Reading Modern Czech Logic

winter semester 2024/25

Department of Logic, Faculty of Arts, Charles University

Tuesdays 17:30 - 19, Room C116, Celetná 20

Teacher: Zuzana Haniková

So far, all participant speak Czech so sessions run in Czech. Please let me know if you plan to attend and are not a speaker of Czech.

Assessment

To obtain a "zápočet" from this course, participants need to present a paper from the course material (or another material if they wish, to be discussed with me) taking roughly 60 to 120 minutes, and to attend at least half of the sessions that will have taken place.

Course material

Here are some works relevant to the seminar, arranged by their authors.

Rieger's works
1. Sur le problème de nombres naturels. Infinitistic methods. Proc. Symp. Foundations Math., Warsaw, 1959, 225-233.
2. A contribution to Gödel’s axiomatic set theory II. Czechoslovak Mathematical Journal 9(1), 1959, 1-49.
3. O některých základních otázkách matematické logiky. Časopis pro pěstování matematiky 81(3), 1956, 342-351.
Hájek's works
1. Sets, semisets, models. Axiomatic Set Theory. Proc. Symp. in Pure Mathematics of the AMS. The AMS, 1971, 67-82
2. Why semisets? Comment. Math. Univ. Carolinae 14(3), 1973, 397-420.
3. Automatic listing of important observational statements I. Kybernetika 9(3), 1973, 187–205; II. Kybernetika 9(4), 1973, 251-271; III. Kybernetika 10(3), 1974, 95–124.
Vopěnka's works
1. Mathematics in the Alternative Set Theory. Teubner Verlag, Leipzig, 1979.
2. Úvod do matematiky v alternatívnej teórii množín. Alfa, Bratislava, 1989.
Sochor's works
1. Real classes in the ultrapower of hereditarily finite sets. Comment. Math. Univ. Carolinae 16(4), 1975, 637-640.
2. The alternative set theory. Set Theory and Hierarchy Theory. A Memorial Tribute to Andrzej Mostowski. Springer Verlag, 1976, 259-271.
3. Differential calculus in the alternative set theory. Set Theory and Hierarchy Theory V., LNM 619, Springer, 1977.
4. Metamathematics of the Alternative Set Theory I, II, III. Comment. Math. Univ. Carolinae 20(4), 1979, 697–722; 23(1), 1982, 55–79; 24(1), 1983, 137–154.

Remarks: Rieger's paper (3) should be taken as supplementary. His (1) advertises the results of (2) and provides motivations and a lot of context, so anyone presenting it will need to explore the context of (2) (to be discussed). Hájek's (1) assumes the background of The Theory of Semisets. His (2) is relatively standalone. As for Vopěnka's books, we will agree on which sections will be presented.

Assignation of works to be presented

Rieger's (2): David Roubínek; Sochor's (2): Jan Urbánek. Sochor's (3): Filip Jankovec; Sochor's (4): Jiří Rýdl; Zich's works: Martin Quirinus Putzer.

Some supplementary material

(not for presentation, in no particular order)
The Ph.D. thesis of Eliška Pecinová Ladislav Svante Rieger (1916-1963). This dissertation provides plenty of data on varius circumstances of Rieger's life, plus Pecinová's appraisal of each of his works.
Rieger's obituary by Karel Čulík.
Hájek's "K nedožitým šedesátinám Ladislava Riegra" .
Vítězslav Švejdar's paper "Modern Czech Logic: Vopěnka and Hájek, history and background", Journal of Applied Logics — IfCoLoG Journal of Logics and their Applications, 5(6):1261–1271, 2018, preprint available from his website.
Thematic block of papers on Petr Vopěnka in Filosofický časopis 4/2016, edited by Kateřina Trlifajová.
My paper Vopěnkova alternativní teorie množin v matematickém kánonu 20. století., Filosofický časopis 3/2022.

Sessions

#1 1.10.2024
Introduction. Rieger's seminar, Vopěnka's successive seminars, Hájek's seminar. Timeframe of interest (roughly 1950 to 1989). Disclaimers: we will not focus on TIL, on the works of Zich et al. (the earliest employees of the dpt. of logic), on Gentzen, etc. Collections as objects. Von Neumann ordinals as obtained by set-theoretic constructions. Two axiomatizations of the theory of hereditarily finite sets (to be continued).
source: Vopěnka's Mathematics in the AST, chapter I, section 1.

• 8.10.2024
NO SESSION.

#2 15.10.2024
1. Axiomatic theory of hereditarily finite sets. (a) The schema of regularity in Vopěnka's Mathematics in the AST is logically equivalent to the foundation schema. (b) ZF \ (inf) + ("Each set is Tarski finite") proves induction schema. (c) ZF \ (inf) + neg(inf) proves that each set is Tarski finite. (d) The axiomatization in Mathematics in the AST, I.1 (extensionality, empty set, set successors, induction schema) proves axioms of ZF \ (inf) and proves that each set is Tarski finte and Dedekind finite. 2. We stated but did not prove Łoś ultraproduct theorem. 3. Basic idea of the Ackermann interpretation of ZFfin in PA.

• 22.10.2024
NO SESSION. ("Děkanský sportovní den".)

#3 29.10.2024
Presentation: Zich's works (Martin Quirinus Putzer).
Zich's dissertation On the Boundary of Simply Connected Regions -- an end of a simply connected region is a prime end iff the limit cut of every sequence of cuts pertaining to the end in question is a singleton. Zich philosophically in between of Carnap and Wittgenstein and criticism of Gödel's theorems. Zich's habilitation on logic with exclusively interpreted quantifiers. Complex-valued logic. Feedback to Zich's pre-WW2 works (K. Reach, J. Kalicki, A. Church, A. N. Kolmogorov, J. Hintikka). Glimpse of Zich's work after WW2.
Some of Zich works.

O hranici oblasti jednoduše souvislé. Dissertation, Faculty of Science (Přírodovědecká fakulta), Charles University, 1931. Available in print from the library of the Department of Logic.
Příspěvek k theorii celých čísel a jednojednoznačných zobrazení. (habilitation thesis) Rozpravy II. třídy České akademie LVIII(II), 1948, 1-12.
Výrokový počet s komplexními hodnotami. Česká mysl 34(3-4), 1938, 189-196.

#4 5.11.2024
Ackermann interpretation (ZF_fin in PA). Source material: Kaye and Wong, On Interpretations of Arithmetic and Set Theory, 2007.
Ideas from Rieger's papers (1) and (3).(ZH)
Presentation: Rieger: A contribution to Gödel’s axiomatic set theory II. Normal dyadic model; p-adic numbers; GB_fin. (David Roubínek)
A brief exposition of p-adic numbers by MacDuffee.

#5 12.11.2024
Presentation: Hájek:why semisets (ZH). The "semiset" notion. No proper semisets in GB. Possible axioms for a theory (of sets and classes) admitting proper semisets. Relative interpretations (and possibly conservativity) as a main guiding principle in the development of the theory of semisets (Vopěnka, Hájek and others).

#6 19.11.2024
Presentation: basic ideas of Vopěnka's alternative set theory, as in Vopěnka's (1) (ZH). Theory of hereditarily finite sets (revision), axioms. The class of natural numbers. Schema of existence of classes. Vopěnka's definition of an infinite class. Ax. of existence of proper semisets. The class of finite natural numbers. Axiom of prolongation.

#7 26.11.2024
Presentation Sochor's (2) (Jan Urbánek). Personal memory of Antonín Sochor. Reasons why Cantor's set theory succeeded in constructing the world of mathematics. Motivations for alternatives. A comparison with nonstandard methods. Axioms of the Alternative set theory. Possibilities of formulating the axiom of cardinalities and of choice. So-called absolute natural numbers.

#8 3.12.2024
Presentation: Sochor's (3) (Filip Jankovec). Construction of an ultrapower in AST, saturated models, statement of an isomorphism between V and ultrapower of V, differential calculus in AST, non-standard analysis in AST, non-standard analysis cannot be fully avoided in AST (unlike in ZFC).

#9 10.12.2024
Presentations: finish Sochor's (2) (Jan Urbánek). Interpreting the AST in ZF (with CH). Start Sochor's (4) , part I (Jiří Rýdl). Various versions of AST axioms.

#10 17.12.2024
Presentation: Sochor's (4) , part I (Jiří Rýdl). Axioms of the AST and their variants. Interpreting KM_fin in the AST.
Presentation: various philosophical references in Vopěnka's writings. In particular, references to Husserl's Crisis in Mathematics in the AST; some mentions of "late Heidegger" in Meditace o základech vědy; and Rezek's lectures at the Faculty of Mathematics and Physics, later published as Husserlova věcnost (with an afterword by Vopěnka) (ZH).

#11 7.1.2025
Plan: introduction to the GUHA method of data analysis (ZH).