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Math. Logic"],"published-print":{"date-parts":[[2023,7]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    We show that Hechler\u2019s forcings for adding a tower and for adding a mad family can be represented as finite support iterations of Mathias forcings with respect to filters and that these filters are\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$${\\mathcal {B}}$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mi>B<\/mml:mi>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    -Canjar for any countably directed unbounded family\u00a0\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$${\\mathcal {B}}$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mi>B<\/mml:mi>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    of the ground model. In particular, they preserve the unboundedness of any unbounded scale of the ground model. Moreover, we show that\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$${\\mathfrak {b}}=\\omega _1$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>b<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>\u03c9<\/mml:mi>\n                              <mml:mn>1<\/mml:mn>\n                            <\/mml:msub>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    in every extension by the above forcing notions.\n                  <\/jats:p>","DOI":"10.1007\/s00153-023-00861-x","type":"journal-article","created":{"date-parts":[[2023,2,12]],"date-time":"2023-02-12T00:04:20Z","timestamp":1676160260000},"page":"811-830","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Towers, mad families, and unboundedness"],"prefix":"10.1007","volume":"62","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4710-8241","authenticated-orcid":false,"given":"Vera","family":"Fischer","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Marlene","family":"Koelbing","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Wolfgang","family":"Wohofsky","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2023,2,12]]},"reference":[{"key":"861_CR1","unstructured":"Blass, A.: Finite support iterations of $$\\sigma $$-centered forcing notions. MathOverflow (2011). http:\/\/mathoverflow.net\/questions\/84129"},{"key":"861_CR2","unstructured":"Fischer, V.V.: The consistency of arbitrarily large spread between the bounding and the splitting numbers. ProQuest LLC, Ann Arbor. Thesis (Ph.D.)\u2014York University (Canada) (2008)"},{"key":"861_CR3","unstructured":"Fischer, V., Koelbing, M., Wohofsky, W.: Refining systems of mad families. Accepted for publication at the Isr. J. Math"},{"key":"861_CR4","unstructured":"Guzm\u00e1n, O., Hru\u0161\u00e1k, M., Mart\u00ednez-Celis, A.: Canjar filters II. In: Proceedings of the 2014 RIMS Meeting on Reflection Principles and Set Theory of Large Cardinals. Kyoto, Japan, vol. 1895, pp. 59\u201367 (2014)"},{"key":"861_CR5","doi-asserted-by":"publisher","DOI":"10.1016\/j.aim.2021.107805","volume":"386","author":"O Guzm\u00e1n","year":"2021","unstructured":"Guzm\u00e1n, O., Kalajdzievski, D.: The ultrafilter and almost disjointness numbers. Adv. Math. 386, 107805 (2021)","journal-title":"Adv. Math."},{"key":"861_CR6","unstructured":"Guichardaz, F.: Forcing over ord-transitive models. Thesis (Ph.D.)\u2013Albert-Ludwigs-Universit\u00e4t Freiburg, Germany (2019)"},{"key":"861_CR7","doi-asserted-by":"crossref","unstructured":"Hechler, S.H.: Short complete nested sequences in $$\\beta \\!N\\backslash \\!N$$ and small maximal almost-disjoint families. Gen. Topol. Appl. 2, 139\u2013149 (1972)","DOI":"10.1016\/0016-660X(72)90001-3"},{"issue":"3","key":"861_CR8","doi-asserted-by":"publisher","first-page":"880","DOI":"10.1016\/j.apal.2013.11.003","volume":"165","author":"M Hru\u0161\u00e1k","year":"2014","unstructured":"Hru\u0161\u00e1k, M., Minami, H.: Mathias\u2013Prikry and Laver\u2013Prikry type forcing. Ann. Pure Appl. Log. 165(3), 880\u2013894 (2014)","journal-title":"Ann. Pure Appl. Log."},{"issue":"3","key":"861_CR9","doi-asserted-by":"publisher","first-page":"909","DOI":"10.2307\/2274464","volume":"55","author":"H Judah","year":"1990","unstructured":"Judah, H., Shelah, S.: The Kunen\u2013Miller chart (Lebesgue measure, the Baire property, Laver reals and preservation theorems for forcing). J. Symb. Log. 55(3), 909\u2013927 (1990)","journal-title":"J. Symb. Log."},{"issue":"1","key":"861_CR10","first-page":"299","volume":"121","author":"FD Tall","year":"1994","unstructured":"Tall, F.D.: $$\\sigma $$-centred forcing and reflection of (sub)metrizability. Proc. Am. Math. Soc. 121(1), 299\u2013306 (1994)","journal-title":"Proc. Am. Math. 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