Iterated forcing with finitely additive measures: applications of probability to forcing theory
The method of finitely additive measures along finite support iterations was introduced by Saharon Shelah in 2000 (see [She00]) to show that, consistently, cov(N ) may have countable cofinality. In 2019, Jakob Kellner, Saharon Shelah and Anda Tanasie ˇ (see [KST19]) improved the method: they achieve...
- Autores:
-
Uribe Zapata, Andrés Felipe
- Tipo de recurso:
- Fecha de publicación:
- 2023
- Institución:
- Universidad Nacional de Colombia
- Repositorio:
- Universidad Nacional de Colombia
- Idioma:
- eng
- OAI Identifier:
- oai:repositorio.unal.edu.co:unal/84529
- Palabra clave:
- 510 - Matemáticas
510 - Matemáticas::519 - Probabilidades y matemáticas aplicadas
510 - Matemáticas::511 - Principios generales de las matemáticas
Forcing (Teoría de los modelos)
Trayectoria aleatoria
Iterated forcing
Probability
Finitely additive measure
Consistency results
Null set
Intersection number
Cardinal invariant
Singular cardinal
Cichon’s diagram
Forcing iterado
Probabilidad
Medida finitamente aditiva
Resultados de consistencia
Conjunto nulo
Numero de intersección
Cardinal invariante
Cardinal singular
diagrama de Cicho´n
- Rights
- openAccess
- License
- Reconocimiento 4.0 Internacional
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| dc.title.eng.fl_str_mv |
Iterated forcing with finitely additive measures: applications of probability to forcing theory |
| dc.title.translated.spa.fl_str_mv |
Forcing iterado con medidas finitamente aditivas: aplicaciones de la probabilidad a la teoría del forcing |
| title |
Iterated forcing with finitely additive measures: applications of probability to forcing theory |
| spellingShingle |
Iterated forcing with finitely additive measures: applications of probability to forcing theory 510 - Matemáticas 510 - Matemáticas::519 - Probabilidades y matemáticas aplicadas 510 - Matemáticas::511 - Principios generales de las matemáticas Forcing (Teoría de los modelos) Trayectoria aleatoria Iterated forcing Probability Finitely additive measure Consistency results Null set Intersection number Cardinal invariant Singular cardinal Cichon’s diagram Forcing iterado Probabilidad Medida finitamente aditiva Resultados de consistencia Conjunto nulo Numero de intersección Cardinal invariante Cardinal singular diagrama de Cicho´n |
| title_short |
Iterated forcing with finitely additive measures: applications of probability to forcing theory |
| title_full |
Iterated forcing with finitely additive measures: applications of probability to forcing theory |
| title_fullStr |
Iterated forcing with finitely additive measures: applications of probability to forcing theory |
| title_full_unstemmed |
Iterated forcing with finitely additive measures: applications of probability to forcing theory |
| title_sort |
Iterated forcing with finitely additive measures: applications of probability to forcing theory |
| dc.creator.fl_str_mv |
Uribe Zapata, Andrés Felipe |
| dc.contributor.advisor.none.fl_str_mv |
Mejía Guzmán, Diego Alejandro Parra Londoño, Carlos Mario |
| dc.contributor.author.none.fl_str_mv |
Uribe Zapata, Andrés Felipe |
| dc.contributor.researchgate.spa.fl_str_mv |
0000-0003-2463-1360 |
| dc.subject.ddc.spa.fl_str_mv |
510 - Matemáticas 510 - Matemáticas::519 - Probabilidades y matemáticas aplicadas 510 - Matemáticas::511 - Principios generales de las matemáticas |
| topic |
510 - Matemáticas 510 - Matemáticas::519 - Probabilidades y matemáticas aplicadas 510 - Matemáticas::511 - Principios generales de las matemáticas Forcing (Teoría de los modelos) Trayectoria aleatoria Iterated forcing Probability Finitely additive measure Consistency results Null set Intersection number Cardinal invariant Singular cardinal Cichon’s diagram Forcing iterado Probabilidad Medida finitamente aditiva Resultados de consistencia Conjunto nulo Numero de intersección Cardinal invariante Cardinal singular diagrama de Cicho´n |
| dc.subject.lemb.none.fl_str_mv |
Forcing (Teoría de los modelos) Trayectoria aleatoria |
| dc.subject.proposal.eng.fl_str_mv |
Iterated forcing Probability Finitely additive measure Consistency results Null set Intersection number Cardinal invariant Singular cardinal Cichon’s diagram |
| dc.subject.proposal.spa.fl_str_mv |
Forcing iterado Probabilidad Medida finitamente aditiva Resultados de consistencia Conjunto nulo Numero de intersección Cardinal invariante Cardinal singular diagrama de Cicho´n |
| description |
The method of finitely additive measures along finite support iterations was introduced by Saharon Shelah in 2000 (see [She00]) to show that, consistently, cov(N ) may have countable cofinality. In 2019, Jakob Kellner, Saharon Shelah and Anda Tanasie ˇ (see [KST19]) improved the method: they achieved some new generalizations and applications, such as separating the left side of Cichon’s ´ diagram with b < cov(N ). In this thesis, based on probability theory tools and the articles cited above, we develop a general theory of iterated forcing using finitely additive measures. For this purpose, we introduce two new notions: on the one hand, we define a new linkedness property, which we call “µ-FAM-linked” and, on the other hand, we generalize the notion of intersection number to forcing notions, which justifies the limit steps of our iteration theory. Finally, we apply our theory to prove in detail the consistency of cf(cov(N )) = ℵ0, and some separations of Cichon’s ´ diagram where cov(N ) is singular. In particular, we obtain a new constellation of Cichon’s diagram ´ separating the left side with cov(N ) singular |
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2023 |
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2023-08-10T19:36:14Z |
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2023-08-10T19:36:14Z |
| dc.date.issued.none.fl_str_mv |
2023-01 |
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Trabajo de grado - Maestría |
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info:eu-repo/semantics/masterThesis |
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Universidad Nacional de Colombia |
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Repositorio Institucional Universidad Nacional de Colombia |
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https://repositorio.unal.edu.co/ |
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Universidad Nacional de Colombia Repositorio Institucional Universidad Nacional de Colombia |
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eng |
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Hannah Arendt. The Human Condition. University of Chicago Press, Chicago IL, 2018. Noga Alon and Joel H. Spencer. The Probabilistic Method. Wiley Publishing, 4th edition, 2016. Tomek Bartoszynski. On covering of real line by null sets. Pacific J. Math., 131(1):1– 12, 1988 Jorg Brendle, Miguel A. Cardona, and Diego A. Mejıa. Filter-linkedness and its effect on preservation of cardinal characteristics. Ann. Pure Appl. Logic, 172(1):102856, 2021 Tomek Bartoszynski, Jaime I. Ihoda, and Saharon Shelah. The cofinality of cardinal invariants related to measure and category. The Journal of Symbolic Logic, 54(3):719– 726, 1989 Jorg Brendle and Haim Judah. Perfect sets of random reals. Israel Journal of Mathe- matics, 83:153–176, 1993 Tomek Bartoszynski and Haim Judah. Set theory: On the Structure of the Real Line. A K Peters, Ltd., Wellesley, MA, 1995. Tomek Bartoszynski and H Judah. Measure and category. In Handbook of Set Theory. Vols. 1, 2, 3, volume 2. 2010. William Blake. The marriage of Heaven and Hell. Camden Hotten, London, 1868. Andreas Blass. Combinatorial cardinal characteristics of the continuum. In Handbook of Set Theory. Vols. 1, 2, 3, pages 395–489. Springer, Dordrecht, 2010. J. L. Bell and M. Machover. A Course in Mathematical Logic. North-Holland Pub- lishing Co., Amsterdam-New York-Oxford, 1977. J. L. Borges. Obras completas. Emecé, 1984. K. P. S. Bhaskara Rao and M. Bhaskara Rao. Theory of Charges: A Study of Finitely Additive Measures, volume 109 of Pure and Applied Mathematics. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983. Jorg Brendle. Larger cardinals in Cichon’s diagram. Israel Journal of Mathematics, 56(3):795–810, 1991. Kai Lai Chung. A Course in Probability Theory. Academic Press, 1974. Miguel A. Cardona and Diego A. Mejía. On cardinal characteristics of Yorioka ideals. Math. Log. Q., 65(2):170–199, 2019. Miguel A. Cardona and Diego A. Mejía. Forcing constellations of Cichon’s diagram by using the Tukey order. Kyoto Daigaku Surikaiseki Kenkyusho Kokyuroku, 22(13):14– 47, 2022. arXiv:2203.00615. Paul J. Cohen. Set Theory and the Continuum Hypothesis. W. A. Benjamin, Inc., New York-Amsterdam, 1966. Ryszard Engelking and Monika Karłowicz. Some theorems of set theory and their topological consequences. Fund. Math., 57:275–285, 1965. P. Erdos. Some remarks on the theory of graphs. Bulletin of the American Mathemat- ical Society, 53(4):292 – 294, 1947. Martin Goldstern, Jakob Kellner, Diego Alejandro Mejía, and Saharon Shelah. Preser- vation of splitting families and cardinal characteristics of the continuum. Israel Jour- nal of Mathematics, 246:73–129, 2021 Martin Goldstern, Jakob Kellner, Diego Alejandro Mejía, and Saharon Shelah. Ci- chon’s maximum without large cardinals. J. Eur. Math. Soc. (JEMS), 24(11):3951– 3967, 2022. Martin Goldstern, Jakob Kellner, and Saharon Shelah. Cichon’s maximum. Ann. of Math., 190(1):113–143, 2019. Martin Goldstern, Diego Alejandro Mejía, and Saharon Shelah. The left side of Ci- chon’s diagram. Proc. Amer. Math. Soc., 144(9):4025–4042, 2016. Steven Givant and Halmos Paul. Introduction to Boolean Algebras, volume 1 of Un- dergraduate Texts in Mathematics. Springer, 2009. Lorenz J. Halbeisen. Combinatorial Set Theory. Springer Monographs in Mathemat- ics. Springer, 2019. David Hilbert. The Foundations of Mathematics. Harvard University Press, 1927. Horowitz Haim. and Saharon Shelah. Saccharinity with ccc. arXiv:1610.02706, 2016 Thomas Jech. Set Theory, the Third Millennium Edition, Revised and Expanded. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2003 Alexander S. Kechris. Classical Descriptive Set Theory. Springer New York, NY, 1995. J.L Kelley. Meaures on boolean algebras. J. Symbolic Logic, 64(2):737–746, 1959. Ashutosh Kumar and Saharon Shelah. On possible restrictions of the null ideal. Jour- nal of Mathematical Logic, 19(02):1950008, 2019. Jakob Kellner, Saharon Shelah, and Anda R. Tanasie. Another ordering of the ten car- dinal characteristics in Cichon’s diagram. Comment. Math. Univ. Carolin., 60(1):61– 95, 2019. Kenneth Kunen. Set Theory, an Introduction to Independence Proofs, volume 102 of Studies in Logic and the Foundations of Mathematics. North-Holland Publishing Co., Amsterdam-New York, 1980. Kenneth Kunen. Set Theory, volume 34 of Studies in Logic (London). College Publi- cations, London, 2011. Kenneth Kunen. The Foundation of Mathematics. College Publications, London, 2012. Luc Lauwers. Purely finitely additive measures are non-constructible objects. Working Papers of Department of Economics, Leuven, 2010. Diego A. Mejíıa. Matrix iterations with vertical support restrictions. In Proceedings of the 14th and 15th Asian Logic Conferences, pages 213–248. World Sci. Publ., Hack- ensack, NJ, 2019. arXiv:1803.05102. Diego A. Mejíıa. Forcing and combinatorics of names. Kyoto Daigaku Surikaiseki Kenkyusho Kokyuroku, 2164:34–49, 2020. http://hdl.handle.net/2433/ 261449. Arnold W. Miller. Some properties of measure and category. American Mathematical Society, 266(1):93–114, 1981. Arnold W. Miller. The Baire category theorem and cardinals of countable cofinality. The Journal of Symbolic Logic, 47(2):275–288, 1982. Yiannis N. Moschovakis. Descriptive Set Theory, volume 100 of Studies in Logic and the Foundations of Mathematics. North-Holland Publishing Co., Amsterdam, 1980. Diego Alejandro Mejía and Ismael E. Rivera-Madrid. Absoluteness theorems for ar- bitrary Polish spaces. Rev. Colombiana Mat., 53(2):109–123, 2019. John C. Oxtoby. Measure and Category. Graduate Texts in Mathematics. Springer, 2013. Janusz Pawlikowski. Adding dominating reals with Baire-bounding posets. American Mathematical Society, 123(2):540–547, 1992. Assaf Rinot. The Engelking-Karłowicz Theorem, and a useful corollary. Personal blog, Sep. 29, 2012. https://blog.assafrinot.com/?p=2054 Maxwell Rosenlicht. Introduction to Analysis, volume 1. Dover Publications, 1968. Sheldon M. Ross. A First Course in Probability. Prentice Hall, Upper Saddle River, N.J., fifth edition, 1998. Saharon Shelah. The future of set theory. In Set theory of the reals, pages 1–12. Bar-Ilan Univ, 1993. Saharon Shelah. Covering of the null ideal may have countable cofinality. Fund. Math., 166(1-2):109–136, 2000. J. R Shoenfield. Martin’s axiom. American Mathematical Monthly, 82(6):610–617, 1975 Saharon Shelah and Haim Judah. Adding dominating reals with the random algebra. American Mathematical Society, 119(1):267–273, 1993. R.M. Solovay and Tennenbaum S. Iterated Cohen extensions and Souslin’s problem. Annals of Mathematics, 94(2):201–245, 1971. John Truss. Connections between different amoeba algebras. Fundamenta Mathemat- icae, 130(2):137–155, 1988. Carlos Uzcátegui Aylwin and Carlos Augusto Di Prisco. Una Introducción a la Teoría Descriptiva de Conjuntos. Ediciones Uniandes-Universidad de los Andes, 2020. |
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Reconocimiento 4.0 Internacionalhttp://creativecommons.org/licenses/by-nc/4.0/info:eu-repo/semantics/openAccesshttp://purl.org/coar/access_right/c_abf2Mejía Guzmán, Diego Alejandro7f850bfbf0ceb6a9c44d167c85f61feaParra Londoño, Carlos Mario83a7ffab01e6c634826118974a200beaUribe Zapata, Andrés Felipe86255b19aa9b3492ae064cfc39076deb0000-0003-2463-13602023-08-10T19:36:14Z2023-08-10T19:36:14Z2023-01https://repositorio.unal.edu.co/handle/unal/84529Universidad Nacional de ColombiaRepositorio Institucional Universidad Nacional de Colombiahttps://repositorio.unal.edu.co/The method of finitely additive measures along finite support iterations was introduced by Saharon Shelah in 2000 (see [She00]) to show that, consistently, cov(N ) may have countable cofinality. In 2019, Jakob Kellner, Saharon Shelah and Anda Tanasie ˇ (see [KST19]) improved the method: they achieved some new generalizations and applications, such as separating the left side of Cichon’s ´ diagram with b < cov(N ). In this thesis, based on probability theory tools and the articles cited above, we develop a general theory of iterated forcing using finitely additive measures. For this purpose, we introduce two new notions: on the one hand, we define a new linkedness property, which we call “µ-FAM-linked” and, on the other hand, we generalize the notion of intersection number to forcing notions, which justifies the limit steps of our iteration theory. Finally, we apply our theory to prove in detail the consistency of cf(cov(N )) = ℵ0, and some separations of Cichon’s ´ diagram where cov(N ) is singular. In particular, we obtain a new constellation of Cichon’s diagram ´ separating the left side with cov(N ) singularEn el año 2000, Saharon Shelah introdujo un método que utiliza medidas finitamente aditivas a lo largo de iteraciones de soporte finito para demostrar que, consistentemente, el cubrimiento del ideal nulo puede tener cofinalidad contable. En 2019, Jakob Kellner, Saharon Shelah y Anda R. Tanasie mejoraron el método: lograron algunas generalizaciones y nuevas aplicaciones. En esta tesis, basada en las herramientas de la teoría de la probabilidad y los trabajos mencionados anteriormente, desarrollamos una teoría general de forcing iterado utilizando medidas finitamente aditivas. Para ello, introducimos dos nociones nuevas: por un lado, definimos una nueva propiedad de ligadura, que llamamos "FAM-ligadura'' y, por otro lado, generalizamos la idea de número de intersección a nociones de forcing, que justifica los pasos límite de nuestra teoría de iteraciones. Finalmente, aplicamos nuestro enfoque para probar en detalle la consistencia de que el cubrimiento del ideal nulo puede tener cofinalidad contable y obtenemos algunas separaciones del diagrama de Cichoń donde el cubrimiento del ideal nulo es singular. En particular, obtenemos una nueva constelación del diagrama de Cichoń separando el lado izquierdo y permitiendo que el cubrimiento del ideal nulo sea singular. (texto tomado de la fuente)MaestríaMagister en ciencias matemáticasÁrea Curricular en Matemáticas190 páginasapplication/pdfengUniversidad Nacional de ColombiaMedellín - Ciencias - Maestría en Ciencias - MatemáticasFacultad de CienciasMedellín, ColombiaUniversidad Nacional de Colombia - Sede Medellín510 - Matemáticas510 - Matemáticas::519 - Probabilidades y matemáticas aplicadas510 - Matemáticas::511 - Principios generales de las matemáticasForcing (Teoría de los modelos)Trayectoria aleatoriaIterated forcingProbabilityFinitely additive measureConsistency resultsNull setIntersection numberCardinal invariantSingular cardinalCichon’s diagramForcing iteradoProbabilidadMedida finitamente aditivaResultados de consistenciaConjunto nuloNumero de intersecciónCardinal invarianteCardinal singulardiagrama de Cicho´nIterated forcing with finitely additive measures: applications of probability to forcing theoryForcing iterado con medidas finitamente aditivas: aplicaciones de la probabilidad a la teoría del forcingTrabajo de grado - Maestríainfo:eu-repo/semantics/masterThesisinfo:eu-repo/semantics/acceptedVersionTexthttp://purl.org/redcol/resource_type/TMHannah Arendt. The Human Condition. University of Chicago Press, Chicago IL, 2018.Noga Alon and Joel H. Spencer. The Probabilistic Method. Wiley Publishing, 4th edition, 2016.Tomek Bartoszynski. On covering of real line by null sets. Pacific J. Math., 131(1):1– 12, 1988Jorg Brendle, Miguel A. Cardona, and Diego A. Mejıa. Filter-linkedness and its effect on preservation of cardinal characteristics. Ann. Pure Appl. Logic, 172(1):102856, 2021Tomek Bartoszynski, Jaime I. Ihoda, and Saharon Shelah. The cofinality of cardinal invariants related to measure and category. The Journal of Symbolic Logic, 54(3):719– 726, 1989Jorg Brendle and Haim Judah. Perfect sets of random reals. Israel Journal of Mathe- matics, 83:153–176, 1993Tomek Bartoszynski and Haim Judah. Set theory: On the Structure of the Real Line. A K Peters, Ltd., Wellesley, MA, 1995.Tomek Bartoszynski and H Judah. Measure and category. In Handbook of Set Theory. Vols. 1, 2, 3, volume 2. 2010.William Blake. The marriage of Heaven and Hell. Camden Hotten, London, 1868.Andreas Blass. Combinatorial cardinal characteristics of the continuum. In Handbook of Set Theory. Vols. 1, 2, 3, pages 395–489. Springer, Dordrecht, 2010.J. L. Bell and M. Machover. A Course in Mathematical Logic. North-Holland Pub- lishing Co., Amsterdam-New York-Oxford, 1977.J. L. Borges. Obras completas. Emecé, 1984.K. P. S. Bhaskara Rao and M. Bhaskara Rao. Theory of Charges: A Study of Finitely Additive Measures, volume 109 of Pure and Applied Mathematics. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983.Jorg Brendle. Larger cardinals in Cichon’s diagram. Israel Journal of Mathematics, 56(3):795–810, 1991.Kai Lai Chung. A Course in Probability Theory. Academic Press, 1974.Miguel A. Cardona and Diego A. Mejía. On cardinal characteristics of Yorioka ideals. Math. Log. Q., 65(2):170–199, 2019.Miguel A. Cardona and Diego A. Mejía. Forcing constellations of Cichon’s diagram by using the Tukey order. Kyoto Daigaku Surikaiseki Kenkyusho Kokyuroku, 22(13):14– 47, 2022. arXiv:2203.00615.Paul J. Cohen. Set Theory and the Continuum Hypothesis. W. A. Benjamin, Inc., New York-Amsterdam, 1966.Ryszard Engelking and Monika Karłowicz. Some theorems of set theory and their topological consequences. Fund. Math., 57:275–285, 1965.P. Erdos. Some remarks on the theory of graphs. Bulletin of the American Mathemat- ical Society, 53(4):292 – 294, 1947.Martin Goldstern, Jakob Kellner, Diego Alejandro Mejía, and Saharon Shelah. Preser- vation of splitting families and cardinal characteristics of the continuum. Israel Jour- nal of Mathematics, 246:73–129, 2021Martin Goldstern, Jakob Kellner, Diego Alejandro Mejía, and Saharon Shelah. Ci- chon’s maximum without large cardinals. J. Eur. Math. Soc. (JEMS), 24(11):3951– 3967, 2022.Martin Goldstern, Jakob Kellner, and Saharon Shelah. Cichon’s maximum. Ann. of Math., 190(1):113–143, 2019.Martin Goldstern, Diego Alejandro Mejía, and Saharon Shelah. The left side of Ci- chon’s diagram. Proc. Amer. Math. Soc., 144(9):4025–4042, 2016.Steven Givant and Halmos Paul. Introduction to Boolean Algebras, volume 1 of Un- dergraduate Texts in Mathematics. Springer, 2009.Lorenz J. Halbeisen. Combinatorial Set Theory. Springer Monographs in Mathemat- ics. Springer, 2019.David Hilbert. The Foundations of Mathematics. Harvard University Press, 1927.Horowitz Haim. and Saharon Shelah. Saccharinity with ccc. arXiv:1610.02706, 2016Thomas Jech. Set Theory, the Third Millennium Edition, Revised and Expanded. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2003Alexander S. Kechris. Classical Descriptive Set Theory. Springer New York, NY, 1995.J.L Kelley. Meaures on boolean algebras. J. Symbolic Logic, 64(2):737–746, 1959.Ashutosh Kumar and Saharon Shelah. On possible restrictions of the null ideal. Jour- nal of Mathematical Logic, 19(02):1950008, 2019.Jakob Kellner, Saharon Shelah, and Anda R. Tanasie. Another ordering of the ten car- dinal characteristics in Cichon’s diagram. Comment. Math. Univ. Carolin., 60(1):61– 95, 2019.Kenneth Kunen. Set Theory, an Introduction to Independence Proofs, volume 102 of Studies in Logic and the Foundations of Mathematics. North-Holland Publishing Co., Amsterdam-New York, 1980.Kenneth Kunen. Set Theory, volume 34 of Studies in Logic (London). College Publi- cations, London, 2011.Kenneth Kunen. The Foundation of Mathematics. College Publications, London, 2012.Luc Lauwers. Purely finitely additive measures are non-constructible objects. Working Papers of Department of Economics, Leuven, 2010.Diego A. Mejíıa. Matrix iterations with vertical support restrictions. In Proceedings of the 14th and 15th Asian Logic Conferences, pages 213–248. World Sci. Publ., Hack- ensack, NJ, 2019. arXiv:1803.05102.Diego A. Mejíıa. Forcing and combinatorics of names. Kyoto Daigaku Surikaiseki Kenkyusho Kokyuroku, 2164:34–49, 2020. http://hdl.handle.net/2433/ 261449.Arnold W. Miller. Some properties of measure and category. American Mathematical Society, 266(1):93–114, 1981.Arnold W. Miller. The Baire category theorem and cardinals of countable cofinality. The Journal of Symbolic Logic, 47(2):275–288, 1982.Yiannis N. Moschovakis. Descriptive Set Theory, volume 100 of Studies in Logic and the Foundations of Mathematics. North-Holland Publishing Co., Amsterdam, 1980.Diego Alejandro Mejía and Ismael E. Rivera-Madrid. Absoluteness theorems for ar- bitrary Polish spaces. Rev. Colombiana Mat., 53(2):109–123, 2019.John C. Oxtoby. Measure and Category. Graduate Texts in Mathematics. Springer, 2013.Janusz Pawlikowski. Adding dominating reals with Baire-bounding posets. American Mathematical Society, 123(2):540–547, 1992.Assaf Rinot. The Engelking-Karłowicz Theorem, and a useful corollary. Personal blog, Sep. 29, 2012. https://blog.assafrinot.com/?p=2054Maxwell Rosenlicht. Introduction to Analysis, volume 1. Dover Publications, 1968.Sheldon M. Ross. A First Course in Probability. Prentice Hall, Upper Saddle River, N.J., fifth edition, 1998.Saharon Shelah. The future of set theory. In Set theory of the reals, pages 1–12. Bar-Ilan Univ, 1993.Saharon Shelah. Covering of the null ideal may have countable cofinality. Fund. Math., 166(1-2):109–136, 2000.J. R Shoenfield. Martin’s axiom. American Mathematical Monthly, 82(6):610–617, 1975Saharon Shelah and Haim Judah. Adding dominating reals with the random algebra. American Mathematical Society, 119(1):267–273, 1993.R.M. Solovay and Tennenbaum S. Iterated Cohen extensions and Souslin’s problem. Annals of Mathematics, 94(2):201–245, 1971.John Truss. Connections between different amoeba algebras. Fundamenta Mathemat- icae, 130(2):137–155, 1988.Carlos Uzcátegui Aylwin and Carlos Augusto Di Prisco. Una Introducción a la Teoría Descriptiva de Conjuntos. 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