the use of which is commonly held indispensable for mathematical practice. The modern turn towards computational methods, which can be witnessed to have a strong influence on contemporary foundational studies, encourages a reassessment within a constructive framework of the methodological intricacies that go along with invocations of maximality principles. The common thread that can be followed through the chapters of this thesis is explained by the attempt to put the widespread use of ideal objects under constructive scrutiny. It thus walks the tracks of a revised Hilbert's programme which has inspired a reapproach to constructive algebra by finitary means, and for which Scott's entailment relations have already shown to provide a vital and utmost versatile tool. In this thesis several forms of the Kuratowski-Zorn Lemma are introduced and proved equivalent over constructive set theory; the notion of Jacobson radical is brought from com- mutative
    [Show full text]