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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
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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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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
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Bernhard Banaschewski 《Mathematical Logic Quarterly》1992,38(1):383-385
This note shows that for the proof of the existence and uniqueness of the algebraic closure of a field one needs only the Boolean Ultrafilter Theorem. 相似文献
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Imre Leader 《Journal of Combinatorial Theory, Series A》2012,119(2):382-396
A finite set X in some Euclidean space Rn is called Ramsey if for any k there is a d such that whenever Rd is k-coloured it contains a monochromatic set congruent to X. This notion was introduced by Erd?s, Graham, Montgomery, Rothschild, Spencer and Straus, who asked if a set is Ramsey if and only if it is spherical, meaning that it lies on the surface of a sphere. This question (made into a conjecture by Graham) has dominated subsequent work in Euclidean Ramsey theory.In this paper we introduce a new conjecture regarding which sets are Ramsey; this is the first ever ‘rival’ conjecture to the conjecture above. Calling a finite set transitive if its symmetry group acts transitively—in other words, if all points of the set look the same—our conjecture is that the Ramsey sets are precisely the transitive sets, together with their subsets. One appealing feature of this conjecture is that it reduces (in one direction) to a purely combinatorial statement. We give this statement as well as several other related conjectures. We also prove the first non-trivial cases of the statement.Curiously, it is far from obvious that our new conjecture is genuinely different from the old. We show that they are indeed different by proving that not every spherical set embeds in a transitive set. This result may be of independent interest. 相似文献
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In the setting of ZF, i.e., Zermelo–Fraenkel set theory without the Axiom of Choice (AC), we study partitions of Russell‐sets into sets each with exactly n elements (called n ‐ary partitions), for some integer n. We show that if n is odd, then a Russell‐set X has an n ‐ary partition if and only if |X | is divisible by n. Furthermore, we establish that it is relative consistent with ZF that there exists a Russell‐set X such that |X | is not divisible by any finite cardinal n > 1 (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
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Kyriakos Keremedis 《Mathematical Logic Quarterly》2001,47(2):205-210
We show that the axiom of choice AC is equivalent to the Vector Space Kinna‐Wagner Principle, i.e., the assertion: “For every family 𝒱= {Vi : i ∈ k} of non trivial vector spaces there is a family ℱ = {Fi : i ∈ k} such that for each i ∈ k, Fi is a non empty independent subset of Vi”. We also show that the statement “every vector space over ℚ has a basis” implies that every infinite well ordered set of pairs has an infinite subset with a choice set, a fact which is known not to be a consequence of the axiom of multiple choice MC. 相似文献
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It is shown that AC(ℝ), the axiom of choice for families of non‐empty subsets of the real line ℝ, does not imply the statement PW(ℝ), the powerset of ℝ can be well ordered. It is also shown that (1) the statement “the set of all denumerable subsets of ℝ has size 2 ” is strictly weaker than AC(ℝ) and (2) each of the statements (i) “if every member of an infinite set of cardinality 2 has power 2 , then the union has power 2 ” and (ii) “ℵ(2 ) ≠ ℵω” (ℵ(2 ) is Hartogs' aleph, the least ℵ not ≤ 2 ), is strictly weaker than the full axiom of choice AC. 相似文献
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For a regular cardinal κ with κ <κ = κ and κ ≤ λ , we construct generically (forcing by a < κ‐closed κ +‐c. c. p. o.‐set ℙ0) a subset S of {x ∈ P κ λ : x ∩ κ is a singular ordinal} such that S is stationary in a strong sense (F IAκ λ ‐stationary in our terminology) but the stationarity of S can be destroyed by a κ +‐c. c. forcing ℙ* (in V ℙ) which does not add any new element of P κ λ . Actually ℙ* can be chosen so that ℙ* is κ‐strategically closed. However we show that such ℙ* itself cannot be κ‐strategically closed or even <κ‐strategically closed if κ is inaccessible. (© 2005 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(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
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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
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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
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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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We show that the Axiom of Choice is equivalent to each of the following statements: (i) A product of closures of subsets of topological spaces is equal to the closure of their product (in the product topology); (ii) A product of complete uniform spaces is complete. 相似文献