diff --git a/iii/forcing/01.tex b/iii/forcing/01.tex new file mode 100644 index 0000000..b85ac89 --- /dev/null +++ b/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} diff --git a/iii/forcing/main.tex b/iii/forcing/main.tex index 291cab8..1674e8a 100644 --- a/iii/forcing/main.tex +++ b/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} diff --git a/iii/lc/01.tex b/iii/lc/01.tex new file mode 100644 index 0000000..74e308b --- /dev/null +++ b/iii/lc/01.tex @@ -0,0 +1,90 @@ +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} diff --git a/iii/lc/main.tex b/iii/lc/main.tex index d89cd62..6281ac2 100644 --- a/iii/lc/main.tex +++ b/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} diff --git a/util.sty b/util.sty index 20b14a5..e6217ac 100644 --- a/util.sty +++ b/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}}