Christine Gaßner,  University of Greifswald,  gassnerc@uni-greifswald.de

The starting point for this project was a question (2018) on the relationship between a principle of choice introduced by Wilhelm ACKERMANN (1935) and the well-ordering theorem in second-order logic. To show that the second-order axioms given by HAC are valid in the basic second-order FRAENKEL model, various necessary (mathematical) assumptions, definitions, examples and proofs as well as several overviews and summaries are provided.

The semantics used are based on ideas by  Peter SCHREIBER and were introduced by Günter ASSER.

2024

   Gaßner: Das Projekt The Axiom AC in HPL, (seminar talk: 1), Greifswald 2024

   Gaßner: AC and the Independence of the Law of Trichotomy in Second-Order Henkin Logic (arXiv, researchgate)

   Gaßner: Permutation Models of Second Order (preprint)

   Gaßner: Relationships between Principles of Choice in Second-Order Henkin Structures (preprint)

   Gaßner: AC and the Independence of WO in Second-Order Henkin Logic, Part II (preprint)

   Gaßner: AC and the Independence of WO in Second-Order Henkin Logic, Part I  (preprint)

   Gaßner: Second-Order Henkin Structures and the Axiom of Choice. A first draft  (pages 1 - 9 of the summary)

• _{A survey of the first known relationships between the Axiom of Choice and other statements in second-order predicate logic was presented in a talk by Günter ASSER at the Mathematical Research Institute of Oberwolfach in 1985. Certain information mentioned by Stewart SHAPIRO in two notes of his book Foundations without foundationalism: a case for second-order logic maybe result from a discussion after this talk at the meeting in Oberwolfach. Inspired by questions raised in this context by Paolo MANCOSU and Stewart SHAPIRO in 2018, we will re-examine the meaning of the Axiom of Choice in second-order predicate logic with Henkin interpretation. The idea of expanding my originally planned article into a book comes from Peter SCHUSTER. In this way, most of the known results could be taken into account. Moreover, at the suggestion of  Michael RATHJEN, the visibility of summaries of the results has been improved. As a result, there are now some separate sections for overviews and summaries.}

   Gaßner: A simple permutation model of HPL and the independence of the principle WOP from HAC (pdf),  Arbeitstreffen  Rathjen -- Gaßner , Kloster 2024

   Rathjen, M.: Contributions to  the independence of HAC from WOP,  Arbeitstreffen  Rathjen -- Gaßner, Leeds & Kloster 2024

   Gaßner: AC in HPL. Ein Querschnitt (slides 1, 2, 3), Meeting   Misselbeck-Wessel -- Gaßner,   Alfried Krupp Wissenschaftskolleg, Greifswald 2024

2023

    Gaßner: Second-Order Henkin Structures and the Axiom of Choice. A first draft of the first part. (contents)

2022

   Gaßner: Second-Order Henkin Semantics and the Axiom of Choice (slides 1,60,78,79, abstract), Colloquium Logicum  2022, Konstanz 2022 

   Gaßner: Second-Order Logic and Russell's Axiom of Choice (slides 1,8,73,74), DMV Annual Meeting 2022, Berlin 2022

2021

   Gaßner: The Axiom of Choice: Different Formulations, Meeting  Tamminga  (Institut für Philosophie) --  Institut für Mathematik und Informatik, Greifswald 2021

   Gaßner: Videos for the lectures Mathematical Logic including second-order permutation models  (see: 1, 2 ,3), WiSe 2021/22 

2019

   Gaßner: Das Auswahlaxiom (seminar), WiSe 2019/20

   Gaßner: Das Auswahlaxiom und die Prädikatenlogik zweiter Stufe (slides), Meeting   Cordes  --  Reinmuth  -- Gaßner ,  Institut für Philosophie, Greifswald 2019  

2016

   Gaßner: BSS RAM's with ν-Oracle and the Axiom of Choice (slides),  Colloquium Logicum 2016, Hamburg 2016

2014

   Gaßner: Mathematical Logic including second-order permutation models (lectures), SoSe 2014

2012

   Gaßner: On the axiom of choice and infinite graphs  (abstract),  Arbeitstreffen  Spreen  -- Schlicht  -- Diener  -- Gaßner  -- Schwende,  Greifswald 2012

1994

   Gaßner: The Axiom of Choice in Second-Order Predicate Logic, Mathematical Logic Quarterly 40 (1994), 533–546. DOI: 10.1002/malq.19940400410

1985

   Asser, G.: Logische Abhängigkeiten im Prädikatenkalkül der zweiten Stufe (Tagungsbericht, slides 1,9,10), Oberwolfach 1985 

   Gaßner: Das Auswahlaxiom im Prädikatenkalkül zweiter Stufe (Dissertation 1984, location, pdf, slides 1,12, defense of thesis 1985), Greifswald 1985

1981, ...  

    Asser, G.: Einführung in die mathematische Logik (Teubner), Teil I: Aussagenkalkül, 1967; Teil II: Prädikatenkalkül erster Stufe, 1972; Teil III: Prädikatenlogik höherer Stufe, 1981

_{Jouko Väänänen:  SECOND-ORDER AND HIGHER-ORDER LOGIC}

_{Benjamin Siskind, Paolo Mancosu, and Stewart Shapiro:  A NOTE ON CHOICE PRINCIPLES IN SECOND-ORDER LOGIC}

_{Kameryn J. Williams: The Structure of Models of Second-order Set Theories}

_{Stephen Mackereth:  Infinity, Choice, and Hume's Principle}

120 years of choice, Leeds 2024

   Gaßner: Das Auswahlaxiom im Prädikatenkalkül zweiter Stufe (Dissertation 1984)

   Schleinitz, W.: Axiomatisierung mehrsortiger Aussagenkalküle (Dissertation 1981)

   Soyka, D.: Metamathematische Methoden in der konstuktiven Mathematik  (Dissertation 1981)

   Weese, M.: Entscheidbarkeit der Theorie der Booleschen Algebren in Sprachen mit Mächtigkeitsquantoren  (Dissertation 1978)

   Schreiber, P.: Untersuchungen über die Modelle der Typentheorie (Dissertation 1965)

First published on September 28, 2024

Updated on December 04, 2024