Annals of Mathematical Logic最新文献

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The model theory of ‘R-formal’ fields “r -形式”场的模型理论

Annals of Mathematical Logic

Pub Date : 1980-12-01 DOI: 10.1016/0003-4843(80)90012-1

Bill Jacob

Let K be a field, and let W(K) denote its Witt ring of Quadratic Forms. It is well-known in the theory of Quadratic Forms that the orders of K correpond in a one to one way with all ring surjections $W(K) → Z$ . In particular, a field L is formally real over an ordered field K if and only if there is a homomorphism $ϕ_{1} : W(L)→ Z$ which extends the given ‘signature’ $ϕ_{K} : W(K)→ Z$ . (E.g. $ϕ_{K} = ϕ_{1}, i_{∗}, where i_{∗} : W(K)1 → W(L)$ is the functinal map.)

Using the above, one may discuss the usual theory of formally real and real closed fields in terms of Witt rings, Knebusch in [6] has, in the above setting, given a remarkable new proof of the uniqueness of real closures. One might ask what happens when the $Z$ above is replaced by some other ring R? That is the subject of this present note. In particular, we shall prove some algebraic and model theoretic analogues of well-known results for real closed fields, where the above $Z$ is replaced by some finitely generated reduced Witt ring.

设K是一个域，设W(K)表示它的二次型威特环。众所周知，在二次型理论中，K的阶与所有的环上射W(K)→Z以一对一的方式对应。特别地，域L在有序域K上是形式实的，当且仅当存在一个扩展给定的“签名”的同态态(): W(L)→Z。(例如:K = 1, i *，其中i *: W(K)1→W(L)是函数映射。)在此基础上，我们可以讨论Witt环的形式实域和实闭域的一般理论，在此背景下，Knebusch在[6]中给出了实闭域唯一性的新证明。有人可能会问，当上面的Z被另一个环R取代时会发生什么?这就是本文的主题。特别地，我们将证明实闭场的一些已知结果的代数和模型理论类似，其中上面的Z被一些有限生成的约简Witt环所取代。

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引用次数: 1

Thin collections of sets of projective ordinals and analogs of L L的射影序数和类似物集合的细集合

Annals of Mathematical Logic

Pub Date : 1980-12-01 DOI: 10.1016/0003-4843(80)90010-8

Howard Becker

引用次数: 26

Simple unstable theories 简单不稳定理论

Annals of Mathematical Logic

Pub Date : 1980-12-01 DOI: 10.1016/0003-4843(80)90009-1

S. Shelah

引用次数: 201

The model theory of ‘R-formal’ fields “r -形式”场的模型理论

Annals of Mathematical Logic

Pub Date : 1980-12-01 DOI: 10.1016/0003-4843(80)90012-1

B. Jacob

引用次数: 1

On the elementary theory of quadruples of vector spaces 关于向量空间的四重组的基本理论

Annals of Mathematical Logic

Pub Date : 1980-12-01 DOI: 10.1016/0003-4843(80)90011-X

Walter Baur

引用次数: 19

Countable models of ω1-categorical theories in admissible languages 可容许语言中ω - 1-直言理论的可数模型

Annals of Mathematical Logic

Pub Date : 1980-11-01 DOI: 10.1016/0003-4843(80)90023-6

Henry A. Kierstead

引用次数: 6

Perfect-set forcing for uncountable cardinals 不可数基数的完美设置强制

Annals of Mathematical Logic

Pub Date : 1980-11-01 DOI: 10.1016/0003-4843(80)90021-2

Akihiro Kanamori

引用次数: 18

Interpretations of Heyting's arithmetic—An analysis by means of a language with set symbols 何亭算术的解读——用集合符号的语言分析

Annals of Mathematical Logic

Pub Date : 1980-11-01 DOI: 10.1016/0003-4843(80)90018-2

Martin Stein

Well-known interpretations of Heyting's arithmetic of all finite types are the Diller-Nahm λ-interpretation [1] and Kreisel's modified realizability, subsequently called mr-interpretation [4]. For both interpretations one can define hybrids λ q resp. mq.

In Section 4 a chain of interpretations—called M-interpretations—is defined (it was introduced in [6], filling the “gap” between λ-interpretation and mr-interpretation.

In this paper it is shwon that it is possible to prove in one stroke the soundness resp. characterization theorems for all interpretations of HA_{ω 〈〉} (Heyting's arithmetic of all finite types with functionals for coding finite sequences). This is done by means of interpretations of systems which contain set-symbols. For these so called M-interpretations, soundness-resp. characterization theorems can be proved simultaneously (Theorem 2.51. Special translations of set symbols and of the formula (λωϵW)A — this means, special decisions about the size of the set W; see Sections 3 and 4 — yield the corresponding results for all interpretations of HA_ω〈〉 mentioned.

The terminology of set theoretical language — we consider an extension of HA_ω〈〉 by an extensively weak fragment only, which leads to a conservative extension of HA_ω〈〉 — is of good use for studying realizing terms of different interpretations: if HA_ω<>⊢A, A^M∃υ ∀w A_M, and ⊢A_M[t_M, w] by soundness theorem for M-interpretations, there exists a simple operation which maps $v ̄ to t ̄_{mr}$ , the realizing term for modified realizability. For interpretations of Heyting's arithmetic — λ-interpretation. M-interpretations and mr-interpretation — this leads to the following stability result for existence theorems: if $∃λ A and t_{^} resp. t_{MM}$ is the term computed by λ-interpretation. resp. M-interpretation, with $∃A[t_{M}]$ , then — using extensional equality and ω-rule for equations — we can prove that $t_{λ} = t_{M} = t_{mr}$ (Section 5).

所有有限类型的Heyting算法的著名解释是Diller-Nahm λ-解释[1]和Kreisel的修正可实现性，后来被称为mr-解释[4]。对于这两种解释，我们都可以定义混合λ q resp。mq。在第4节中，定义了一个解释链——称为m -解释(于2010年引入)，填补了λ-解释和mr-解释之间的“空白”。在本文中，证明了可以一次性地证明其完备性。HAω < >的所有解释的表征定理(编码有限序列的泛函的所有有限类型的Heyting算法)。这是通过对包含集合符号的系统的解释来实现的。对于这些所谓的“m -解释”，即“健全-回应”。表征定理可以同时被证明(定理2.51)。集合符号和公式(λωϵW)A的特殊转换-这意味着，关于集合W大小的特殊决定;参见第3节和第4节，对上述所有的HAω < >的解释都得到相应的结果。集合理论语言的术语——我们只考虑广义弱片段对HAω < >的扩展，它会导致HAω < >的保守扩展——对于研究不同解释的实现项很有用:如果HAω<> & a, AM，并且通过m -解释的稳健性定理，存在一个简单的操作，可以将v´映射到t´mr，即修改可实现性的实现项。对于何亭的算术解释——λ解释。m -解释和mr-解释-这导致了存在定理的如下稳定性结果:如果∃λ A和t^ resp。tMM是λ解释计算的项。分别地。使用∃A[tM]进行m解释，然后-使用方程的扩展等式和ω规则-我们可以证明tλ = tM = tmr(第5节)。

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引用次数: 11

Derived rules related to a constructive theory of metric spaces in intuitionistic higher order arithmetic without countable choice 导出无可数选择的直觉高阶算术度量空间构造理论的规则

Annals of Mathematical Logic

Pub Date : 1980-11-01 DOI: 10.1016/0003-4843(80)90019-4

Susumu Hayashi

引用次数: 5

Concerning the consistency of the Souslin hypothesis with the continuum hypothesis 关于苏斯林假设与连续统假设的一致性

Annals of Mathematical Logic

Pub Date : 1980-11-01 DOI: 10.1016/0003-4843(80)90022-4

Keith J. Devlin

引用次数: 4

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