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Shelah's creature forcing is a very powerful method, with wide range of applications. The method also has some applications in ZFC, let's quote a few of them that I am aware of:

(1) In A partition theorem Shelah proves a very general infinitary Ramsey theorem in ZFC, which is parallel to the Galvin-Prikry theorem and the Carlson-Simpson theorem.

(2) In Partition theorems from creatures and idempotent ultrafilters, creature forcing is used to prove some Ramsey type theorems. As an application of their general method, new proof of Carlson-Simpson theorem is given. See also Creature forcing and topological Ramsey spaces

(3) In Ramsey theorems for product of finite sets with submeasures creature forcing is used to prove a parametrized partition theorem on products of finite sets equipped with submeasures. It improves the results of DiPrisco, Llopis, and Todorcevic.

Question 1. What other ZFC examples are available whose proofs is given by creature forcing?

Question 2. Are there any applications of creature forcing in proving ZFC results, beyond those in Ramsey type theorems?

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    $\begingroup$ Creature forcing is the stuff of nightmares. $\endgroup$
    – Asaf Karagila ♦
    Jul 20, 2019 at 12:53
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    $\begingroup$ Science fiction - double feature - Dr. X will build a creature... $\endgroup$
    – Goldstern
    Jul 20, 2019 at 23:58

2 Answers 2

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In Norms on possibilities II: more ccc ideals on $2^{\omega}$ (published version https://doi.org/10.1515/JAA.1997.103), Rosłanowski and Shelah use creature forcings to construct many ccc $\sigma$-ideals with properties similar to those of the ideal of meager / null sets, answering a question of Kunen.

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Also check: MR3563073 Dobrinen, Natasha Creature forcing and topological Ramsey spaces. Topology Appl. 213 (2016), 110–126; DOI: 10.1016/j.topol.2016.08.008, arXiv: 1509.06402.

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    $\begingroup$ and MR2146637 Goldstern, Martin; Shelah, Saharon Clones from creatures. Trans. Amer. Math. Soc. 357 (2005), no. 9, 3525–3551. (not exactly ZFC but a consequence of CH) $\endgroup$
    – Andrzej
    Jul 21, 2019 at 16:02
  • $\begingroup$ and MR1838340 Bartoszynski, Tomek; Shelah, Saharon Strongly meager sets do not form an ideal. J. Math. Log. 1 (2001), no. 1, 1–34. $\endgroup$
    – Andrzej
    Jul 21, 2019 at 16:06
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    $\begingroup$ On the importance of typesetting. The first comment read to me as if the paper's title was "Saharon Clones from Creatures". $\endgroup$
    – Asaf Karagila ♦
    Jul 23, 2019 at 23:24

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