Lectures 02

iii/forcing/01.tex

@@ -0,0 +1,98 @@
+\subsection{???}
+Independence results are found across mathematical disciplines.
+\begin{enumerate}
+    \item The \emph{parallel postulate} is independent from the other four postulates of Euclidean geometry.
+    It states that for any given point not on a line, there is a unique line passing through that point that does not intersect the given line.
+    In the 19th century, it was shown that the other four postulates are satisfied by hyperbolic geometry, but this postulate is not satisfied.
+    This shows that the other four axioms are insufficient to prove the parallel postulate.
+    \item Let \( \varphi \) be the statement in the language of fields describing the existence of a square root of 2.
+    We know that \( \mathbb Q \) is a field satisfying \( \neg\varphi \), and \( \mathbb Q[\sqrt{2}] \) satisfies \( \varphi \).
+    The fields \( \mathbb Q \) and \( \mathbb Q[\sqrt{2}] \) are models of the theory of fields, one of which satisfies \( \varphi \), and one of which satisfies \( \neg\varphi \).
+    This shows that the theory of fields does not prove \( \varphi \) or \( \neg\varphi \).
+    A similar result holds for the statement \( \varphi \) that says that there are no roots of \( x^4 = -1 \).
+    \item G\"odel's incompleteness theorem implies that there must always be an independence result in a sufficiently powerful consistent set theory.
+\end{enumerate}
+We will show that there are other independence results in set theory that are not self-referential like the G\"odel incompleteness theorems.
+\begin{theorem}[Cantor]
+    \( \abs{\mathbb N} < \abs{\mathcal P(\mathbb N)} \).
+\end{theorem}
+The continuum hypothesis is that there are no sets of cardinality strictly between \( \abs{\mathbb N} \) and \( \abs{\mathcal P(N)} = \abs{\mathbb R} \).
+\begin{definition}
+    The \emph{continuum hypothesis} \( \mathsf{CH} \) states that if \( X \subseteq \mathbb P(\mathbb N) \) is infinite, then either \( \abs{X} = \abs{\mathbb N} \) or \( \abs{X} = \abs{\mathcal P(\mathbb N)} \), or equivalently,
+    \[ 2^{\aleph_0} = \aleph_1 \]
+\end{definition}
+Progress was made on the continuum hypothesis in the 19th and 20th centuries.
+\begin{enumerate}
+    \item In 1883, Cantor showed that any closed subset of \( \mathbb R \) satisfies \( \mathsf{CH} \).
+    \item In 1916, Alexandrov and Hausdorff showed that any Borel set of \( \mathbb R \) satisfies \( \mathsf{CH} \).
+    \item In 1930, Suslin strengthened this result to analytic subsets of \( \mathbb R \).
+    \item In 1938, G\"odel showed that if \( \mathsf{ZF} \) is consistent, then so is \( \mathsf{ZFC} + \mathsf{CH} \).
+    \item However, in 1963, Cohen showed that if \( \mathsf{ZF} \) is consistent, then so is \( \mathsf{ZFC} + \neg\mathsf{CH} \).
+\end{enumerate}
+In this course, we will prove results (iv) and (v), thus establishing the independence of the continuum hypothesis from \( \mathsf{ZFC} \).
+
+\subsection{Systems of set theory}
+The language of set theory \( \mathcal L = \mathcal L_\in \) is a first-order predicate logic with equality and membership as primitive relations.
+We assume the existence of infinitely many variables \( v_1, v_2, \dots \) denoting sets.
+We will only use the logical connectives \( \vee \) and \( \neg \) as well as the existential quantifier \( \exists \).
+Conjunction, implication, and universal quantification can be defined in terms of disjunction, negation, and existential quantification.
+
+We say that an occurrence of a variable \( x \) is \emph{bound} in a formula \( \varphi \) if is in a quantifier \( \exists x \) or lies in the scope of such a quantifier.
+An occurrence is called \emph{free} if it is not bound.
+We write \( FV(\varphi) \) for the set of free variables of \( \varphi \).
+We will write \( \varphi(u_1, \dots, u_n) \) to emphasise the dependence of \( \varphi \) on its free variables \( u_1, \dots, u_n \).
+By doing so, we will allow ourselves to freely change the names of the free variables, and assume that substituted variables are free.
+The syntax \( \varphi(u_0, \dots, u_n) \) does not imply that \( u_i \) occurs freely, or even at all.
+
+The axioms of set theory are as follows.
+% TODO: Add them!
+
+Some common set theories are as follows.
+\begin{itemize}
+    \item \emph{Zermelo set theory} \( \mathsf{Z} \) consists of axioms (i) to (viii).
+    Axioms (ix) and (ix') are equivalent relative to \( \mathsf{Z} \).
+    \item \emph{Zermelo--Fraenkel set theory} \( \mathsf{ZF} \) consists of axioms (i) to (ix).
+    Axioms (x) and (x') are equivalent relative to \( \mathsf{ZF} \).
+    \item \emph{Zermelo--Fraenkel set theory with choice} \( \mathsf{ZFC} \) consists of axioms (i) to (x).
+    \item \emph{Zermelo--Fraenkel set theory without power set} \( \mathsf{ZF}^- \) consists of axioms (i) to (vii), with the axiom of collection (ix') instead of replacement (ix); it has been shown that (ix) is weaker than (ix') in the presence of axioms (i) to (vii).
+    \item \emph{Zermelo--Fraenkel set theory with choice and without power set} \( \mathsf{ZFC}^- \) consists of axioms (i) to (vii), with the axiom of collection (ix') and the well-ordering principle (x').
+\end{itemize}
+In this course, our main metatheory will be \( \mathsf{ZF} \), and we will be explicit about the use of choice.
+
+We say that a class \( X \) is \emph{definable} over \( M \) if there exists a formula \( \varphi \) and sets \( a_1, \dots, a_n \in M \) such that for all \( z \in M \), we have \( z \in X \) if and only if \( \varphi(z, a_1, \dots, a_n) \).
+A class is \emph{proper} over \( M \) if it is not a set in \( M \).
+In this course, we will assume that all mentioned classes are definable.
+For example, the universe class \( V = \qty{x \mid x = x} \), the Russell class \( R = \qty{x \mid x \notin x} \), and the class of ordinals are all definable.
+Any set is a definable class.
+Classes are heavily dependent on the underlying model: if \( M = 2 \) then \( \mathrm{Ord} = 2 = M \), and if \( M = 3 \cup \qty{1} \) then \( \mathrm{Ord} = 3 \neq M \).
+
+\subsection{Adding defined functions}
+Often in set theory, we use symbols such as \( 0, 1, \subseteq, \cap, \wedge, \forall \); they do not exist in our language.
+\begin{definition}
+    Suppose that \( \mathcal L \subseteq \mathcal L' \) and \( T \) is a set of sentences in \( \mathcal L \).
+    We say that \( P \) is a \emph{defined \( n \)-ary predicate} symbol over \( T \) if there is a formula \( \varphi \) in \( \mathcal L \) such that
+    \[ T \vdash \forall x_1, \dots, x_n.\, (P(x_1, \dots, x_n) \iff \varphi(x_1, \dots, x_n)) \]
+    Similarly, we say that \( f \) is a \emph{defined \( n \)-ary function} symbol over \( T \) if there is a formula \( \varphi \) in \( \mathcal L \) such that
+    \[ f(x_1, \dots, x_n) = y \text{ if and only if } T \vdash \varphi(x_1, \dots, x_n, y) \]
+    and
+    \[ T \vdash \forall x_1, \dots, x_n.\, \exists! y.\, \varphi(x_1, \dots, x_n, y) \]
+    We say that a set of sentences \( T' \) of \( \mathcal L' \) is an \emph{extension by definitions} of \( T \) over \( \mathcal L \) when \( T' = T \cup S \) and \( S = \qty{\varphi_s \mid s \in \mathcal L' \setminus \mathcal L'} \) and each \( \varphi_s \) is a definition of \( s \) in the language \( \mathcal L \) over \( T \).
+\end{definition}
+The following, among other things, are defined over \( \mathsf{ZF} \).
+\[ 0 \quad 1 \quad \subseteq \quad \cap \quad \mathcal P \quad \bigcup \]
+\begin{theorem}
+    Suppose that \( \mathcal L \subseteq \mathcal L' \), and that \( T \) is a set of \( \mathcal L \)-sentences and \( T' \) is an extension by definitions of \( T \) to \( \mathcal L' \).
+    Then
+    \begin{enumerate}
+        \item (conservativity) If \( \varphi \) is a sentence of \( \mathcal L \), then \( T \vdash \varphi \iff T' \vdash \varphi \).
+        \item (abbreviations) If \( \varphi \) is a formula of \( \mathcal L' \), then there exists a formula \( \hat\varphi \) of \( \mathcal L \) whose free variables are exactly those of \( \varphi \), such that \( T' \vdash \forall x.\, (\varphi \iff \hat\varphi) \).
+    \end{enumerate}
+\end{theorem}
+\begin{example}
+    The intersection \( a \cap b \) can be defined as the unique set \( c \) such that
+    \[ \forall x\. (x \in c \iff x \in a \wedge x \in b) \]
+    This definition makes sense only if there is a unique \( c \) satisfying this formula \( \varphi(c) \).
+    If
+    \[ M = \qty{a, c, d, \qty{a}, \qty{a, b}, \qty{a, b, c}, \qty{a, b, d}} \]
+    then it is easy to check that both \( \qty{a} \) and \( \qty{a, d} \) satisfy \( \varphi \), so intersection cannot be defined.
+\end{example}

iii/forcing/main.tex

@@ -10,7 +10,7 @@
 
 \tableofcontentsnewpage{}
 
-% \section{Definitions and examples}
-% \input{01_definitions_and_examples.tex}
+\section{???}
+\input{01.tex}
 
 \end{document}

iii/lc/01.tex

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+Modern set theory largely concerns itself with the consequences of the incompleteness phenomenon.
+Given any `reasonable' set theory \( T \), then G\"odel's first incompleteness theorem shows that there is \( \varphi \) such that \( T \nvdash \varphi \) and \( T \nvdash \neg\varphi \).
+To be `reasonable', the set of axioms must be computably enumerable, among other similar restrictions.
+In particular, G\"odel's second incompleteness theorem shows that \( T \nvdash \Con(T) \), where \( \Con(T) \) is the statement that \( T \) is consistent.
+Hence,
+\[ \qty{\psi \mid T \vdash \psi} \subsetneq \qty{\psi \mid T + \varphi \vdash \psi} \]
+We might say
+\[ T <_{\text{consequence}} T + \varphi \]
+so \( T \) has strictly fewer consequences than \( T + \varphi \).
+Modern set theory is about understanding the relation \( \leq_{\text{consequence}} \) and other similar relations.
+It turns out that large cardinal axioms are the most natural hierarchy that we can use to measure the strength of set theories.
+
+In this course we will not provide a definition for the notion of `large cardinal', but we will provide an informal description.
+A \emph{large cardinal property} is a formula \( \Phi \) such that \( \Phi(\kappa) \) implies that \( \kappa \) is a very large cardinal, so large that its existence cannot be proven in \( \mathsf{ZFC} \).
+A \emph{large cardinal axiom} is an axiom of the form \( \exists \kappa.\, \Phi(\kappa) \), which we will abbreviate \( \Phi \mathrm{C} \).
+We begin with some non-examples.
+
+\begin{enumerate}
+    \item \( \kappa \) is called an \emph{\( \aleph \) fixed point} if \( \kappa = \aleph_\kappa \).
+    Note that, for example, \( \omega \), \( \omega_1 \), and \( \aleph_\omega \) are not \( \aleph \) fixed points.
+    However, we have the following result.
+    We say that \( F : \mathrm{Ord} \to \mathrm{Ord} \) is \emph{normal} if \( \alpha \leq \beta \) implies \( F(\alpha) \leq F(\beta) \), and if \( \lambda \) is a limit, \( F(\lambda) = \bigcup_{\alpha < \lambda} F(\alpha) \).
+    One can show that every normal ordinal operation has arbitrarily large fixed points, and in particular that these fixed points may be enumerated by the ordinals.
+    In particular, since the operation \( \alpha \mapsto \aleph_\alpha \) is normal, it admits fixed points.
+    \item Let \( \Phi(\kappa) \) be the property
+    \[ \kappa = \aleph_\kappa \wedge \Con(\mathsf{ZFC}) \]
+    Clearly \( \Phi \mathrm{C} \) implies \( \Con(\mathsf{ZFC}) \), so \( \mathsf{ZFC} \nvdash \Phi \mathrm{C} \).
+    We would like our large cardinal axioms to be unprovable by \( \mathsf{ZFC} \) because of their size, not because of any other arbitrary reasons that we may attach to these axioms.
+\end{enumerate}
+
+One source of large cardinal axioms is as follows.
+Consider the ordinal \( \omega \); it is much larger than any ordinal smaller than it.
+We can consider properties that encapsulate the notion that \( \omega \) is much larger than any smaller ordinal, and use these as large cardinal properties.
+
+\begin{enumerate}
+    \item If \( n < \omega \), then \( n^+ < \omega \), where \( n^+ \) is the cardinal successor of \( n \).
+    We define
+    \[ \Lambda(\kappa) \iff \forall \alpha (\alpha < \kappa \to \alpha^+ < \kappa) \]
+    where \( \alpha^+ \) is the least cardinal strictly larger than \( \alpha \).
+    Then, \( \Lambda(\kappa) \) holds precisely when \( \kappa \) is a limit cardinal.
+    These need not be very large, for example, \( \aleph_\omega \) is a limit cardinal, and the existence of this cardinal is proven by \( \mathsf{ZFC} \).
+    \item If \( n < \omega \), then \( 2^n < \omega \), where \( 2^n \) is the size of the power set of \( n \).
+    \[ \Sigma(\kappa) \iff \forall \alpha (\alpha < \kappa \to 2^\alpha < \kappa) \]
+    where \( 2^\alpha \) is the cardinality of \( \mathcal P(\alpha) \).
+    Such cardinals are called \emph{strong limit cardinals}.
+    We will show that these exist in all models of \( \mathsf{ZFC} \).
+    Similarly to the aleph hierarchy, we can define the \emph{beth} hierarchy, denoted \( \beth_\alpha \).
+    This is given by
+    \[ \beth_0 = \aleph_0;\quad \beth_{\alpha + 1} = 2^{\beth_\alpha};\quad \beth_{\lambda} = \bigcup_{\alpha < \lambda} \beth_\alpha \]
+    Cantor's theorem shows that \( \aleph_\alpha \leq \beth_\alpha \), and the continuum hypothesis is the assertion that \( \aleph_1 = \beth_1 \).
+    Note that \( \kappa \) is a strong limit cardinal if and only if \( \kappa = \beth_\lambda \) for some limit ordinal \( \lambda \).
+    In particular, \( \mathsf{ZFC} \vdash \Sigma \mathrm{C} \).
+    \item If \( s : n \to \omega \) for \( n < \omega \), then \( \sup(s) = \bigcup \operatorname{ran}(s) < \omega \).
+    This gives rise to the following definition.
+    \begin{definition}
+        Let \( \lambda \) be a limit ordinal.
+        We say that \( C \subseteq \lambda \) is \emph{cofinal} or \emph{unbounded} if \( \bigcup C = \lambda \).
+        We define the \emph{cofinality} of \( \lambda \), denoted \( \cf(\lambda) \), to be the cardinality of the smallest cofinal subset.
+        If \( \lambda \) is a cardinal, then \( \cf(\lambda) \leq \lambda \).
+        If this inequality is strict, the cardinal is called \emph{singular}; if this is an equality, it is called \emph{regular}.
+    \end{definition}
+    Then \( \omega \) is a regular cardinal.
+    Note that \( \aleph_1 \) is also regular, since countable unions of countable sets are countable.
+    This argument generalises to all succcessor cardinals, so all successor cardinals \( \aleph_{\alpha + 1} \) are regular.
+    The cardinal \( \aleph_\omega \) is not regular, as it is the union of \( \qty{\aleph_n \mid n \in \mathbb N} \), which is a subset of \( \aleph_\omega \) of cardinality \( \aleph_0 \), giving \( \cf(\aleph_\omega) = \aleph_0 \).
+    Note also that the cofinality of \( \aleph_{\aleph_\omega} \) is also \( \aleph_0 \).
+    Limit cardinals are often singular.
+\end{enumerate}
+
+Motivated by these examples of properties of \( \omega \), we make the following definition.
+
+\begin{definition}
+    A cardinal \( \kappa \) is called \emph{weakly inacessible} if it is an uncountable regular limit, and \emph{(strongly) inaccessible} if it is an uncountable regular strong limit.
+    We write \( \operatorname{WI}(\kappa) \) to denote that \( \kappa \) is weakly inaccessible, and \( \operatorname{I}(\kappa) \) if \( \kappa \) is inaccessible.
+\end{definition}
+
+To argue that these are large cardinal properties, we will show that they are very large, and that the existence of such cardinals cannot be proven in \( \mathsf{ZFC} \).
+Note that we cannot actually prove this statement; if \( \mathsf{ZFC} \) were inconsistent, it would prove every statement.
+Whenever we write statements such as \( \mathsf{ZFC} \nvdash \mathrm{WIC} \), it should be interpreted to mean `if \( \mathsf{ZFC} \) is consistent, it does not prove \( \mathrm{IC} \)'.
+
+\begin{proposition}
+    Weakly inaccessible cardinals are aleph fixed points.
+\end{proposition}
+\begin{proof}
+    Suppose \( \kappa \) is weakly inacessible but \( \kappa < \aleph_\kappa \).
+    Fix \( \alpha \) such that \( \kappa = \aleph_\alpha \), then \( \alpha < \kappa \).
+    As \( \kappa \) is a limit cardinal, \( \alpha \) must be a limit ordinal.
+    But then \( \aleph_\alpha = \bigcup_{\beta < \alpha} \aleph_\beta \), so in particular, the set \( \qty{\aleph_\beta \mid \beta < \alpha} \) is cofinal in \( \kappa \), but this set is of size \( \abs{\alpha} < \kappa \).
+    Hence \( \kappa \) is singular, contradicting regularity.
+\end{proof}

iii/lc/main.tex

@@ -10,7 +10,7 @@
 
 \tableofcontentsnewpage{}
 
-% \section{Definitions and examples}
-% \input{01_definitions_and_examples.tex}
+\section{???}
+\input{01.tex}
 
 \end{document}

util.sty

@@ -221,6 +221,8 @@
 \DeclareMathOperator{\mSpec}{mSpec}
 \DeclareMathOperator{\Proj}{Proj}
 \DeclareMathOperator{\res}{res}
+\DeclareMathOperator{\Con}{Con}
+\DeclareMathOperator{\cf}{cf}
 % https://github.com/wspr/unicode-math/issues/457
 \AtBeginDocument{%
 \newcommand{\dashrightarrow}{\mathrel{\rightdasharrow}}