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., accepted.

37- Expressive power of infinitary logic and absolute co-Hopfianity, with M. Asgharzadeh and S. Shelah. Illinois J. Math., accepted.

38- On group well represented as automorphic groups of groups, with M. Asgharzadeh, D. Herden and S. Shelah. J. Commut. Algebra (Dedicated to Laszlo Fuchs for his 100th birthday), accepted.

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

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

41- Graphs represented by Ext, with M. Asgharzadeh and S. Shelah.

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

43- Representing the language of a topos as quotient of the category of spans, with A.R Shir Ali Nasab.

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

45- No universal graphs at uncountable regular cardinals.

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

47- PFA for \aleph_1-sized posets, Prikty-type proper forcing, and the size of the continuum, 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- Naturality and definability III, with M. Asgharzadeh and S. Shelah.  

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.

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

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).

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.