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There are four papers by Vladimir Alfeevich Kuznetsov, which discuss the above titled topic:

(1) Some problems in set theory from the standpoint of a formal system G+ of Gentzen type. (Russian) Akad. Nauk Ukrainy Inst. Mat. Preprint 1992, no. 21, 51 pp.

(2) The forcing method from the point of view of a formal system G+ of Gentzen type. (Russian) Akad. Nauk Ukrain. SSR Inst. Mat. Preprint 1991, no. 19, 59 pp.

(3) Classical descriptive set theory from the point of view of the Gentzen-type formal system G+. (Russian) Akad. Nauk Ukrain. SSR Inst. Mat. Preprint 1990, no. 40, 63 pp.

(4) Set theory from the point of view of the Gentzen-type formal system G+. (Russian) Akad. Nauk Ukrain. SSR Inst. Mat. Preprint 1990, no. 13, 63 pp.

The papers are in Russian, and I even could not find any access to them. Also their review in Mathscinet do not say much about the results of the papers.

I wonder to know if someone can explain in some details the results obtained in the above papers, in particular paper no. 2.

Also I am interested in finding some reference in English related to the above papers.

Thanks in advance.

Remark. As an example, the review of the second paper in Mathscinet is as follows:

"This paper is devoted to a systematic presentation of the forcing method in a nontraditional focus: we present proofs of theorems in the form of trees of a formal system $G^+$ of Gentzen type in the Zermelo-Fraenkel and Morse axiom systems. The proofs presented contain practically all the nuances of set-theoretic constructions.''

See here for the review of the papers.

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  • $\begingroup$ (+1) It is very interesting. But my search on key words in this thread often leads me to unrelated topics! Maybe because proof theorists and modal logic use the term "forcing" in another context. $\endgroup$
    – Morteza Azad
    Commented Jan 25, 2016 at 5:21
  • $\begingroup$ @Mohammad Golshani: What is your motivation for asking this question? As can be seen by your "Remark" the paper presents "proofs of theorems" (about forcing, I presume) "in the form of trees of a formal system $G^+$ of Gentzen type" (natural deduction?) "in the Zermelo-Frankel and Morse axiom systems." If $G^+$ is a system of natural deduction then you might want to take a look at Avigad's paper "Forcing in Proof Theory" and make the necessary translations between Hilbert-type deduction systems and systems of natural deduction to get the desired results. $\endgroup$
    – Thomas Benjamin
    Commented Feb 12, 2016 at 17:41

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