Peter Holy

**E-Mail:** peter.holy@tuwien.ac.at

I am a senior lecturer at the Technical University of Vienna.

My office is DA05G02, on the fifth floor of the green area in the Freihaus building.

**Research Interests:** I am working in set theory with a focus on forcing, definability and large cardinals, as well as second order set theory and weak forms of the axiom of choice.

Publications

24: **(with Jonathan Schilhan) ***The ordering principle and the axiom of dependent choice*.

In preparation.

23: **(with Philipp Lücke and Sandra Müller) ***Outward compactness*.

Submitted, 2024. pdf talk slides

22: **(with Philipp Schlicht, Christopher Turner and Philip Welch) ***Asymmetric Cut and Choose Games*.

Bulletin of Symbolic Logic, 2023. pdf talk slides (general maths audience) talk slides (set theory audience) talk slides (general maths audience)

21: **(with Brent Cody) ***Ideal Operators and Higher Indescribability*.

Journal of Symbolic Logic 88, no. 2, pp. 835-873, 2023. pdf

20: **(with Rodrigo Freire) ***An ordinal-connection axiom as a weak form of global choice under the GCH*.

Archive for Mathematical Logic, 2022. pdf

19: **(with Rodrigo Freire) ***An axiomatic approach to forcing in a general setting*.

Bulletin of Symbolic Logic 28, no. 3, pp. 427-450, 2022. pdf

18: *Ramsey-like Operators*.

Fundamenta Mathematicae 259, pp. 111-149, 2022. pdf talk slides

17: **(with Marlene Koelbing, Philipp Schlicht and Wolfgang Wohofsky) ***Ideal Topologies in Higher Descriptive Set Theory*.

Annals of Pure and Applied Logic 173, no. 4, 36 pages, 2022. pdf talk slides 1 talk slides 2 talk slides 3 talk slides 4

16: **(with Philipp Lücke) ***Small Models, Large Cardinals, and Induced Ideals*.

Annals of Pure and Applied Logic 172, no. 2, 50 pages, 2021. pdf talk slides 1 talk slides 2

15: **(with Philipp Lücke and Ana Njegomir) ***Characterizing Large Cardinals through Neeman's pure side condition forcing*.

Fundamenta Mathematicae 252, pp. 53-102, 2021. pdf talk slides

14: **(with Victoria Gitman, Joel Hamkins, Philipp Schlicht and Kameryn Williams) ***The exact Strength of the Class Forcing Theorem*.

Journal of Symbolic Logic 85, no. 3, pp. 869-905, 2020. pdf talk slides

13: **(with Philipp Lücke and Ana Njegomir) ***Small Embedding Characterizations for Large Cardinals*.

Annals of Pure and Applied Logic 170, no. 2, pp. 251-271, 2019. pdf talk slides

12: **(with Regula Krapf and Philipp Schlicht) ***Sufficient conditions for the forcing theorem, and turning proper classes into sets*.

Fundamenta Mathematicae 246, pp. 27-44, 2019. pdf

11: **(with Philipp Schlicht) ***A Hierarchy of Ramsey-like cardinals*.

Fundamenta Mathematicae 242, pp 49-74, 2018. pdf

10: **(with Regula Krapf and Philipp Schlicht) ***Characterizations of Pretameness and the Ord-cc*.

Annals of Pure and Applied Logic 169, no. 8, pp. 775-802, 2018. pdf

9: **(with Philipp Lücke) ***Simplest possible locally definable Wellorders.*

Fundamenta Mathematicae 236, pp. 101-139, 2017. pdf talk slides

8: **(with Regula Krapf, Philipp Lücke, Ana Njegomir and Philipp Schlicht) ***Class Forcing, the Forcing Theorem and Boolean Completions*.

Journal of Symbolic Logic 81, no. 4, pp. 1500-1530, 2016. pdf talk slides

7: *Sigma_1-Wellorders without Collapsing*.

Archive for Mathematical Logic 54, issue 3, pp 453-462, 2015. pdf talk slides

6: **(with David Asperó and Philipp Lücke) ***Forcing lightface Definable Wellorders without the GCH.*

Annals of Pure and Applied Logic 166, no. 5, pp. 553-582, 2015. pdf, David Asperó's talk slides

5: **(with Philip Welch and Liuzhen Wu) ***Local Club Condensation and L-likeness.*

Journal of Symbolic Logic 80, no. 4, pp. 1361-1378, 2015. pdf, talk slides

4: **(with Sy-David Friedman and Philipp Lücke) ***Large Cardinals and Lightface Definable Wellorders without GCH.*

Journal of Symbolic Logic 80, issue 1, pp 251-284, 2015. pdf, talk slides

3: **(with Philipp Lücke) ***Locally Sigma_1-definable Wellorders of H(kappa^+).*

Fundamenta Mathematicae 226, pp 221-236, 2014. pdf, talk slides

2: **(with Sy-David Friedman) ***A Quasi Lower Bound on the Consistency Strength of PFA.*

Transactions of the AMS 366, pp 4021-4065, 2014. pdf, talk slides

1: **(with Sy-David Friedman) ***Condensation and Large Cardinals.*

Fundamenta Mathematicae 215, no. 2, pp 133-166, 2011. pdf, info

**Dissertation:*** Condensation and Large Cardinals.*

(2010, advisor: Sy D. Friedman) pdf, info

**Masters Thesis:*** Absoluteness Results in
Set Theory.*

(2007, advisor: Sy D. Friedman) pdf

Unpublished Notes

*Interesting thin sets and Nash-Williams cardinals.*

(2024) pdf

**(with Philipp Lücke)** *A countable support iteration for the tree property at aleph_2 and related properties.*

(2020) pdf

*A short forcing argument for the proper forcing axiom using Magidor's characterization of supercompactness.*

(2019) pdf

**(with Regula Krapf and Philipp Schlicht)** *Separation in Class Forcing Extensions.*

(2017) pdf

*A generalization of the notion of bounded degree for infinite graphs.*

(2014) pdf

*PFA and Class Forcing.*

(2013) pdf

*Canonical Function Coding over a Stationary Set.*

(2013) pdf, info

*Condensation and Large Cardinals - A Simplified Version of my Dissertation.*

(2013) pdf, info

Talks

*Compactness of strong logics*

TU Wien Algebra seminar, Vienna, 15.03.2024

*Infinite Cut and Choose games*

TU Wien Algebra seminar, Vienna, 04.11.2022 --> pdf

*Ulam style Cut and Choose games*

VCU Analysis, Logic and Physics Seminar, 08.10.2021 --> pdf

*A surprising equality between two infinities (p=t)*

Habilitation Colloquium, Bonn, 31.01.2019

*Surreal Numbers*

Basic Notions Seminar, Bonn, 25.10.2016 (following slides by Yurii Khomskii)

*Forcing: How to prove Unprovability*

Basic Notions Seminar, Bonn, 18.11.2015 (Part II of a two-part talk, part I by Regula Krapf).

*The Outer Model Programme*

Norwich Pure Maths Seminar, 28.04.2014 --> pdf

** Compactness Principles**

*Large Cardinal Compactness*

Seventh Workshop on generalised Baire spaces, Bristol, 10.02.2024 --> pdf

*A short talk on Outward Compactness*

KGRC set theory seminar, Vienna, 22.11.2022

** Weak forms of Global Choice**

*A weak form of global choice under the GCH*

KGRC set theory seminar, Vienna, 10.01.2023 and 17.01.2023 (two part talk)

** Infinite Games**

*Asymmetric Cut and Choose games*

Barcelona set theory seminar, 25.05.2022; New York set theory seminar, 11.11.2022 --> pdf

** Large Cardinals**

*Ramsey-like Operators*

Torino/Udine logic seminar, 20.11.2020; Set Theory in the UK workshop, 04.12.2020; Kurt Gödel Research Center, 17.12.2020 --> pdf

*Uniform Large Cardinal Characterisations and Ideals up to measurability*

New York Set Theory Seminar, 10.07.2020 --> pdf

*Small Models, Large Cardinals, and Large Cardinal Ideals*

Bonn Logic Seminar, 23.10.2018; European Set Theory Conference, Vienna, 05.07.2019 --> pdf

*Characterizing Large Cardinals through Forcing*

Set Theory Today Conference, Vienna, 10.09.2018 --> pdf

*A Hierarchy of Ramsey-like cardinals*

Bristol Set Theory Seminar, 07.03.2017

*Small embedding characterizations for large cardinals, and internal large cardinals*

Bonn Logic Seminar, 09.05.2016; Young Set Theory Workshop, Copenhagen, 16.06.2016; Kurt Gödel Research Center, 06.04.2017 --> pdf

** Generalized or Higher Descriptive Set Theory**

*Ideal Topologies*

Bristol Logic and Set Theory Seminar, 10.06.2020 --> pdf

*Generalised Topologies on 2^kappa, Silver forcing and Diamond*

Udine Logic Seminar, 29.05.2020 --> pdf

*Ideal Topologies*

Udine Logic Seminar, 01.04.2020 --> pdf

*The Edinburgh Topology*

Amsterdam Academy Colloquium on Generalised Baire Spaces, 24.08.2018 (Part I of a two-part talk, part II by Wolfgang Wohofsky) --> pdf

** General Audience Talks**

*Mathematische Unentscheidbarkeit (Undecidability in Mathematics)*

Antrittsvorlesung, Dies Academicus, University of Bonn, 04.12.2019 --> pdf (in German)

** Class Forcing**

*The exact Strength of the Class Forcing Theorem*

Bonn Logic Seminar, 23.10.2017 --> pdf

*How to and how not to turn proper classes into sets*

Bonn Logic Seminar, 24.10.2016

*Class Forcing and Pretameness*

Hamburg Logic Seminar, 30.05.2016

*Failures of the Forcing Theorem*

European Set Theory Conference, Cambridge, 25.08.2015 --> pdf

*Non-pretame Class Forcing and the Forcing Theorem, the axioms of ZFC and non-definable Class Forcing*

Kurt Gödel Research Center, Vienna, 27.05.2015

*The Dark Side of Class Forcing*

Bristol Logic Seminar, 10.03.2015

** Locally Definable Wellorders**

*Forcing lightface definable Wellorders without the GCH*

Bonn Logic Seminar, 09.11.2015

*Delta^1_1 subsets of the generalized Baire Space*

Amsterdam workshop on Set Theory, 03.11.2014 --> pdf

*Simplest Possible Wellorders of H(kappa^+)*

Winter School in Abstract Analysis, section Set Theory and Topology, Hejnice, 26.01.2014; Bonn Logic Seminar, 19.02.2014; Bristol Logic Seminar, 18.03.2014

*Locally Sigma_1-definable wellorders of H(kappa^+)*

ESI Workshop on Forcing and Large Cardinals, Vienna, 23.09.2013

*Large Cardinals and lightface definable Wellorders without the GCH*

Kurt Gödel Research Center, Vienna, 10.01.2013; Bristol Logic Seminar, 11.03.2013 --> pdf

** Condensation and the Outer Model Programme**

*Condensation does not imply Square*

Inner and Outer Model Theory Workshop, Bristol, 06.07.2014; Bonn Logic Seminar, 20.04.2015; Münster Set Theory Seminar, 11.06.2015; Logic Colloquium, Helskinki, 03.08.2015 --> pdf

*The Outer Model Programme*

Oxford Logic Seminar, 07.02.2013 --> pdf

*L-like Models with Large Cardinals and a quasi lower Bound on the Consistency Strength of PFA.*

PhD Colloquium Paderborn, 13.09.2012; Bristol Logic Seminar, 28.11.2012; Bonn Logic Seminar, 08.04.2013; Young Set Theory Workshop Oropa, 12.07.2013 --> pdf

Teaching

**Current Teaching**

*at the Technical University of Vienna:*

autumn 2024:

Algebra 2 - Lecture and Exercises

**Past Teaching**

*at the Technical University of Vienna:*

spring 2024: Algebra (with Michael Pinsker)

autumn 2023: Algebra 2 - Lecture (with Michael Pinsker) and Exercises

spring 2023: Linear Algebra 2

*at the University of Udine:*

autumn 2021: PhD course Infinite games and (generic) large cardinals

*at the University of Bonn:*

autumn 2019: Models of Set Theory 2 -- Iterated forcing and its applications (lecture and exercises)

spring 2019:

Exercises for Basic Features of Mathematics II

Seminar in Set Theory -- Large Cardinals (with Peter Koepke)

autumn 2018:

Exercises for Basic Features of Mathematics I

spring 2018:

Exercises for Basic Features of Mathematics II

Seminar in Logic -- Large Cardinals and Strong Logics

autumn 2017:

Exercises for Basic Features of Mathematics I

Seminar in Set Theory -- Constructibility (with Peter Koepke, Philipp Lücke and Philipp Schlicht)

spring 2017: Exercises for Models of Set Theory

autumn 2015: Exercises for Elements of Mathematics

spring 2015:

Seminar in Logic -- Constructibility (with Peter Koepke)

Exercises for Introduction to Mathematical Logic

Logic Bachelor thesis seminar (with Peter Koepke and Philipp Lücke)

autumn 2014: Exercises for Elements of Mathematics

*at the University of Bristol:*

2013/2014: Mathematics Tutorials for 1st year students (Analysis, Group Theory, Number Theory)

2012/2013: Mathematics Tutorials for 1st year students (Analysis, Group Theory, Number Theory)

*at the Kurt Gödel Research Center for Mathematical Logic, Vienna:*

autumn 2011:

Reading Course in Set Theory

Exercises for Introduction to Mathematical Logic.

spring 2011: Exercises for Axiomatic Set Theory 1.

spring 2010: Exercises for Axiomatic Set Theory 1.

spring 2009: Exercises for Axiomatic Set Theory 1.

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Update log here.