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Many different uniformity numbers of Yorioka ideals

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Abstract

Using a countable support product of creature forcing posets, we show that consistently, for uncountably many different functions the associated Yorioka ideals’ uniformity numbers can be pairwise different. In addition we show that, in the same forcing extension, for two other types of simple cardinal characteristics parametrised by reals (localisation and anti-localisation cardinals), for uncountably many parameters the corresponding cardinals are pairwise different.

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Notes

  1. The proof, due to Yorioka [19, Lemma 3.4], is far from trivial. An alternative proof follows from [8, Theorem 3.12], where it is proved that \({\text {add}}(\mathcal {I}_f)\) is above some uncountable cardinal characteristic (without using the fact that \(\mathcal {I}_f\) is an ideal itself).

  2. The notation \(x\in ^\infty y\) naturally denotes \(\exists ^\infty \, i:x(i)\in y(i)\). We originally used \(\notin ^*\) instead of \(\notin ^\infty \), but this turned out to be unnecessarily confusing alongside our use of \(\in ^*\), since \(\notin ^\infty \) is not the negation of \(\in ^*\).

  3. A brief note on notation: \(\mathfrak {c}^\forall _{c,h}\) and \(\mathfrak {c}^\exists _{c,h}\) were used in [10, 12, 14,15,16] with the meaning of covering \(\prod c\) with slaloms, where the relations for covering are \(\in ^*\) and \(\in ^\infty \), respectively. The notation \(\mathfrak {v}^\forall _{c,h}\) and \(\mathfrak {v}^\exists _{c,h}\) is intended to be read as avoiding or evading such coverings.

  4. By similar methods, it can also be proved that \({\text {cof}}(\mathcal {M}) = \sup (\{ \mathfrak {d}\} \cup \{\mathfrak {c}^\exists _{c,h} \mid c \in \omega ^\omega \})\) and, whenever h goes to infinity, \({\text {add}}(\mathcal {N}) = \min (\{ \mathfrak {b}\} \cup \{\mathfrak {v}^\forall _{c,h} \mid c \in \omega ^\omega \})\) and \({\text {cof}}(\mathcal {N}) = \sup (\{ \mathfrak {d}\} \cup \{\mathfrak {c}^\forall _{c,h} \mid c \in \omega ^\omega \})\).

  5. Note that the \(c'\) we work with here and in the subsequent proof is such that \(\prod c'\) already is a family of slaloms, which reduces the complexity of the Tukey connection (since we do not have to bijectively map sets to their cardinalities).

  6. The usual creature forcing notation (as in e. g. [9]) defines the set of possibilities more abstractly as \({\text {poss}}(p, {\le }k) := \prod _{\ell \le k} p(\ell )\) and defines \(p \wedge \eta \) as a condition with an extended trunk (a concept which we did not deem necessary to introduce in our paper). Since working with possibilities \(\eta \) as sequences of singletons suffices for our proofs and is conceptually easier, we instead opted for this simpler definition.

  7. Note that bigness is equivalent to the concept of completeness in the sense of [10].

  8. For the first inequality, recall that \(\frac{x}{2}\le \lfloor x\rfloor \) iff \(x\ge 1\).

  9. We can simplify property (S5) by restricting it to \(\ell =1\): Assume the statement holds for some \(\ell ' > 1\). Let \(f'_\alpha = f_\alpha \circ \mathrm {pow}_{\ell '}\). Then \(f_{b_\alpha ,g_\alpha } \le ^* f'_\alpha \) and \(f'_\alpha \ll g_{c_\alpha ,h_\alpha }\) still holds (by the definition of \(\ll \)). This is why we can work with \(f'_\alpha \) instead of \(f_\alpha \), in which case property (S5) is already satisfied for \(\ell =1\) and property (S6) remains true for \(f'_\alpha \).

  10. Concretely, there are \(p_0 \in \mathbb {Q}\) and \(\nu < \kappa _\alpha \) such that \(p_0 \Vdash |\dot{F}| = \nu \). So we just replace \(\dot{F}\) by some \(\mathbb {Q}\)-name \(\dot{F}'\) for a subset of \(\prod c_\alpha \) of size \(\le \nu \) such that \(p_0\Vdash \dot{F}'=\dot{F}\).

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Authors and Affiliations

  1. Institute of Discrete Mathematics and Geometry, TU Wien, Wiedner Hauptstraße 8–10/104, 1040, Wien, Austria

    Lukas Daniel Klausner

  2. Faculty of Science, Shizuoka University, 836 Ohya, Suruga-ku, Shizuoka City, Shizuoka Prefecture, 422-8529, Japan

    Diego Alejandro Mejía

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Correspondence to Diego Alejandro Mejía.

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The first author was supported by the Austrian Science Fund (FWF) project P29575 “Forcing Methods: Creatures, Products and Iterations”. The second author was supported by the Austrian Science Fund (FWF) project I3081 “Filters, Ultrafilters and Connections with Forcing”, the grant no. IN201711 of Dirección Operativa de Investigación – Institución Universitaria Pascual Bravo, and by the Grant-in-Aid for Early Career Scientists 18K13448, Japan Society for the Promotion of Science. We are grateful to Martin Goldstern and Teruyuki Yorioka for giving helpful comments and suggestions to improve our paper.

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Klausner, L.D., Mejía, D.A. Many different uniformity numbers of Yorioka ideals. Arch. Math. Logic 61, 653–683 (2022). https://doi.org/10.1007/s00153-021-00809-z

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