共查询到20条相似文献，搜索用时 15 毫秒
1.
Karl-Heinz Diener 《Mathematical Logic Quarterly》1993,39(1):492-514
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. 相似文献
2.
Karl‐Heinz Diener 《Mathematical Logic Quarterly》2000,46(4):563-568
We will prove that some so‐called union theorems (see [2]) are equivalent in ZF0 to statements about the transitive closure of relations. The special case of “bounded” union theorems dealing with κ‐hereditary sets yields equivalents to statements about the transitive closure of κ‐narrow relations. The instance κ = ω1 (i. e., hereditarily countable sets) yields an equivalent to Howard‐Rubin's Form 172 (the transitive closure Tc(x) of every hereditarily countable set x is countable). In particular, the countable union theorem (Howard‐Rubin's Form 31) and, a fortiori, the axiom of countable choice imply Form 172. 相似文献
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Wolfgang Degen 《Mathematical Logic Quarterly》2001,47(2):197-204
We shall investigate certain statements concerning the rigidity of unary functions which have connections with (weak) forms of the axiom of choice. 相似文献
4.
Douglas Bridges Fred Richman Peter Schuster 《Proceedings of the American Mathematical Society》2000,128(9):2749-2752
A weak choice principle is introduced that is implied by both countable choice and the law of excluded middle. This principle suffices to prove that metric independence is the same as linear independence in an arbitrary normed space over a locally compact field, and to prove the fundamental theorem of algebra. 相似文献
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Wolfgang Degen 《Mathematical Logic Quarterly》2000,46(3):313-334
We shall investigate certain set‐theoretic pigeonhole principles which arise as generalizations of the usual (finitary) pigeonhole principle; and we shall show that many of them are equivalent to full AC. We discuss also several restricted cases and variations of those principles and relate them to restricted choice principles. In this sense the pigeonhole principle is a rich source of weak choice principles. It is shown that certain sequences of restricted pigeonhole principles form implicational hierarchies with respect to ZF. We state also several open problems in order to indicate the extent to which the whole subject of pigeonhole principles requires further exploration. 相似文献
6.
Athanassios Tzouvaras 《Mathematical Logic Quarterly》1993,39(1):454-460
We show that there are universes of sets which contain descending ?-sequences of length α for every ordinal α, though they do not contain any ?-cycle. It is also shown that there is no set universe containing a descending ?-sequence of length On. MSC: 03E30; 03E65. 相似文献
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Paul Howard Kyriakos Keremedis Jean E. Rubin Adrienne Stanley 《Mathematical Logic Quarterly》2000,46(1):3-16
We study the relationship between the countable axiom of choice and the Tychonoff product theorem for countable families of topological spaces. 相似文献
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J. W. Degen 《Mathematical Logic Quarterly》1994,40(1):111-124
Our main contribution is a formal definition of what could be called a T-notion of infinity, for set theories T extending ZF. Around this definition we organize some old and new notions of infinity; we also indicate some easy independence proofs. Mathematics Subject Classification: 03E25, 03E20. 相似文献
10.
Marcel Ern 《Mathematical Logic Quarterly》2001,47(2):211-222
We introduce the notion of constructive suprema and of constructively directed sets. The Axiom of Choice turns out to be equivalent to the postulate that every supremum is constructive, but also to the hypothesis that every directed set admits a function assigning to each finite subset an upper bound. The Axiom of Multiple Choice (which is known to be weaker than the full Axiom of Choice in set theory without foundation) implies a simple set‐theoretical induction principle (SIP), stating that any system of sets that is closed under unions of well‐ordered subsystems and contains all finite subsets of a given set must also contain that set itself. This is not provable without choice principles but equivalent to the statement that the existence of joins for constructively directed subsets of a poset follows from the existence of joins for nonempty well‐ordered subsets. Moreover, we establish the equivalence of SIP with several other fundamental statements concerning inductivity, compactness, algebraic closure systems, and the exchange between chains and directed sets. 相似文献
11.
Omar De la Cruz Eric J. Hall Paul Howard Kyriakos Keremedis Jean E. Rubin 《Mathematical Logic Quarterly》2008,54(6):652-665
We study statements about countable and well‐ordered unions and their relation to each other and to countable and well‐ordered forms of the axiom of choice. Using WO as an abbreviation for “well‐orderable”, here are two typical results: The assertion that every WO family of countable sets has a WO union does not imply that every countable family of WO sets has a WO union; the axiom of choice for WO families of WO sets does not imply that the countable union of countable sets is WO. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
12.
Bernhard Banaschewski 《Mathematical Logic Quarterly》1994,40(4):478-480
In the present note we give a direct deduction of the Axiom of Choice from the Maximal Ideal Theorem for commutative rings with unit. Mathematics Subject Classification: 03E25, 04A25, 13A15. 相似文献
13.
Paul Howard Kyriakos Keremedis Jean E. Rubin Adrienne Stanley Eleftherios Tatchtsis 《Mathematical Logic Quarterly》2001,47(3):423-431
We study the relationship between various properties of the real numbers and weak choice principles. 相似文献
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John L. Bell 《Mathematical Logic Quarterly》2008,54(2):194-201
A weak form of intuitionistic set theory WST lacking the axiom of extensionality is introduced. While WST is too weak to support the derivation of the law of excluded middle from the axiom of choice, we show that bee.ng up WST with moderate extensionality principles or quotient sets enables the derivation to go through. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
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Kyriakos Keremedis 《Mathematical Logic Quarterly》2000,46(4):569-571
We show that for every we ordered cardinal number m the Tychonoff product 2m is a compact space without the use of any choice but in Cohen's Second Mode 2ℝ is not compact. 相似文献
18.
Working in the context of restricted forms of the Axiom of Choice, we consider the problem of splitting the ordinals below λ of cofinality θ into λ many stationary sets, where θ < λ are regular cardinals. This is a continuation of [4] (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
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Paul Howard Kyriakos Keremedis Jean E. Rubin Adrienne Stanley 《Mathematical Logic Quarterly》2000,46(2):219-232
The axiom of multiple choice implies that metric spaces are paracompact but the reverse implication cannot be proved in set theory without the axiom of choice. 相似文献