PhD thesis

The effects of adding a real to models of set theory.

Papers

1- Shelah's strong covering property and CH in V[r], with E. Eslami, Math. Log. Q. 58 (2012), no. 3, 153-158.

2- Independence of higher Kurepa hypotheses, with Sy D. Friedman, Arch. Math. Logic 51 (2012), no. 5-6, 621-633.

3- Almost Souslin Kurepa trees, Proc. Amer. Math. Soc. 141 (2013), no. 5, 1821-1826.

4- Killing the GCH everywhere with a single real, with Sy D. Friedman, J. Symbolic Logic 78 (2013), no 3, 803-823.

5- Killing GCH everywhere by a cofinality-preserving forcing notion over a model of GCH, with Sy D. Friedman, Fund. Math. 223 (2013), no 2, 171-193.

6- The foundation axiom and elementary self-embeddings of the universe, with A. S. Daghighi, J. D. Hamkins, and E. Jeřabek,  in: Infinity, Computability, and Metamathematics: Festschrift celebrating the 60th birthdays of Peter Koepke and Philip Welch (S. Geschke, B. Lowe, and P. Schlicht, eds.), College Publications, London, 2014, pp. 89-112.

7- More on almost Souslin Kurepa trees, Proc. Amer. Math. Soc. 142 (2014), no 10, 3631-3634.

8- Adding a lot of Cohen reals by adding a few I, with M. Gitik, Trans. Amer. Math. Soc. 367 (2015), no. 1, 209-229.

9- Adding a lot of Cohen reals by adding a few II, with M. Gitik, Fund. Math. 231 (2015), 209-224.

10- Collapsing the cardinals of HOD, with  J. Cummings and Sy D. Friedman,  J. Math. logic.15 (2015), no. 2, 1550007, 32 pp.

11- On Foreman's maximality principle, with Y. Hayut, J. Symbolic Logic. 81 (2016), no 4, 1344-1356.

12-  On cuts in ultraproducts of linear orders I, with S. Shelah, J. Math. Log. 16 (2016), no. 2, 1650008, 34 pp.

13- HOD, V and the GCH,  J. Symbolic Logic. 82 (2017), no. 1, 224-246.

14- An Easton like theorem in the presence of Shelah cardinals, Arch. Math. Logic 56 (2017), no. 3-4, 273-287.

15- A Groszek-Laver pair of undistinguishable E₀ classes, with V. Kanovei and V. Lyubetsky, Math. Log. Q. 63 (2017), no. 1-2, 19-31.

16- The tree property on a countable segment of successors of singular cardinals, with Y. Hayut, Fund. Math. 240 (2018), no 2, 199-204.

17- The tree property at  double successors of singular cardinals of uncountable cofinality, with R. Mohammadpour, Ann. Pure Appl. Logic 169 (2018), no. 2, 164-175.

18- The tree property at the successor of a singular limit of measurable cardinals,  Arch. Math. Logic 57 (2018), no. 1-2, 3-25.

19- On a question of Silver about gap-two cardinal transfer principles, with Sh. Mohsenipour, Arch. Math. Logic 57 (2018), no. 1-2, 27-35.

20- Adding a lot of random reals by adding a few, with M. Gitik, Fund. Math. 241 (2018), no 1, 97-108.

21- On cuts in ultraproducts of linear orders II, with S. Shelah, J. Symbolic Logic. 83 (2018), no 1, 29-39.

22- The generalized Kurepa hypothesis at singular cardinals, Period. Math. Hungar. 78 (2019), no. 2, 200-202.

23- The Special Aronszajn tree property, with Y. Hayut, J. Math. Log. 20 (2020), no. 1, 2050003, 26 pp.

24- The tree property at double successors of singular cardinals of uncountable cofinality with infinite gaps, with A. Poveda, Ann. Pure Appl. Logic 172 (2021), no. 1, 102853.

25- Specializing trees and answer to a question of Williams, with S. Shelah, J. Math. Log. 21 (2021), no. 1, 2050023, 20 pp.

26- (Weak) diamond can fail at the least inaccessible cardinal, Fund. Math. 256 (2022) , no 2, 113-129.

27- The Keisler-Shelah isomorphism theorem and the continuum hypothesis, with S. Shelah, Fund. Math. 260 (2023), no. 1, 59-66.

28- The Keisler-Shelah isomorphism theorem and the continuum hypothesis II, with S. Shelah. Monatsh. Math. 201 (2023), no. 3, 789-801.

29- Cardinal collapsing and product forcing, with R. Mohammadpour, RIMS Kokyuroku, no. 2261, 12-15.

30- On slow minimal reals I, with S. Shelah. Proc. Am. Math. Soc. 151, No. 10, 4527-4536 (2023).

31- Co-Hopfian and boundedly endo-rigid mixed abelian groups, with M. Asgharzadeh and S. Shelah, Pacific J. Math. 327 (2023), no. 2, 183-232.

32- Unlimited accumulation by the Shelah's PCF operator, Period. Math. Hungar. 88 (2024), no. 1, 273-280.

33- Completeness of the Godel-Lob provability logic for the filter sequence of normal measures, with R. Zoghifard, J. Symbolic Logic. 89 (2024), no 1, 163-174.

 34- Usuba's principle UB_\lambda can fail at singular cardinals, with S. Shelah,  J. Symbolic Logic. 89 (2024), no 1, 195-203.

35- Adding highly generic subsets of \omega_2, with E. Eslami and R. Hoseini Naveh., MLQ Math. Log. Q. 70 (2024), no. 1, 126-133.

36- On C^s_n(κ) and the Juhasz-Kunen question, with S. Shelah. Notre Dame J. Form. Log. 65 (2024), no. 4, 481-500.

37- Expressive power of infinitary logic and absolute co-Hopfianity, with M. Asgharzadeh and S. Shelah. Illinois J. Math. 69 (2025), no. 2, 269-302.

38- Representing the language of a topos as quotient of the category of spans, with A. Shiralinasab Langari. Theory Appl. Categ. 44 (2025), Paper No. 27, 805-825.

39- On group well represented as automorphic groups of groups, with M. Asgharzadeh, D. Herden and S. Shelah (Dedicated to Laszlo Fuchs for his 100th birthday). J. Commut. Algebra 17 (2025), no. 3, 251-270.

40- Naturality and definability III, with M. Asgharzadeh and S. Shelah. Fund. Math. 271 (2025), no. 2, 131-156.

41- Forcing with invarinat measures, with N. Ackerman, C. Freer, M. Mirabi and R. Patel.,    Log. Univers. 19 (2025), no. 4, 799-840.

42- Graphs represented by Ext, with M. Asgharzadeh and S. Shelah.  Forum Math. 38 (2026), no. 1, 215-241.

43- Quite free p-groups with trivial duality, with M. Asgharzadeh and S. Shelah. Monatsh. Math., accepted.

44- The proper forcing axiom for \aleph_1-sized posets, \omega_1-linked symmetrically proper forcing, and the size of the continuum, with D. Aspero.  J. Math. Log., accepted.

45- Failure of singular compactness for Hom, with M. Asgharzadeh and S. Shelah.  Bull. Lond. Math. Soc., accepted.

46- Definable tree property for uncountable regular cardinals, with M. Mirabi.

47- The special Aronszajn tree property at \aleph_2 and GCH, with D. Aspero.

48- Kaplansky test problems for R-modules in ZFC, with M. Asgharzadeh and S. Shelah.

49- On the classification of definable ccc forcing notions, with H. Horowitz and S. Shelah.

50- Shelah's partition functions and the Hales-Jewett numbers, with M. Mirabi.

51- A generic absoluteness principle consistent with large continuum.

52- On a question of Hamkins and Lowe on the modal logic of collapse forcing, with W. Mitchell.

53- NNR revisited, with S. Shelah.

54- Cohen generic structures with functions, with N. Ackerman and M. Mirabi.

55- Adding highly undefinable sets over L, with S. Shelah.

56- The tree property at all regular even cardinals,

57- Adding abraham clubs and \alpha-properness, with R. Hoseini Naveh.

58- Combinatorial and number-theoretic properties of generic reals, with W. Brian.

59- The measuring principle and the continuum hypothesis, with S. Shelah.

60- No universal graphs at uncountable regular cardinals.

Notes

1- Woodin's surgery method.

2- An introduction to forcing.

3- Singular cofinality conjecture and a question of Gorelic.

4- On a question of Zadrozny.

5-  Power set at \aleph_\omega: On a theorem of Woodin.

6- All uncountable regular cardinals can be inaccessible in HOD.

7- Notes on countably generated complete Boolean algebras.

8- Adding many random reals may add many Cohen reals.

9- On a theorem of Magidor.

10- On the notions of cut, dimension and transcendence degree for models of ZFC.

11- Fraisse limit via forcing.

12- Changing measurable into small accessible cardinals.

13- Strongly compct diagonal Prikry forcing.

14- Two remarks on Merimovich's model of the total failure of GCH

15- The Abraham-Shelah \Delta^2_2-well-ordering of the reals

16- On the number of cofinalities of cuts in ultraproducts of linear orders

Talk slides:

1- Singular Cardinals Problem (IPM Colloquium, 2015).

2- On the birth of set theoretic algebra (Algebra day, 2018).

3- Diophantine approximation of Cohen reals (7th annual conference of Iranian Association for Logic, 2020).

4- Completeness of the provability logic GL with respect to the filter sequence of normal measures (8th annual conference of Iranian Association for Logic, 2021).

5- Shelah's Black Boxes (IPM Colloquium, 2024).

6- Questions that haunt my nights, while Shelah might answer by dawn (Conference in honor of Saharon Shelah's 80th birthday, 2025)

Students (PhD):

1- Rouholah Hoseini Naveh: ``Forcing with elementary substructures'' (in Farsi), 2024.

 Students (MSc):

1- Rahman Mohammadpour ``The Modal Logic of Forcing and Hamkins' Maximality Principle'' (in Farsi), 2015.

2- Zahra A. Biglou: ``Around Vaught's conjecture'' (in Farsi), 2018.

3- Zakieh Zakeri: ``Topics in Cardinal Arithmetic'' (in Farsi), 2020.