Spencer Unger

I am an assistant professor in the Department of Mathematics at University of Toronto. For research related email, I can be reached at sunger (at) math (dot) toronto (dot) edu. For university related email, including undergraduate teaching, I can be reached at spencer (dot) unger (at) utoronto (dot) ca.

Research interests

My research is in set theory and mathematical logic. I am interested in questions in combinatorial set theory and measurable combinatorics.


All graduate and undergraduate course information is available on Quercus


From July 1, 2017 until June 30, 2021, I am partially supported by NSF DMS-1700425.


Papers are listed in roughly chronological order

Fragility and indestructibility of the tree property, Archive for Mathematical Logic 51 (2012) 635-645.

Aronszajn Trees and the successors of a singular cardinal, Archive for Mathematical Logic 52 (2013) 483-496.

(Joint with Dima Sinapova) Combinatorics at aleph_omega, Annals of Pure and Applied Logic 165 (2014) 996-1007.

A model of Cummings and Foreman revisited, Annals of Pure and Applied Logic 165 (2014) 1813-1831.

Fragility and indestructibility II, Annals of Pure and Applied Logic 166 (2015) 1110-1122.

(Joint with Dima Sinapova) Scales at aleph_omega, Israel Journal of Math, 209 (2015) 463-486.

The tree property below aleph_{omega times 2}, Annals of Pure and Applied Logic, 167 (2016) 247-261.

Iterating along a Prikry sequence, Fundamenta Mathematicae, 232 (2016).

(Joint with Andrew Marks) Borel measurable paradoxical decompositions via matchings, Advances in Mathematics, 289 (2016) 397-410.

(Joint with Dima Sinapova) Modified extender based forcing, Journal of Symbolic Logic, 81 (2016) 1432-1443.

(Joint with Will Boney) Large cardinal axioms from tameness in AECs,Proceedings of the American Mathematical Society, 145 (2017) 4517-4532.

(Joint with Dima Sinapova) The tree property at aleph_{omega^2+1} and aleph_{omega^2+2}, Journal of Symbolic Logic, 82 (2018) 669-682. The published version contains a false proof of Lemma 4.1. Here is a correct proof.

(Joint with Yair Hayut) The strong tree property and weak square, Mathematical Logic Quarterly, 63 (2016) 150-154.

(Joint with Omer Ben-Neria and Chris Lambie-Hanson) Diagonal supercompact Radin forcing, Annals of Pure and Applied Logic 171 (2020).

(Joint with Omer Ben-Neria) Homogeneous changes in cofinalities with applications to HOD, Journal of Mathematical Logic, 17 (2017).

(Joint with Andrew Marks) Borel circle squaring, Annals of Mathematics, 168 (2017) 581-605.

Successive failures of approachability, accepted for publication in Israel journal of mathematics.

(Joint with Yair Hayut) Stationary reflection, submitted.

(Joint with Omer Ben-Neria, Moti Gitik and Itay Neeman) On the powersets of singular cardinals in HOD, Proceedings of the American Mathematical Society 148 (2020), 1777-1789.

(Joint with Clinton Conley and Andrew Marks) Measurable realizations of abstract systems of congruences, Forum of Math, Sigma 8 (2020) e10.

(Joint with James Cummings, Yair Hayut, Menachem Magidor, Itay Neeman and Dima Sinapova) The ineffable tree property and the failure of the singular cardinals hypothesis, Transactions of the American Mathematical Society, 373 (2020).

(Joint with James Cummings, Yair Hayut, Menachem Magidor, Itay Neeman and Dima Sinapova) The tree property at the two immediate successors of a singular cardinal, accepted for publication in the Journal of Symbolic Logic.

(Joint with Omer Ben-Neria and Yair Hayut) Reflection and the failure of SCH, submitted.

Appalchian set theory

I took notes for two Appalachian set theory workshops as a graduate student, one given by Itay Neeman and the other by Moti Gitik. The webpage for those workshops is here and the notes can be found by scrolling down the page. The notes were published in a nice volume by Cambridge University Press.

Research notes

Gitik's Gap 2 short extender forcing

Gitik's Gap 2 short forcing with collapses

Solovay's model

Compactness for the chromatic number at aleph_{omega_1 +1}

Old pages from CMU and UCLA

For information about my time a UCLA including the graduate student seminar that I organized, some logic courses that I taught and the forcing class in the 2013 Logic Summer School that I taught, click here.

For some very old seminar notes and information about summer calculus classes that I taught at CMU, click here.