Given a Polish space X and a countable collection of analytic hypergraphs on X, I consider the σ-ideal generated by Borel anticliques for the hypergraphs in the family. It turns out that many of the quotient posets are proper. I investigate the forcing properties of these posets, certain natural operations on them, and prove some related dichotomies.

Let I be an F σ -ideal on natural numbers. We characterize the ultrafilters which are generic over the model L for the poset of I -positive sets of natural numbers ordered by inclusion.

I describe several ways in which forcing arguments can be used to yield clean and conceptual proofs of nonreducibility, ergodicity and other results in the theory of analytic equivalence relations. In particular, I present simple Borel equivalence relations $E, F$ such that a natural proof of nonreducibility of $E$ to $F$ uses the independence of the Singular Cardinal Hypothesis at $\aleph_\omega$.

I describe several ways in which forcing arguments can be used to yield clean and conceptual proofs of nonreducibility, ergodicity and other results in the theory of analytic equivalence relations. In particular, I present simple Borel equivalence relationsE, Fsuch that a natural proof of nonreducibility ofEtoFuses the independence of the Singular Cardinal Hypothesis at ℵω.

With every σ-ideal I on a Polish space we associate the σ-ideal I* generated by the closed sets in I. We study the forcing notions of Borel sets modulo the respective σ-ideals I and I* and find connections between their forcing properties. To this end, we associate to a σ-ideal on a Polish space an ideal on a countable set and show how forcing properties of the forcing depend on combinatorial properties of the ideal. We also study the 1—1 or (...) constant property of σ-ideals, i.e., the property that every Borel function defined on a Borel positive set can be restricted to a positive Borel set on which it either 1—1 or constant. We prove the following dichotomy: if I is a σ-ideal generated by closed sets, then either the forcing P I adds a Cohen real, or else I has the 1—1 or constant property. (shrink)

I prove several natural preservation theorems for the countable support iteration. This solves a question of Rosłanowski regarding the preservation of localization properties and greatly simplifies the proofs in the area.

The main topics of this thesis are cardinal invariants, P -points and MAD families. Cardinal invariants of the continuum are cardinal numbers that are bigger than $\aleph _{0}$ and smaller or equal than $\mathfrak {c}.$ Of course, they are only interesting when they have some combinatorial or topological definition. An almost disjoint family is a family of infinite subsets of $\omega $ such that the intersection of any two of its elements is finite. A MAD family is a maximal almost (...) disjoint family. An ultrafilter $\mathcal {U}$ on $\omega $ is called a P-point if every countable $\mathcal {B\subseteq U}$ there is $X\in $ $\mathcal {U}$ such that $X\setminus B$ is finite for every $B\in \mathcal {B}.$ This kind of ultrafilters has been extensively studied, however there is still a large number of open questions about them.In the preliminaries we recall the principal properties of filters, ultrafilters, ideals, MAD families and cardinal invariants of the continuum. We present the construction of Shelah, Mildenberger, Raghavan, and Steprāns of a completely separable MAD family under $\mathfrak {s\leq a}.$ None of the results in this chapter are due to the author.The second chapter is dedicated to a principle of Sierpiński. The principle $\left $ of Sierpiński is the following statement: There is a family of functions $\left \{ \varphi _{n}:\omega _{1}\longrightarrow \omega _{1}\mid n\in \omega \right \} $ such that for every $I\in \left [ \omega _{1}\right ] ^{\omega _{1}}$ there is $n\in \omega $ for which $\varphi _{n}\left [ I\right ] =\omega _{1}.$ This principle was recently studied by Arnie Miller. He showed that this principle is equivalent to the following statement: There is a set $X=\left \{ f_{\alpha }\mid \alpha \alpha $ then $f_{\beta }\cap g$ is infinite. Miller showed that the principle of Sierpiński implies that non $\left =\omega _{1}.$ He asked if the converse was true, i.e., does non $\left =\omega _{1}$ imply the principle $\left $ of Sierpiński? We answer his question affirmatively. In other words, we show that non $\left =\omega _{1}$ is enough to construct an $\mathcal {IE}$ -Luzin set. It is not hard to see that the $\mathcal {IE}$ -Luzin set we constructed is meager. This is no coincidence, because with the aid of an inaccessible cardinal, we construct a model where non $\left =\omega _{1}$ and every $\mathcal {IE}$ -Luzin set is meager.The third chapter is dedicated to a conjecture of Hrušák. Michael Hrušák conjectured the following: Every Borel cardinal invariant is either at most non $\left $ or at least cov $\left $. Although the veracity of this conjecture is still an open problem, we were able to obtain some partial results: The conjecture is false for “Borel invariants of $\omega _{1}^{\omega }$ ” nevertheless, it is true for a large class of definable invariants. This is part of a joint work with Michael Hrušák and Jindřich Zapletal.In the fourth chapter we present a survey on destructibility of ideals and MAD families. We prove several classic theorems, but we also prove some new results. For example, we show that every almost disjoint family of size less than $\mathfrak {c}$ can be extended to a Cohen indestructible MAD family is equivalent to $\mathfrak {b=c}$. A MAD family $\mathcal {A}$ is Shelah–Steprāns if for every $X\subseteq \left [ \omega \right ] ^{<\omega }\setminus \left \{ \emptyset \right \} $ either there is $A\in \mathcal {I}\left $ such that $s\cap A\neq \emptyset $ for every $s\in X$ or there is $B\in \mathcal {I}\left $ that contains infinitely many elements of X $ denotes the ideal generated by $\mathcal {A}$ ). This concept was introduced by Raghavan which is connected to the notion of “strongly separable” introduced by Shelah and Steprāns. We prove that Shelah–Steprāns MAD families have very strong indestructibility properties: Shelah–Steprāns MAD families are indestructible for “many” definable forcings that does not add dominating reals. According to the author’s best knowledge, this is the strongest notion that has been considered in the literature so far. In spite of their strong indestructibility, Shelah–Steprāns MAD families can be destroyed by a ccc forcing that does not add unsplit or dominating reals. We also consider some strong combinatorial properties of MAD families and show the relationships between them.The fifth chapter is one of the most important chapters in the thesis. A MAD family $\mathcal {A}$ is called $+$ -Ramsey if every tree that branches into $\mathcal {I}\left $ -positive sets has an $\mathcal {I}\left $ -positive branch. Michael Hrušák’s first published question is the following: Is there a $+$ -Ramsey MAD family? It was previously known that such families can consistently exist. However, there was no construction of such families using only the axioms of ZFC. We solve this problem by constructing such a family without any extra assumptions.In the fourth and fifth chapters, we introduce several notions of MAD families, in the sixth chapter we prove several implications and non implications between them. We construct several MAD families with different properties.In the seventh chapter we build models without P -points. We show that there are no P -points after adding Silver reals either iteratively or by the side by side product. These results have some important consequences: The first one is that is its possible to get rid of P -points using only definable forcings. This answers a question of Michael Hrušák. We can also use our results to build models with no P -points and with arbitrarily large continuum, which was also an open question. These results were obtained with David Chodounský.prepared by Osvaldo Guzmán GonzálezE-mail : [email protected] : https://arxiv.org/abs/1810.09680. (shrink)

This thesis divides naturally into two parts, each concerned with the extent to which the theory of $L$ can be changed by forcing.The first part focuses primarily on applying generic-absoluteness principles to how that definable sets of reals enjoy regularity properties. The work in Part I is joint with Itay Neeman and is adapted from our paper Happy and mad families in $L$, JSL, 2018. The project was motivated by questions about mad families, maximal families of infinite subsets of $\omega (...) $ of which any two have only finitely many members in common. We begin, in the spirit of Mathias, by establishing a strong Ramsey property for sets of reals in the Solovay model, giving a new proof of Törnquist’s theorem that there are no infinite mad families in the Solovay model.In Chapter 3 we stray from the main line of inquiry to briefly study a game-theoretic characterization of filters with the Baire Property.Neeman and Zapletal showed, assuming roughly the existence of a proper class of Woodin cardinals, that the boldface theory of $L$ cannot be changed by proper forcing. They call their result the Embedding Theorem, because they conclude that in fact there is an elementary embedding from the $L$ of the ground model to that of the proper forcing extension. With a view toward analyzing mad families under $\mathsf {AD}^+$ and in $L$ under large-cardinal hypotheses, in Chapter 4 we establish triangular versions of the Embedding Theorem. These are enough for us to use Mathias’s methods to show that there are no infinite mad families in $L$ under large cardinals and that $\mathsf {AD}^+$ implies that there are no infinite mad families. These are again corollaries of theorems about strong Ramsey properties under large-cardinal assumptions and $\mathsf {AD}^+$, respectively. Our first theorem improves the large-cardinal assumption under which Todorcevic established the nonexistence of infinite mad families in $L$. Part I concludes with Chapter 5, a short list of open questions.In the second part of the thesis, we undertake a finer analysis of the Embedding Theorem and its consistency strength. Schindler found that the the Embedding Theorem is consistent relative to much weaker assumptions than the existence of Woodin cardinals. He defined remarkable cardinals, which can exist even in L, and showed that the Embedding Theorem is equiconsistent with the existence of a remarkable cardinal. His theorem resembles a theorem of Harrington–Shelah and Kunen from the 1980s: the absoluteness of the theory of $L$ to ccc forcing extensions is equiconsistent with a weakly compact cardinal. Joint with Itay Neeman, we improve Schindler’s theorem by showing that absoluteness for $\sigma $ -closed $\ast $ ccc posets—instead of the larger class of proper posets—implies the remarkability of $\aleph _1^V$ in L. This requires a fundamental change in the proof, since Schindler’s lower-bound argument uses Jensen’s reshaping forcing, which, though proper, need not be $\sigma $ -closed $\ast $ ccc in that context. Our proof bears more resemblance to that of Harrington–Shelah than to Schindler’s.The proof of Theorem 6.2 splits naturally into two arguments. In Chapter 7 we extend the Harrington–Shelah method of coding reals into a specializing function to allow for trees with uncountable levels that may not belong to L. This culminates in Theorem 7.4, which asserts that if there are $X\subseteq \omega _1$ and a tree $T\subseteq \omega _1$ of height $\omega _1$ such that X is codable along T, then $L$ -absoluteness for ccc posets must fail.We complete the argument in Chapter 8, where we show that if in any $\sigma $ -closed extension of V there is no $X\subseteq \omega _1$ codable along a tree T, then $\aleph _1^V$ must be remarkable in L.In Chapter 9 we review Schindler’s proof of generic absoluteness from a remarkable cardinal to show that the argument gives a level-by-level upper bound: a strongly $\lambda ^+$ -remarkable cardinal is enough to get $L$ -absoluteness for $\lambda $ -linked proper posets.Chapter 10 is devoted to partially reversing the level-by-level upper bound of Chapter 9. Adapting the methods of Neeman, Hierarchies of forcing axioms II, we are able to show that $L$ -absoluteness for $\left |\mathbf {R}\right |\cdot \left |\lambda \right |$ -linked posets implies that the interval $[\aleph _1^V,\lambda ]$ is $\Sigma ^2_1$ -remarkable in L.prepared by Zach Norwood.E-mail: [email protected]. (shrink)

In our papers on the rationality of magic, we distinghuished, for purposes of analysis, three levels of rationality. First and lowest (rationalitYl) the goal directed action of an agent with given aims and circumstances, where among his circumstances we included his knowledge and opinions. On this level the magician's treatment of illness by incantation is as rational as any traditional doctor's blood-letting or any modern one's use of anti-biotics. At the second level (rationalitY2) we add the element of rational thinking (...) or thinking which obeys some set of explicit rules, a level which is not found in magic in general, though it is sometimes given to specific details of magical thinking within the magical thought-system. It was the late Sir Edward E. Evans-Pritchard who observed that when considering magic in detail the magician may be as consistent or critical as anyone else; but when considering magic in general, or any system of thought in general, the magician could not be critical or even comprehend the criticism. Evans-Pritchard went even further: he was sceptical as to whether it could be done in a truly consistent manner: one cannot be critical of one's own system, he thought. On this level (rationalitY2) of discussion we have explained (earlier) why we prefer to wed Evans Pritchard's view of the magician's capacity for piece-meal rationality to Sir James Frazer's view that magic in general is pseudo-rational because it lacks standards of rational thinking. (shrink)

Analysis of various Buddhist philosophical views.

Introduction : La philosophie en Égypte -- La philosophie chretienne arabe contemporaine : Kanawati et Youssef Karam -- Le discours éthique en Égypte -- Naguib Mahfouz et la philosophie -- La philosophie et la revolution en Égypte.

Verse work, with autocommentary, on Jain metaphysics and onthology.

This article utilises Gadamerian hermeneutics method and Freirean theory of the purpose of Religious Education to explore how Religious Education can contribute to achieving United Nations Sustainable Development Goal (SDG) 4, emphasising education for sustainable development. The study contends that Religious Education in Lesotho occupies a distinctive position in the education system, surpassing other countries in its extensive integration. Due to historical factors, Religious Education is taught in nearly all religiously affiliated schools, comprising about 90% of all educational institutions in (...) Lesotho, and even in certain public schools. The curriculum of Religious Education in Lesotho aligns directly with the educational objectives of SDG 4. Additionally, Lesotho’s Religious Education corresponds to three levels of school education for sustainable development: whole school approach, classroom approach, and community approach. Nevertheless, the study asserts that for Religious Education in Lesotho to better align with education for sustainable development, various enhancements are required. Firstly, inclusivity across all religious traditions is essential, moving beyond a sole focus on Christianity. Secondly, there is a need for wider dissemination and implementation of Religious Education in all schools, irrespective of their affiliation. Lastly, a shift from traditional pedagogies to transformative approaches is recommended to boost the effectiveness of Religious Education in advancing sustainable development.Contribution: This article contributes by analysing Lesotho’s case and its implications for enhancing Religious Education teaching to effectively address SDGs. It broadens existing knowledge on Religious Education’s potential as a platform for sustainable development and provides recommendations for its improved implementation in a specific context. (shrink)

Ser modern és ser racionalista. Aquest llibre repassa amb detall les tesis de Descartes, la seva relació amb els contemporanis i les conseqüències de les seves teories.

L'utilitarisme té com a criteri bàsic aconseguir el màxim bé per al màxim nombre possible de persones. L'utilitarisme dóna molta importància a la recerca de la felicitat personal. Aquest llibre en descriu les tesis bàsiques i respon a les principals acusacions que ha rebut.

Study on the commentary of seven treatises of Buddhist logic (sapta pramanaśastra) of Dharmakīrti, 7th cent.

Study on the commentary of seven pramāṇaśastra (Buddhist logic) by Acharya Dharmakīrti, 7th century.

jild-i 1. Az ʻAbd al-Ḥamīd Kātib (59 Sh.) tā Abū al-Ḥasan Masʻūdī (336 Sh.) -- jild-i 2. Az Abū al-Ḥasan ʻĀmirī (291 Sh.) tā Abū al-Faz̤l Bayhaqī (465 Sh.) -- jild-i 3. Az Nāṣir Khusraw (383 Sh.) tā Sadīd al-Dīn ʻAwfī (616 Sh.) -- jild-i 4. Az Najm al-Dīn Rāzī (553 Sh.) tā Fakhr al-Muḥaqqiqīn (748 Sh.) -- jild-i 5. Az ʻUbayd Zākānī (670 Sh.) tā Jalāl al-Dīn Sayūṭī (884 Sh.) -- jild-i 6. Az Rūzbihān Khunjī (825 Sh.) tā Shaykh (...) Ḥurr ʻĀmilī (1071 Sh.) -- jild-i 7. Az ʻAlāmah Majlisī (1006 Sh.) tā Muḥammad Bāqir Shafaq (1223 Sh.) -- jild-i 8. Az Muḥammad Ṣādiq Marvazī (1147 Sh.) tā Mīrzā Fatḥʻalī Ākhūndzādah (1289 Sh.) -- jild-i 9. Az Mīrzā-yi Shīrāzī Avval (1194 Sh.) tā Ākhūnd Khurāsānī (1290 Sh.) -- jild-i 10. Az ʻAbd Allāh Māzandarānī (1219 Sh.) tā Saʻd Zaghlūl (1306 Sh.) -- jild-i 11. Az ʻImād al-ʻUlamāʼ Khalkhālī (1237 Sh.) tā ʻAbd al-ʻAzīz S̲aʻālibī (1323 Sh.) -- jild-i 12. Az Āyat Allāh Burūjirdī (1254 Sh.) tā Muḥammad Amīn Rasūlʹzādah (1333 Sh.) -- jild-i 13. Az ʻAbd al-Karīm Zanjānī (1263 Sh.) tā Amīn al-Ḥusaynī (1353 Sh.) -- jild-i 14. Az Jamālzādah (1274 Sh.) tā Mahdī Bāzargān (1373) -- jild-i 15. Az Muḥammad Ḥamīd Allāh (1286 Sh.) tā Muṭahharī (1358 Sh.) -- jild-i 16. Az Farīdūn Ādamīyat (1299 Sh.) tā Muṣṭafá Khumaynī (1356 Sh.) -- jild-i 17. Az Samīr Amīn (1310 Sh.) tā Ḥasan Ḥanafī -- jild-i 18. Az Sayyid Muḥammad Bāqir Ṣadr (1314 Sh.) tā Rāshid al-Ghanūshī -- jild-i 19. Az ʻAlī Awmlīl (1321 Sh.) tā Ṭāriq Ramaz̤ān. (shrink)