该杂志发表研究论文，偶尔也会发表关于数理逻辑的调查或阐述。只要数理逻辑的方法发挥重要作用，也欢迎其他相关领域的贡献，例如理论计算机科学或哲学。因此，该杂志面向对数理逻辑在各自领域的应用及其与其他研究领域的互动感兴趣的逻辑学家和数学家、计算机科学家和哲学家。

The Archive for Mathematical Logic is a journal dedicated to research in mathematical logic, primarily publishing research papers and occasionally featuring surveys or expository articles on the subject. The journal welcomes contributions from other related fields, particularly theoretical computer science and philosophy, as long as they significantly utilize methods from mathematical logic. Thus, it targets logicians, mathematicians, computer scientists, and philosophers who are interested in the applications of mathematical logic in their respective fields and its interactions with other areas of research.

《Archive for Mathematical Logic》是一本专注于数理逻辑研究的期刊，主要发表研究论文，偶尔也会刊登关于数理逻辑的综述或阐述性文章。该期刊欢迎任何在数理逻辑方法上具有重要贡献的相关领域的研究，尤其是理论计算机科学和哲学。因此，它的目标读者包括对数理逻辑在各自领域的应用及其与其他研究领域的互动感兴趣的逻辑学家、数学家、计算机科学家和哲学家。

Archive for Mathematical Logic

Alice Medvedev

Alexander Van Abel

Variations on the Feferman-Vaught theorem, with applications to $$\prod _p \mathbb {F}_p$$

Feferman-Vaught 定理的变体及其在 $$\prod _p \mathbb {F}_p$$ 中的应用

Gunter Fuchs

On the consistency strength of critical leaps

论临界跃迁的一致性强度

Hyoyoon Lee

Junguk Lee

Relativized Galois groups of first order theories over a hyperimaginary

超虚数域上的一阶理论的相对化伽罗瓦群

Abstract We investigate whether classical combinatorial theorems are provable in ZF. Some statements are not provable in ZF, but they are equivalent within ZF. For example, the following statements (i)–(iii) are equivalent: $$cf({\omega }_1)={\omega }_1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>c</mml:mi> <mml:mi>f</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>ω</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>ω</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> </mml:math> , $${\omega }_1\rightarrow ({\omega }_1,{\omega }+1)^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>ω</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>→</mml:mo> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>ω</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mi>ω</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math> , any family $$\mathcal {A}\subset [{On}]^{&lt;{\omega }}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>⊂</mml:mo> <mml:msup> <mml:mrow> <mml:mo>[</mml:mo> <mml:mrow> <mml:mi>On</mml:mi> </mml:mrow> <mml:mo>]</mml:mo> </mml:mrow> <mml:mrow> <mml:mo>&lt;</mml:mo> <mml:mi>ω</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> of size $${\omega }_1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ω</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> contains a $$\Delta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Δ</mml:mi> </mml:math> -system of size $${\omega }_1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ω</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> . Some classical results cannot be proven in ZF alone; however, we can establish weaker versions of these statements within the framework of ZF, such as $${{\omega }_2}\rightarrow ({\omega }_1,{\omega }+1)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>ω</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>→</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>ω</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mi>ω</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , any family $$\mathcal {A}\subset [{On}]^{&lt;{\omega }}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>⊂</mml:mo> <mml:msup> <mml:mrow> <mml:mo>[</mml:mo> <mml:mrow> <mml:mi>On</mml:mi> </mml:mrow> <mml:mo>]</mml:mo> </mml:mrow> <mml:mrow> <mml:mo>&lt;</mml:mo> <mml:mi>ω</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> of size $${\omega }_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ω</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> contains a $$\Delta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Δ</mml:mi> </mml:math> -system of size $${\omega }_1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ω</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> . Some statements can be proven in ZF using purely combinatorial arguments, such as: given a set mapping $$F:{\omega }_1\rightarrow {[{\omega }_1]}^{&lt;{\omega }}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>F</mml:mi> <mml:mo>:</mml:mo> <mml:msub> <mml:mi>ω</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>→</mml:mo> <mml:msup> <mml:mrow> <mml:mo>[</mml:mo> <mml:msub> <mml:mi>ω</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>]</mml:mo> </mml:mrow> <mml:mrow> <mml:mo>&lt;</mml:mo> <mml:mi>ω</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> , the set $${\omega }_1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ω</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> has a partition into $${\omega }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ω</mml:mi> </mml:math> -many F -free sets. Other statements can be proven in ZF by employing certain methods of absoluteness, for example: given a set mapping $$F:{\omega }_1\rightarrow {[{\omega }_1]}^{&lt;{\omega }}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>F</mml:mi> <mml:mo>:</mml:mo> <mml:msub> <mml:mi>ω</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>→</mml:mo> <mml:msup> <mml:mrow> <mml:mo>[</mml:mo> <mml:msub> <mml:mi>ω</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>]</mml:mo> </mml:mrow> <mml:mrow> <mml:mo>&lt;</mml:mo> <mml:mi>ω</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> , there is an F -free set of size $${\omega }_1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ω</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> , for each $$n\in {\omega }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>ω</mml:mi> </mml:mrow> </mml:math> , every family $$\mathcal {A}\subset {[{\omega }_1]}^{{\omega }}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>⊂</mml:mo> <mml:msup> <mml:mrow> <mml:mo>[</mml:mo> <mml:msub> <mml:mi>ω</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>]</mml:mo> </mml:mrow> <mml:mi>ω</mml:mi> </mml:msup> </mml:mrow> </mml:math> with $$|A\cap B|\le n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>A</mml:mi> <mml:mo>∩</mml:mo> <mml:mi>B</mml:mi> <mml:mo>|</mml:mo> <mml:mo>≤</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:math> for $$\{A,B\}\in {[\mathcal {A}]}^{2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:mo>{</mml:mo> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> <mml:mi>B</mml:mi> <mml:mo>}</mml:mo> </mml:mrow> <mml:mo>∈</mml:mo> <mml:msup> <mml:mrow> <mml:mo>[</mml:mo> <mml:mi>A</mml:mi> <mml:mo>]</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math> has property B . In contrast to statement (5), we show that the following ZFC theorem of Komjáth is not provable from ZF + $$cf({\omega }_1)={\omega }_1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>c</mml:mi> <mml:mi>f</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>ω</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>ω</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> </mml:math> : (6 $$ ^*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mmultiscripts> <mml:mrow/> <mml:mrow/> <mml:mo>∗</mml:mo> </mml:mmultiscripts> </mml:math> ) every family $$\mathcal {A}\subset {[{\omega }_1]}^{{\omega }}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>⊂</mml:mo> <mml:msup> <mml:mrow> <mml:mo>[</mml:mo> <mml:msub> <mml:mi>ω</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>]</mml:mo> </mml:mrow> <mml:mi>ω</mml:mi> </mml:msup> </mml:mrow> </mml:math> with $$|A\cap B|\le 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>A</mml:mi> <mml:mo>∩</mml:mo> <mml:mi>B</mml:mi> <mml:mo>|</mml:mo> <mml:mo>≤</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> for $$\{A,B\}\in {[\mathcal {A}]}^{2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:mo>{</mml:mo> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> <mml:mi>B</mml:mi> <mml:mo>}</mml:mo> </mml:mrow> <mml:mo>∈</mml:mo> <mml:msup> <mml:mrow> <mml:mo>[</mml:mo> <mml:mi>A</mml:mi> <mml:mo>]</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math> is essentially disjoint . A function f is a uniform denumeration on $${\omega }_1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ω</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> iff $${\text {dom}}(f)={\omega }_1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext>dom</mml:mtext> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>f</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>ω</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> </mml:math> , and for every $$1\le {\alpha }&lt;{\omega }_1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>≤</mml:mo> <mml:mi>α</mml:mi> <mml:mo>&lt;</mml:mo> <mml:msub> <mml:mi>ω</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> </mml:math> , $$f({\alpha })$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>(</mml:mo> <mml:mi>α</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> is a function from $${\omega }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ω</mml:mi> </mml:math> onto $${\alpha }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> . It is easy to see that the existence of a uniform denumeration of $${\omega }_1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ω</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> implies $$cf({\omega }_1)={\omega }_1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>c</mml:mi> <mml:mi>f</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>ω</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>ω</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> </mml:math> . We prove that the failure of the reverse implication is equiconsistent with the existence of an inaccessible cardinal.

Tamás Csernák

Lajos Soukup

Infinite combinatorics revisited in the absence of Axiom of choice

无限组合数学在选择公理缺失情况下的再探讨

Abstract We investigate the degree spectra of computable relations on canonically ordered natural numbers $$(\omega ,&lt;)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>ω</mml:mi> <mml:mo>,</mml:mo> <mml:mo>&lt;</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> and integers $$(\zeta ,&lt;)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>ζ</mml:mi> <mml:mo>,</mml:mo> <mml:mo>&lt;</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . As for $$(\omega ,&lt;)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>ω</mml:mi> <mml:mo>,</mml:mo> <mml:mo>&lt;</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , we provide several criteria that fix the degree spectrum of a computable relation to all c.e. or to all $$\Delta _2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Δ</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> degrees; this includes the complete characterization of the degree spectra of so-called computable block functions that have only finitely many types of blocks. Compared to Bazhenov et al. (in: LIPIcs, vol 219, pp 8:1–8:20, 2022), we obtain a more general solution to the problem regarding possible degree spectra on $$(\omega ,&lt;)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>ω</mml:mi> <mml:mo>,</mml:mo> <mml:mo>&lt;</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , answering the question whether there are infinitely many such spectra. As for $$(\zeta ,&lt;)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>ζ</mml:mi> <mml:mo>,</mml:mo> <mml:mo>&lt;</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , we prove the following dichotomy result: given an arbitrary computable relation R on $$(\zeta ,&lt;)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>ζ</mml:mi> <mml:mo>,</mml:mo> <mml:mo>&lt;</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , its degree spectrum is either trivial or it contains all c.e. degrees. This result, and the proof techniques required to solve it, extend the analogous theorem for $$(\omega ,&lt;)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>ω</mml:mi> <mml:mo>,</mml:mo> <mml:mo>&lt;</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> obtained by Wright (Computability 7:349–365, 2018), and provide initial insight to Wright’s question whether such a dichotomy holds on computable ill-founded linear orders. This article is an extended version of Bazhenov et al. (in: LIPIcs, vol 219, pp 8:1–8:20, 2022).

Nikolay Bazhenov

Dariusz Kalociński

Michał Wrocławski

Degrees of relations on canonically ordered natural numbers and integers

规范有序的自然数和整数上的关系度

Ziemowit Kostana

What would the rational Urysohn space and the random graph look like if they were uncountable?

如果乌雷松空间和随机图是无穷的，它们会是什么样子？

Abbreviation	Arch. Math. Logic
Journal Impact	0.45
Quartiles(Global)	MATHEMATICS(Q4)

ISSN	0933-5846, 1432-0665
h-index	32

Publication Information	Publisher: Springer New York，Publishing cycle: 8 issues per year，Journal Type: journal，Open Access Journals: No
Basic data	Year of publication: 1988，Proportion of original research papers: 100.00%，Self Citation Rate:0.00%， Gold OA Rate: 32.39%

Archive for Mathematical Logic

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Variations on the Feferman-Vaught theorem, with applications to $$\prod _p \mathbb {F}_p$$

On the consistency strength of critical leaps

Relativized Galois groups of first order theories over a hyperimaginary

Infinite combinatorics revisited in the absence of Axiom of choice

Degrees of relations on canonically ordered natural numbers and integers

Variations on the Feferman-Vaught theorem, with applications to $$\prod _p \mathbb {F}_p$$

On the consistency strength of critical leaps

Relativized Galois groups of first order theories over a hyperimaginary

Infinite combinatorics revisited in the absence of Axiom of choice

What would the rational Urysohn space and the random graph look like if they were uncountable?