Els Castanyers I Castanyeres
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The relative consistency of the negation of the Axiom of Choice using permuta...
This document discusses the axiom of choice (AC) and related principles in set theory. It makes the following key points:
1. AC allows proving the existence of sets without explicit construction, which some early mathematicians found controversial.
2. AC is equivalent to several other principles like Zorn's lemma and is needed to prove certain paradoxical theorems.
3. Various weaker forms of choice exist, like the axiom of dependent choice, and some principles like the prime ideal theorem do not imply full AC.
4. Models of set theory without AC, called permutation models, can be constructed, showing the consistency of denying AC is independent of Zermelo-Fraenkel set theory
Basic S and L : The existence of an S-space under MA and $\neg$ CH
This document discusses S and L spaces, which are topological spaces that have certain separation and density properties. It summarizes key results about the existence of S and L spaces under different set theory assumptions, such as the existence of S and L spaces under the continuum hypothesis but their destruction under Martin's Axiom and the negation of the continuum hypothesis. The document also outlines proofs of the consistency of the existence of an S space under Martin's Axiom and the negation of the continuum hypothesis.
L Space In ZFC
1. The document discusses S-spaces and L-spaces, which are types of topological spaces that are hereditarily separable or hereditarily Lindelöf.
2. It presents theorems showing the existence or nonexistence of S-spaces and L-spaces under different set theory assumptions, such as the existence of Suslin lines or adding Cohen reals.
3. The canonical form theorem states that S-spaces and L-spaces (when they exist) can be represented as subspaces of 2^ω1, the Tychonoff product of ω1 many 2-point discrete spaces.
LA DEPURADORA I EL MOLÍ
La depuradora es una máquina que limpia el agua residual de una casa o fábrica eliminando la suciedad y desechos antes de devolverla a un río u otro cuerpo de agua. El molino es una máquina que muele o tritura materiales como granos para hacer harina u otros productos. Ambas máquinas cumplen funciones importantes en la industria y el hogar.
Recommended
The relative consistency of the negation of the Axiom of Choice using permuta...
This document discusses the axiom of choice (AC) and related principles in set theory. It makes the following key points:
1. AC allows proving the existence of sets without explicit construction, which some early mathematicians found controversial.
2. AC is equivalent to several other principles like Zorn's lemma and is needed to prove certain paradoxical theorems.
3. Various weaker forms of choice exist, like the axiom of dependent choice, and some principles like the prime ideal theorem do not imply full AC.
4. Models of set theory without AC, called permutation models, can be constructed, showing the consistency of denying AC is independent of Zermelo-Fraenkel set theory
Basic S and L : The existence of an S-space under MA and $\neg$ CH
This document discusses S and L spaces, which are topological spaces that have certain separation and density properties. It summarizes key results about the existence of S and L spaces under different set theory assumptions, such as the existence of S and L spaces under the continuum hypothesis but their destruction under Martin's Axiom and the negation of the continuum hypothesis. The document also outlines proofs of the consistency of the existence of an S space under Martin's Axiom and the negation of the continuum hypothesis.
L Space In ZFC
1. The document discusses S-spaces and L-spaces, which are types of topological spaces that are hereditarily separable or hereditarily Lindelöf.
2. It presents theorems showing the existence or nonexistence of S-spaces and L-spaces under different set theory assumptions, such as the existence of Suslin lines or adding Cohen reals.
3. The canonical form theorem states that S-spaces and L-spaces (when they exist) can be represented as subspaces of 2^ω1, the Tychonoff product of ω1 many 2-point discrete spaces.
LA DEPURADORA I EL MOLÍ
La depuradora es una máquina que limpia el agua residual de una casa o fábrica eliminando la suciedad y desechos antes de devolverla a un río u otro cuerpo de agua. El molino es una máquina que muele o tritura materiales como granos para hacer harina u otros productos. Ambas máquinas cumplen funciones importantes en la industria y el hogar.
RECICLA-JOCS
RECICLA-JOCS is a program in Barcelona that collects toys and games that are no longer wanted by their owners and redistributes them to children in need. The goal is to reduce waste by reusing items that are still in good condition rather than throwing them away, while also helping families who cannot afford new toys give their children gifts at Christmas and other holidays. Donated toys are cleaned, repaired if needed, and delivered to charities and social services to distribute to children.
NADAL 07 (3ª part)
This very short document appears to be wishing someone named Nadal a happy new year in Catalan. It states "Bon Nadal i Feliç Any Nou" which translates to "Merry Christmas and Happy New Year" in Catalan. The document does not provide much other context or information to summarize further in 3 sentences or less.
NADAL 07
Nadal va anar a l'escola com qualsevol altre nen. Allà va aprendre les matèries bàsiques com la lectura, l'escriptura i l'aritmètica, però també va descobrir la seva passió pel tennis. Malgrat que als professors els costava fer-lo concentrar-se en classe, Nadal sempre va ser un estudiant aplicat.
Les Danses
Les Dances es un curso de baile ofrecido durante el año académico 2007-2008. El curso enseña diferentes estilos de baile como el clásico, el jazz y el contemporáneo. Los estudiantes aprenderán las técnicas básicas de cada estilo a través de clases prácticas y teóricas durante el año.
Festa Fi De Curs
La pandemia de COVID-19 ha tenido un impacto significativo en la economía mundial. Muchos países experimentaron fuertes caídas en el PIB y aumentos en el desempleo debido a los cierres generalizados. A medida que se levantan las restricciones, la recuperación económica será gradual a medida que los consumidores y las empresas se readaptan a la "nueva normalidad".
Excursió Bcn
A group of students took an excursion to Barcelona from May 29th to May 30th, 2007. The trip included visits to important landmarks like La Sagrada Familia church and Park Güell, as well as time for shopping and exploring the city. The students learned about Barcelona's history and culture during their educational trip.
Edificis Romans
Los edificios romanos más importantes incluyen arcos de triunfo, acueductos, templos, foros, teatros, circos, anfiteatros, coliseos y termas. También construyeron murallas defensivas alrededor de ciudades como Tarragona y León.
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RECICLA-JOCS
RECICLA-JOCS is a program in Barcelona that collects toys and games that are no longer wanted by their owners and redistributes them to children in need. The goal is to reduce waste by reusing items that are still in good condition rather than throwing them away, while also helping families who cannot afford new toys give their children gifts at Christmas and other holidays. Donated toys are cleaned, repaired if needed, and delivered to charities and social services to distribute to children.
NADAL 07 (3ª part)
This very short document appears to be wishing someone named Nadal a happy new year in Catalan. It states "Bon Nadal i Feliç Any Nou" which translates to "Merry Christmas and Happy New Year" in Catalan. The document does not provide much other context or information to summarize further in 3 sentences or less.
NADAL 07
Nadal va anar a l'escola com qualsevol altre nen. Allà va aprendre les matèries bàsiques com la lectura, l'escriptura i l'aritmètica, però també va descobrir la seva passió pel tennis. Malgrat que als professors els costava fer-lo concentrar-se en classe, Nadal sempre va ser un estudiant aplicat.
Les Danses
Les Dances es un curso de baile ofrecido durante el año académico 2007-2008. El curso enseña diferentes estilos de baile como el clásico, el jazz y el contemporáneo. Los estudiantes aprenderán las técnicas básicas de cada estilo a través de clases prácticas y teóricas durante el año.
Festa Fi De Curs
La pandemia de COVID-19 ha tenido un impacto significativo en la economía mundial. Muchos países experimentaron fuertes caídas en el PIB y aumentos en el desempleo debido a los cierres generalizados. A medida que se levantan las restricciones, la recuperación económica será gradual a medida que los consumidores y las empresas se readaptan a la "nueva normalidad".
Excursió Bcn
A group of students took an excursion to Barcelona from May 29th to May 30th, 2007. The trip included visits to important landmarks like La Sagrada Familia church and Park Güell, as well as time for shopping and exploring the city. The students learned about Barcelona's history and culture during their educational trip.
Edificis Romans
Los edificios romanos más importantes incluyen arcos de triunfo, acueductos, templos, foros, teatros, circos, anfiteatros, coliseos y termas. También construyeron murallas defensivas alrededor de ciudades como Tarragona y León.
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Els Castanyers I Castanyeres
- 1. ELS CASTANYERS I CASTANYERES Curs 2007-08