Universal domination and idealized forcing

Abstract: We introduce the universality property of a definable $σ$-ideal on a Polish space, which, on one hand, can serve as a benchmark for the properness of the associated idealized forcing of positive Borel sets ordered by inclusion, and, on the other hand, unifies many of the results that can be found in Zapletal's book. We show that under mild absoluteness assumptions, it implies properness, various dichotomy theorems, and closure under well-ordered unions in the Solovay model and under $\mathsf{AD}<sup>+$, among other things. All major classes of proper idealized forcings studied in the book have this property. Further, we use this viewpoint to answer a question of Khomskii by showing that the naive idealized forcing for adding an eventually different real or a refining real is not proper below some condition. We also answer a question related to the definability of $σ$-ideals generated by Borel sets due to Kanovei, Sabok, and Zapletal.

• The paper introduces universal pseudo-genericity as a unifying framework that connects domination properties of Borel relations with idealized forcing, properness, and virtual genericity.
• It proves that, under suitable provable Δ¹₂-on-Σ¹₁ definability, weak universality is equivalent to properness and derives preservation, game, dichotomy, and uniformization theorems.
• It identifies important boundaries by showing that true eventually different and refining forcing are not proper, while also producing Fσ counterexamples to broad game-ideal and definability conjectures.

The universality property

The paper develops a unifying framework for the study of idealized forcing, the theory initiated by Zapletal in which one forces with $\mathcal{I}$-positive Borel subsets of a Polish space, ordered by inclusion, for a $\sigma$-ideal $\mathcal{I}$. The central new notion is universal pseudo-genericity. A real $x$ is $\mathcal{I}$-pseudo-generic over a model $M$ if it avoids every $\mathcal{I}$-small Borel set coded in $M$; it is universally pseudo-generic over $M$ if for every forcing notion $\mathbb{Q} \in M$ there exists a $\sigma$0-generic filter $\sigma$1 over $\sigma$2 such that $\sigma$3 remains pseudo-generic over $\sigma$4. The ideal $\sigma$5 has the universality property if every generic pseudo-generic real over a countable model is already universally pseudo-generic; a weaker version quantifies only over $\sigma$6-generic reals.

For a Borel relation $\sigma$7, the associated ideal $\sigma$8 consists of sets that are not $\sigma$9-$\mathcal{I}$0-dominating, and pseudo-genericity coincides with $\mathcal{I}$1-domination over the model. The universality property for $\mathcal{I}$2 is thus a purely combinatorial statement about domination, and the author's stated aim is that it can often be verified without analyzing the internal structure of conditions in $\mathcal{I}$3. The framework is motivated by forthcoming work on canonical models for cardinal invariants of the continuum, but here it serves both as a benchmark for properness and as a device that unifies large portions of Zapletal's monograph.

Properness and virtual genericity

The main structural theorem states that, assuming $\mathcal{I}$4 is provably $\mathcal{I}$5 on $\mathcal{I}$6, the weak universality property is equivalent to $\mathcal{I}$7-properness of $\mathcal{I}$8 (for the class $\mathcal{I}$9 of countable elementary submodels). In fact, it suffices that universality holds for the collapse forcing $x$0. Consequently, an ideal that is provably $x$1 on $x$2 and has the universality property yields a forcing that is proper in every forcing extension. The proof technique is a genericity transfer argument: a putative condition forcing that no suitable $x$3-generic can be added is contradicted by building a $x$4-generic over a forcing extension of $x$5 and then a $x$6-generic over that extension, using upwards correctness of the ideal.

A second structural result is the virtual genericity property: under the universality property, every generically added pseudo-generic real $x$7 over $x$8 is $x$9-generic over some forcing extension of $\mathcal{I}$0. Conversely, assuming $\mathcal{I}$1-correctness, universality is equivalent to the conjunction of $\mathcal{I}$2-properness and virtual genericity. This gives a canonicity statement for the naive forcings $\mathcal{I}$3: every generic $\mathcal{I}$4-dominating real is virtually generic for $\mathcal{I}$5, so these forcings are canonical for adding $\mathcal{I}$6-dominating reals. Concrete instances include: every new generic real over $\mathcal{I}$7 is a Sacks real over some forcing extension of $\mathcal{I}$8, and analogues for unbounded, splitting, and dominating reals.

The paper also provides game characterizations. A two-player game $\mathcal{I}$9 is introduced in which Player I builds conditions in $M$0 and names for $M$1-small Borel sets while Player II builds a $M$2-generic filter; Player II wins if the resulting generic real avoids the union of the small sets. These games are determined, and Player II having winning strategies in all of them characterizes the universality property (and the restricted games characterize weak universality). Variants are given, including a version where the game is played via projection maps between forcing notions.

Preservation theorems

Preservation properties of $M$3 are recast as universality-like statements. For an ideal $M$4 with a provable Borel base, the paper proves that $M$5 preserves that a fixed $M$6-positive set $M$7 stays positive if and only if, for every countable model $M$8 and condition $M$9, one can find a $\mathcal{I}$0-generic $\mathcal{I}$1 over $\mathcal{I}$2 and a $\mathcal{I}$3-pseudo-generic $\mathcal{I}$4 over $\mathcal{I}$5 such that $\mathcal{I}$6 remains $\mathcal{I}$7-pseudo-generic over a suitable extension $\mathcal{I}$8. A global version characterizes preservation of the ideal $\mathcal{I}$9 itself. As corollaries: if every $M$0-generic real stays pseudo-generic over extensions by Cohen reals, then $M$1 preserves Baire category; replacing Cohen by random yields preservation of Lebesgue measure. This recovers, in the general framework, the well-known fact that $M$2 relations (ideals $M$3-generated by closed sets) preserve category.

Dichotomy theorems, determinacy, and the Solovay model

Assuming $M$4 is provably $M$5 on $M$6 (e.g. defined by a Borel base) and has the universality property, the paper derives the standard regularity conclusions of Zapletal's theory under mild absoluteness assumptions. Every analytic set is either contained in a small Borel set or contains an $M$7-positive Borel set (the third dichotomy); under the assumption that every real has a sharp, the conclusion extends to coanalytic sets; under universally Baire absoluteness, to all universally Baire sets (the first dichotomy).

Stronger results hold when the correctness assumptions can be eliminated by working over inner models containing all ordinals. If $M$8 does not inject into the reals, every Suslin set satisfies the first dichotomy, can be uniformized by a Borel function on an $M$9-positive set, and $M$0 is closed under well-ordered unions of uniformly Suslin sets. The same three conclusions hold under $M$1 and in the Solovay model (built over $M$2 from the Lévy collapse of a strong limit $M$3). The $M$4 argument uses the fact that every true $M$5 statement has a Suslin witness, together with a pre-well-ordering argument for closure under well-ordered unions; the Solovay model argument exploits that every set of reals there is $M$6-Borel.

Positive examples

All major classes of proper idealized forcings in Zapletal's book fit the framework:

• Coanalytic porosity ideals have the total universality property, and the associated forcings preserve Baire category. The proof is a direct domination argument using the monotone porosity map.
• Farah–Zapletal ideals, defined via a provable Borel base and a winning strategy for Player II in the Farah–Zapletal game, have the universality property and are provably $M$7 on $M$8. The paper establishes an exact equivalence: an ideal is Farah–Zapletal if and only if it is a game ideal with a simple scrambling tool (in the sense of Kanovei–Sabok–Zapletal) and has the universality property for countable elementary submodels. The author notes it is unclear whether game ideals with simple scrambling tools differ from ordinary game ideals.
• Analytic $M$9-cover ideals $\mathbb{Q} \in M$0 for analytic $\mathbb{Q} \in M$1-ideals $\mathbb{Q} \in M$2 on $\mathbb{Q} \in M$3 have the total universality property; the proof uses Solecki's representation $\mathbb{Q} \in M$4 by a lower semicontinuous submeasure and a submeasure bookkeeping argument. These ideals are also game ideals with simple scrambling tools, hence provably $\mathbb{Q} \in M$5 on $\mathbb{Q} \in M$6.
• Concrete Cichoń-related forcings: the paper gives a short direct proof that the eventual dominance relation $\mathbb{Q} \in M$7 has the total universality property, so dominating forcing $\mathbb{Q} \in M$8 (which increases $\mathbb{Q} \in M$9) is proper, and every generic dominating real is virtually generic for it; combined with Brendle–Hjorth–Spinas, in the Solovay model or under $\sigma$00 every set of reals is either non-dominating or contains a Borel, indeed closed, dominating family. Similarly, a slalom-based eventually different relation $\sigma$01 (increasing $\sigma$02) and a slalom-based localization relation $\sigma$03 (increasing $\sigma$04) have the total universality property, so the associated forcings are proper. The paper also observes a $\sigma$05 relation with weak but not full universality, whose forcing is equivalent to Sacks forcing, and asks whether the two notions can be separated by an $\sigma$06 relation.

Negative results: two improper forcings

The framework also functions as a test: failure of universality can be converted into non-properness. The paper answers a question of Khomskii negatively for two natural forcings.

True eventually different forcing $\sigma$07 is not proper. The construction fixes a family of infinite sets $\sigma$08 with pairwise disjoint sets along chains, infinite intersections over antichains, and full coverage along branches, together with constant functions $\sigma$09 on $\sigma$10. A continuous function $\sigma$11 is built from these data, and a combinatorial argument using Dilworth's theorem produces, for any countable model $\sigma$12, an eventually different real $\sigma$13 over $\sigma$14 such that $\sigma$15 agrees with $\sigma$16 infinitely often for every $\sigma$17. Consequently $\sigma$18 fails to be eventually different over every nontrivial extension $\sigma$19. The stronger conclusion is that $\sigma$20 collapses the continuum to $\sigma$21 below some condition: the set of reals satisfying a certain Borel (in fact $\sigma$22) condition is an $\sigma$23-eventually different family, but any generic member of it codes a countable set containing all ground model reals.

Refining (reaping/unsplitting) forcing is treated similarly. Using partial functions $\sigma$24 whose finite-antichain subfamilies are independent, and a combinatorial antichain lemma, the paper builds a reaping real $\sigma$25 over any countable $\sigma$26 that is split by $\sigma$27 for every $\sigma$28. The associated Borel $\sigma$29-refining family again yields collapse of the continuum to $\sigma$30 below a condition. Under the regularity assumption that every $\sigma$31 set is Baire measurable or $\sigma$32-regular, refining forcing is shown to collapse the continuum to $\sigma$33: a dominating family of strictly increasing functions provides finite-to-one maps $\sigma$34 such that every ground model real is computable from $\sigma$35 for some $\sigma$36. A structural theorem is proved as well: for any Borel refining family $\sigma$37 there is a finite-to-one $\sigma$38 such that some Ellentuck cube is contained in $\sigma$39. Since refining forcing is not proper, it is in particular distinct from Mathias forcing, consistent with the known fact that the Mathias ideal is not $\sigma$40 on $\sigma$41.

Answers to definability questions

The paper gives a negative answer to Question 5.14 of Kanovei, Sabok, and Zapletal, which asked whether every $\sigma$42-ideal generated by Borel sets is a game ideal, and whether every $\sigma$43 on $\sigma$44 $\sigma$45-ideal is a game ideal. Both are refuted by $\sigma$46 relations, hence by ideals $\sigma$47-generated by $\sigma$48 sets:

• There is an $\sigma$49 relation with the total universality property and proper $\sigma$50, whose ideal is not $\sigma$51 on $\sigma$52. The relation is a lottery sum combining Sacks-type domination with eventual dominance, built over a $\sigma$53-complete set $\sigma$54: membership of $\sigma$55 in $\sigma$56 is encoded by positivity of the section $\sigma$57.
• There is an $\sigma$58 relation whose ideal is $\sigma$59 on $\sigma$60 (so the forcing is proper) but is not a game ideal. The construction uses a $\sigma$61 set $\sigma$62 not reducible to the set $\sigma$63 of Borel codes of determined games with a Player II win; a game ideal structure would produce such a reduction, a contradiction. The author credits Gabe Goldberg for the observation that no complete $\sigma$64 set exists.

This shows that universality does not imply the definability hypotheses used elsewhere in the paper, and that the empirical observation that natural proper idealized forcings tend to have provably $\sigma$65 on $\sigma$66 ideals has no general justification. The author also notes that no natural example of a $\sigma$67-ideal generated by Borel sets failing $\sigma$68 on $\sigma$69 was previously known.

Limitations and open questions

Several hypotheses and gaps are acknowledged explicitly. The main theorems require the ideal to be provably $\sigma$70 on $\sigma$71 or, in the dichotomy results, some absoluteness assumption ($\sigma$72-correctness via sharps, universally Baire absoluteness, or working over inner models); whether these can be weakened is tied to the open question of whether the restriction of any Borel-base ideal to some positive set is $\sigma$73 on $\sigma$74. Whether the universality property implies total universality for Borel relations is open, as is the existence of an $\sigma$75 relation separating weak from full universality. The relationship between the slalom forcing $\sigma$76 and the Bartoszyński–Judah forcing $\sigma$77 is unknown, as is the existence of tree dichotomies for the analytic $\sigma$78- and $\sigma$79-dominating families. Whether true eventually different or refining forcing is proper below some condition remains open, as does a characterization of Borel $\sigma$80-refining families and the question of whether the refining ideal is $\sigma$81 on $\sigma$82. The paper also does not determine the homogeneity properties of dominating forcing $\sigma$83, and notes that the claim in Farah–Zapletal that it is equivalent to Laver forcing is only established below the strictly increasing functions. Finally, whether "true localization" forcing $\sigma$84 is improper, by analogy with $\sigma$85, is left unexamined.

Conclusion

The paper introduces universal pseudo-genericity as a property of definable $\sigma$86-ideals that simultaneously characterizes properness of the associated idealized forcing (under provable $\sigma$87 on $\sigma$88 definability), implies virtual genericity, preservation theorems, and dichotomy and uniformization results in the Solovay model and under $\sigma$89, and encompasses all major classes of proper idealized forcings previously studied. Its methodological contribution is that universality can be verified by elementary combinatorial arguments about the defining relation, bypassing structural analysis of forcing conditions. The negative applications — the collapse of true eventually different and refining forcing, answering a question of Khomskii, and the counterexamples to the game-ideal question of Kanovei–Sabok–Zapletal — demonstrate that the property is also an effective obstruction detector. The framework leaves open the precise relationship between universality, total universality, and provable $\sigma$90 on $\sigma$91 definability, which remain the principal unresolved questions of the theory.