Potser la pregunta més senzilla, i alhora fonamental, que es pot fer sobre els nombres realś es: ... more Potser la pregunta més senzilla, i alhora fonamental, que es pot fer sobre els nombres realś es: Quants n'hi ha? Cantor va demostrar que els nombres reals no es poden numerar i va formular l'any 1878 la conjectura següent:
més de n objectes, aleshores en alguna casella hi haurà almenys dos objectes. Per exemple, si rep... more més de n objectes, aleshores en alguna casella hi haurà almenys dos objectes. Per exemple, si repartim 6 coloms en 5 caselles, aleshores en alguna de les caselles hi haurà d'haver almenys 2 coloms. Figura 1: El principi de les caselles. Això normalment es representa així: 6 → (2) 5 . I en general escrivim n → (2) r per indicar que, si distribuïm n objectes en r caselles, aleshores en alguna casella hi haurà almenys 2 objectes.
La demostración habitual del Segundo Teorema de Incompletud de Gödel a partir de teorías débiles ... more La demostración habitual del Segundo Teorema de Incompletud de Gödel a partir de teorías débiles como IΣ 1 es larga y técnicamente intrincada. Raramente se dan todos los detalles y en muchos casos se omiten completamente apelando a la capacidad de lector para completarlos. En la primera parte de este artículo presentamos una guía de los principales puntos técnicos de la demostración habitual del Segundo Teorema de Incompletud de Gödel a partir de teorías débiles. En la segunda parte presentamos una demostración distinta y más simple para la Teoría de Conjuntos de Zermelo-Fraenkel debida a T. Jech ], y observamos que puede ser extendida de forma que englobe teorías débiles, evitando así muchas de las complicaciones técnicas que requieren las demostraciones habituales.
For each natural number n, let C (n) be the closed and unbounded proper class of ordinals α such ... more For each natural number n, let C (n) be the closed and unbounded proper class of ordinals α such that V α is a n elementary substructure of V . We say that κ is a C (n) -cardinal if it is the critical point of an elementary embedding j : V → M, M transitive, with j (κ) in C (n) . By analyzing the notion of C (n) -cardinal at various levels of the usual hierarchy of large cardinal principles we show that, starting at the level of superstrong cardinals and up to the level of rank-into-rank embeddings, C (n) -cardinals form a much finer hierarchy. The naturalness of the notion of C (n)cardinal is exemplified by showing that the existence of C (n) -extendible cardinals is equivalent to simple reflection principles for classes of structures, which generalize the notions of supercompact and extendible cardinals. Moreover, building on results of , we give new characterizations of Vopeňka's Principle in terms of C (n) -extendible cardinals.
We prove that, under suitable assumptions on a category C, the existence of supercompact cardinal... more We prove that, under suitable assumptions on a category C, the existence of supercompact cardinals implies that every absolute epireflective class of objects of C is a small-orthogonality class. More precisely, if L is a localization functor on an accessible category C such that the unit morphism X → LX is an extremal epimorphism for all X, and the class of L-local objects is defined by an absolute formula with parameters, then the existence of a supercompact cardinal above the cardinalities of the parameters implies that L is a localization with respect to some set of morphisms.
We study the preservation of the property of L(R) being a Solovay model under projective ccc forc... more We study the preservation of the property of L(R) being a Solovay model under projective ccc forcing extensions. We compute the exact consistency strength of the generic absoluteness of L(R) under forcing with projective ccc partial orderings and, as an application, we build models in which Martin's Axiom holds for Σ ∼ 1 n partial orderings, but it fails for the Σ ∼ 1 n+1 .
As is well-known, Cantor's continuum problem, namely, what is the cardinality of ? is in... more As is well-known, Cantor's continuum problem, namely, what is the cardinality of ? is independent of the usual ZFC ax- ioms of Set Theory. K. Godel ((12), (13)) suggested that new natural axioms should be found that would settle the problem and hinted at large-cardinal axioms as such. However, shortly after the invention of forcing, it was shown by Levy and Solovay (20) that the prob- lem remains independent even if one adds to ZFC the usual large- cardinal axioms, like the existence of measurable cardinals, or even supercompact cardinals, provided, of course, that these axioms are consistent. While numerous axioms have been proposed that settle the problem-although not always in the same way-from the Axiom of Constructibility to strong combinatorial axioms like the Proper Forcing Axiom or Martin's Maximum, none of them so far has been recognized as a natural axiom and been accepted as an appropriate solution to the continuum problem. In this paper we discuss some heuristic principles, which might be regarded as Meta-Axioms of Set Theory, that provide a criterion for assessing the naturalness of the set-theoretic axioms. Under this criterion we then evaluate several kinds of axioms, with a special emphasis on a class of recently intro- duced set-theoretic principles for which we can reasonably argue that they constitute very natural axioms of Set Theory and which settle Cantor's continuum problem.
Let κ be a cardinal which is measurable after generically adding κ+ω many Cohen subsets to κ and ... more Let κ be a cardinal which is measurable after generically adding κ+ω many Cohen subsets to κ and let G = (κ, E) be the κ-Rado graph. We prove, for 2 ≤ m < ω, that there is a finite value r + m such that the set [κ] m can be partitioned into classes˙Ci : i < r + m¸s uch that for any coloring of any of the classes Ci in fewer than κ colors, there is a copy G * of G in G such that [G * ] m ∩ Ci is monochromatic. It follows that G → (G) m <κ/r + m , that is, for any coloring of [G] m with fewer than κ colors there is a copy G of G such that [G ] m has at most r + m colors. On the other hand, we show that there are colorings of G such that if G is any copy of G then Ci ∩ [G ] m = ∅ for all i < r + m , and hence G
We give a unified presentation of the set-theoretic axioms of generic absoluteness, we survey the... more We give a unified presentation of the set-theoretic axioms of generic absoluteness, we survey the known results regarding their consistency strength and some of their consequences as well as their relationship to other kinds of set-theoretic axioms, and we provide a list of the main open problems.
, Hugh Woodin introduced Ω-logic, an approach to truth in the universe of sets inspired by recent... more , Hugh Woodin introduced Ω-logic, an approach to truth in the universe of sets inspired by recent work in large cardinals. Expository accounts of Ω-logic appear in . In this paper we present proofs of some elementary facts about Ω-logic, relative to the published literature, leading up to the generic invariance of Ωlogic and the Ω-conjecture.
Potser la pregunta més senzilla, i alhora fonamental, que es pot fer sobre els nombres realś es: ... more Potser la pregunta més senzilla, i alhora fonamental, que es pot fer sobre els nombres realś es: Quants n'hi ha? Cantor va demostrar que els nombres reals no es poden numerar i va formular l'any 1878 la conjectura següent:
més de n objectes, aleshores en alguna casella hi haurà almenys dos objectes. Per exemple, si rep... more més de n objectes, aleshores en alguna casella hi haurà almenys dos objectes. Per exemple, si repartim 6 coloms en 5 caselles, aleshores en alguna de les caselles hi haurà d'haver almenys 2 coloms. Figura 1: El principi de les caselles. Això normalment es representa així: 6 → (2) 5 . I en general escrivim n → (2) r per indicar que, si distribuïm n objectes en r caselles, aleshores en alguna casella hi haurà almenys 2 objectes.
La demostración habitual del Segundo Teorema de Incompletud de Gödel a partir de teorías débiles ... more La demostración habitual del Segundo Teorema de Incompletud de Gödel a partir de teorías débiles como IΣ 1 es larga y técnicamente intrincada. Raramente se dan todos los detalles y en muchos casos se omiten completamente apelando a la capacidad de lector para completarlos. En la primera parte de este artículo presentamos una guía de los principales puntos técnicos de la demostración habitual del Segundo Teorema de Incompletud de Gödel a partir de teorías débiles. En la segunda parte presentamos una demostración distinta y más simple para la Teoría de Conjuntos de Zermelo-Fraenkel debida a T. Jech ], y observamos que puede ser extendida de forma que englobe teorías débiles, evitando así muchas de las complicaciones técnicas que requieren las demostraciones habituales.
For each natural number n, let C (n) be the closed and unbounded proper class of ordinals α such ... more For each natural number n, let C (n) be the closed and unbounded proper class of ordinals α such that V α is a n elementary substructure of V . We say that κ is a C (n) -cardinal if it is the critical point of an elementary embedding j : V → M, M transitive, with j (κ) in C (n) . By analyzing the notion of C (n) -cardinal at various levels of the usual hierarchy of large cardinal principles we show that, starting at the level of superstrong cardinals and up to the level of rank-into-rank embeddings, C (n) -cardinals form a much finer hierarchy. The naturalness of the notion of C (n)cardinal is exemplified by showing that the existence of C (n) -extendible cardinals is equivalent to simple reflection principles for classes of structures, which generalize the notions of supercompact and extendible cardinals. Moreover, building on results of , we give new characterizations of Vopeňka's Principle in terms of C (n) -extendible cardinals.
We prove that, under suitable assumptions on a category C, the existence of supercompact cardinal... more We prove that, under suitable assumptions on a category C, the existence of supercompact cardinals implies that every absolute epireflective class of objects of C is a small-orthogonality class. More precisely, if L is a localization functor on an accessible category C such that the unit morphism X → LX is an extremal epimorphism for all X, and the class of L-local objects is defined by an absolute formula with parameters, then the existence of a supercompact cardinal above the cardinalities of the parameters implies that L is a localization with respect to some set of morphisms.
We study the preservation of the property of L(R) being a Solovay model under projective ccc forc... more We study the preservation of the property of L(R) being a Solovay model under projective ccc forcing extensions. We compute the exact consistency strength of the generic absoluteness of L(R) under forcing with projective ccc partial orderings and, as an application, we build models in which Martin's Axiom holds for Σ ∼ 1 n partial orderings, but it fails for the Σ ∼ 1 n+1 .
As is well-known, Cantor&#x27;s continuum problem, namely, what is the cardinality of ? is in... more As is well-known, Cantor&#x27;s continuum problem, namely, what is the cardinality of ? is independent of the usual ZFC ax- ioms of Set Theory. K. Godel ((12), (13)) suggested that new natural axioms should be found that would settle the problem and hinted at large-cardinal axioms as such. However, shortly after the invention of forcing, it was shown by Levy and Solovay (20) that the prob- lem remains independent even if one adds to ZFC the usual large- cardinal axioms, like the existence of measurable cardinals, or even supercompact cardinals, provided, of course, that these axioms are consistent. While numerous axioms have been proposed that settle the problem-although not always in the same way-from the Axiom of Constructibility to strong combinatorial axioms like the Proper Forcing Axiom or Martin&#x27;s Maximum, none of them so far has been recognized as a natural axiom and been accepted as an appropriate solution to the continuum problem. In this paper we discuss some heuristic principles, which might be regarded as Meta-Axioms of Set Theory, that provide a criterion for assessing the naturalness of the set-theoretic axioms. Under this criterion we then evaluate several kinds of axioms, with a special emphasis on a class of recently intro- duced set-theoretic principles for which we can reasonably argue that they constitute very natural axioms of Set Theory and which settle Cantor&#x27;s continuum problem.
Let κ be a cardinal which is measurable after generically adding κ+ω many Cohen subsets to κ and ... more Let κ be a cardinal which is measurable after generically adding κ+ω many Cohen subsets to κ and let G = (κ, E) be the κ-Rado graph. We prove, for 2 ≤ m < ω, that there is a finite value r + m such that the set [κ] m can be partitioned into classes˙Ci : i < r + m¸s uch that for any coloring of any of the classes Ci in fewer than κ colors, there is a copy G * of G in G such that [G * ] m ∩ Ci is monochromatic. It follows that G → (G) m <κ/r + m , that is, for any coloring of [G] m with fewer than κ colors there is a copy G of G such that [G ] m has at most r + m colors. On the other hand, we show that there are colorings of G such that if G is any copy of G then Ci ∩ [G ] m = ∅ for all i < r + m , and hence G
We give a unified presentation of the set-theoretic axioms of generic absoluteness, we survey the... more We give a unified presentation of the set-theoretic axioms of generic absoluteness, we survey the known results regarding their consistency strength and some of their consequences as well as their relationship to other kinds of set-theoretic axioms, and we provide a list of the main open problems.
, Hugh Woodin introduced Ω-logic, an approach to truth in the universe of sets inspired by recent... more , Hugh Woodin introduced Ω-logic, an approach to truth in the universe of sets inspired by recent work in large cardinals. Expository accounts of Ω-logic appear in . In this paper we present proofs of some elementary facts about Ω-logic, relative to the published literature, leading up to the generic invariance of Ωlogic and the Ω-conjecture.
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Papers by Joan Bagaria