The relative consistency of g<cf(Sym(omega))
英文摘要：
We prove the consistency result from the title. By forcing we construct a model of g=aleph_1, b=cf(Sym(omega))=aleph_2.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
相关论文
幂集之下的强分划关系：一致性，Sierpinski是对的吗？II
Strong partition relations below the power set: consistency, was Sierpinski right, II?
本文围绕Galvin关于强分划关系一致性的猜想展开，证否了原猜想，证明了将ω₂替换为大基数2^ω时的一致性，并给出相关迭代引理与定理推广。
math.LO
The consistency of 2^{aleph_{0}}> aleph_{omega} + I(aleph_{2})=I(aleph_{omega})
An omega-coloring is a pair <f,B> where f:[B]^{2} ---> omega. The set B is the field of f and denoted Fld(f). Let f,g be omega-colorings. We say that f realizes the coloring g if there is a one-one function k:Fld(g) ---> Fld(f) such that for all {x,y}, {u,v} in dom(g) we have f({k(x),k(y)}) not= f({k(u),k(v)}) => g({x,y}) not= g({u,v}). We write f~g if f realizes g and g realizes f. We call the ~-classes of omega-colorings with finite fields identities. We say that an identity I is of size r if |Fld(f)|=r for some/all f in I. For a cardinal kappa and f:[kappa]^2 ---> omega we define I(f) to be…
math.LO
Cardinal characteristics at aleph omega
We prove the consistency of the statement $\mathfrak{u}_{\aleph_ω}<2^{\aleph_ω}$. We show that the consistency strength of this statement is exactly a measurable cardinal $μ$ so that $o(μ)=μ^{++}$.
math.LO
Consistency of the existence of a 4-place function on aleph_2 with no independent subset of size k_n
It is consistent that there is a set mapping from the four-tuples of omega_n into the finite subsets with no free subsets of size t_n for some natural number t_n. For any n< omega it is consistent that there is a set mapping from the pairs of omega_n into the finite subsets with no infinite free sets.
math.LO
Possible Size of an ultrapower of omega
Let omega be the first infinite ordinal (or the set of all natural numbers) with the usual order <. In section 1 we show that, assuming the consistency of a supercompact cardinal, there may exist an ultrapower of omega, whose cardinality is (1) a singular strong limit cardinal, (2) a strongly inaccessible cardinal. This answers two questions in [CK], modulo the assumption of supercompactness. In section 2 we construct several lambda-Archimedean ultrapowers of omega under some large cardinal assumptions. For example, we show that, assuming the consistency of a measurable cardinal, there may exi…
math.LO
On CH + 2^{aleph_1}-> (alpha)^2_2 for alpha < omega_2
We prove the consistency of ``CH + 2^{aleph_1} is arbitrarily large + 2^{aleph_1} not-> (omega_1 x omega)^2_2''. If fact, we can get 2^{aleph_1} not-> [omega_1 x omega]^2_{aleph_0}. In addition to this theorem, we give generalizations to other cardinals.
math.LO
更多相关论文收起更多论文
Sigma-Prikry forcing III: Down to Aleph_omega
We prove the consistency of the failure of the singular cardinals hypothesis at $\aleph_ω$ together with the reflection of all stationary subsets of $\aleph_{ω+1}$. This shows that two classic results of Magidor (from 1977 and 1982) can hold simultaneously.
math.LO
On the weak Freese-Nation property of complete Boolean algebras
The following results are proved: (a) In a model obtained by adding aleph_2 Cohen reals, there is always a c.c.c. complete Boolean algebra without the weak Freese-Nation property. (b) Modulo the consistency strength of a supercompact cardinal, the existence of a c.c.c. complete Boolean algebras without the weak Freese-Nation property consistent with GCH. (c) Under some consequences of the negation of 0^#, the weak Freese-Nation property of (P(omega),subseteq) is equivalent to the weak Freese-Nation property of any of C(kappa) or R(kappa) for uncountable kappa. (d) Modulo consistency of (aleph_…
math.LO
On Godel's "Much Weaker" Assumption
Godelian sentences of a sufficiently strong and recursively enumerable theory, constructed in Godel's 1931 groundbreaking paper on the incompleteness theorems, are unprovable if the theory is consistent; however, they could be refutable. These sentences are independent when the theory is so-called omega-consistent; a notion introduced by Godel, which is stronger than (simple) consistency, but ``much weaker'' than soundness. Godel goes to great lengths to show in detail that omega-consistency is stronger than consistency, but never shows, or seems to forget to say, why it is much weaker than so…
math.LO
The spectrum of characters of ultrafilters on omega
We show the consistency of the set of regular cardinals which are the character of some ultrafilter on omega is not convex. We also deal with the set of pi chi-characters of ultrafilters on omega.
math.GNmath.LO
ω-稳定的K-圈与对称子
Symétrons et K-boucles omega-stables
本文在Poizat工作基础上发展ω-稳定K-圈与对称子的模型论，建立不可分解定理的适当版本，并将Lascar分析适配到该情境。
math.LO
Around cofin
We show the consistency of "there is a nice sigma --ideal I on the reals with add(I)= omega_1 which cannot be represented as the union of a strictly increasing sequence of length omega_1 of sigma-subideals". This answers a question by Borodulin-Nadzieja and Glab.
math.LO
On the consistency of the definable tree property on \aleph_1
In this paper we prove the equiconsistency of ``Every omega_1 tree which is first order definable over H_{omega_1} has a cofinal branch'' with the existence of a Pi^1_1 reflecting cardinal. The proof uses a definable version of Ramsey theorem on aleph_1 which is again equiconsistent with a Pi^1_1 reflecting cardinal. We also prove that the addition of $MA$ to the definable tree property increases the consistency strength to that of a weakly compact cardinal. Finally we comment on the generalization to higher cardinals.
math.COmath.LO
The character spectrum of beta(N)
We show the consistency of: the set of regular cardinals which are the character of some ultrafilter on omega can be quite chaotic, in particular not only can be not convex but can have many gaps. We also deal with the set of pi-characters of ultrafilters on omega.
math.GNmath.LO
Evasion and prediction II
A subgroup G \leq Z^omega exhibits the Specker phenomenon if every homomorphism G \to Z maps almost all unit vectors to 0. We give several combinatorial characterizations of the cardinal se, the size of the smallest G \leq Z^omega exhibiting the Specker phenomenon. We also prove the consistency of b < e, where b is the unbounding number and e the evasion number introduced recently by Blass. Finally we show that e \geq \min { aleph_0--e , cov(M) }, where aleph_0--e is the aleph_0--evasion number (the uniformity of the evasion ideal) and cov(M) is the covering number of the meager ideal. All the…
math.LO
MAD Families and SANE Player
We throw some light on the question: is there a MAD family (= a family of infinite subsets of N, the intersection of any two is finite) which is completely separable (i.e. any X subseteq N is included in a finite union of members of the family or include a member of the family). We prove that it is hard to prove the consistency of the negation: (a) if 2^{aleph_0} < aleph_omega, then there is such a family (b) if there is no such families then some situation related to pcf holds whose consistency is large.
math.LO
一个重复点给出 {ω_1} 上的闭无界超滤子
One Repeat Point Gives a Closed, Unbounded Ultrafilter on {ω_1}
本文确定关于 ω_1 的闭无界超滤子成为真正超滤子所需的一致性强度恰为 ZFC 加一个可测基数。
math.LO
Consistency Strengths of Modified Maximality Principles
The Maximality Principle MP is a scheme which states that if a sentence of the language of ZFC is true in some forcing extension V^P, and remains true in any further forcing extension of V^P, then it is true in all forcing extensions of V. A modified maximality principle MP_Gamma arises when considering forcing with a particular class Gamma of forcing notions. A parametrized form of such a principle, MP_Gamma(X), considers formulas taking parameters; to avoid inconsistency such parameters must be restricted to a specific set X which depends on the forcing class Gamma being considered. A strong…
math.LO
选择器证明的性质
Properties of Selector Proofs
本文概述选择器证明的数学性质，包括其与普通可证性、ω规则、相对一致性的关系，迭代选择器证明及选择器复杂度，该概念由Artemov引入且与相对一致性密切相关。
math.LO
The cofinality spectrum of the infinite symmetric group
A group G that is not finitely generated can be written as the union of a chain of proper subgroups. The cofinality spectrum of G, written CF(S), is the set of regular cardinals lambda such that G can be expressed as the union of a chain of lambda proper subgroups. The cofinality of G, written c(G), is the least element of CF(G). We show that it is consistent that CF(S) is quite a bizarre set of cardinals. For example, we prove Theorem (A): Let T be any subset of omega setminus {0}. Then it is consistent that aleph_n in CF(S) if and only if n in T . One might suspect that it is consistent that…
math.GRmath.LO
Classical and Effective Descriptive Complexities of omega-Powers
We prove that, for each non null countable ordinal alpha, there exist some Sigma^0_alpha-complete omega-powers, and some Pi^0_alpha-complete omega-powers, extending previous works on the topological complexity of omega-powers. We prove effective versions of these results. In particular, for each non null recursive ordinal alpha, there exists a recursive finitary language A such that A^omega is Sigma^0_alpha-complete (respectively, Pi^0_alpha-complete). To do this, we prove effective versions of a result by Kuratowski, describing a Borel set as the range of a closed subset of the Baire space by…
cs.CCcs.LOmath.LO
