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159 lines
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159 lines
5.5 KiB
TeX
\lecture{21}{2024-01-18}{}
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\begin{goal}
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We want to show that certain statements are consistent with $\ZFC$
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(or $\ZF$), for instance $\CH$.
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We start with a model $M$ of $\ZFC$.
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Usually we want $M$ to be transitive.
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We want to enlarge $M$ to get a bigger model,
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where our desired statement holds,
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i.e.~add more reals to violate $\CH$.
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However we need to do this in a somewhat controlled way,
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so we can't just do it the way one builds field extensions.
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In particular, when trying to violate $\CH$ we need to make sure that
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we don't collapse cardinals.
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\end{goal}
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\begin{remark}
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The idea behind forcing is clever.
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Unfortunately an easy ``how could I have come up with this myself''-approach
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does not seem to exist.
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\end{remark}
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\begin{remark}
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How can a countable transitive model $M$ even exist?
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$M$ believes some statements that are wrong from the outside perspective.
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For example there exists $\aleph_1^M \in M$
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such that $M \models x = \aleph_1$.
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$\aleph_1^M$ is indeed an ordinal (since being an ordinal is a $\Sigma_0$-statement).
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However $\aleph_1^M$ is countable, since $M$ is countable
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and transitive.
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This is fine.
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(Note that ``$\aleph_1^M$ is uncountable'' is a $\Pi_1$-statement.)
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\end{remark}
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\begin{idea}[The method of \vocab{forcing}]
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Start with $M$, a countable transitive model of $\ZFC$
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and let $\mathbb{P} \in M$ be a partial order,
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where $p \le q$ means that $p$ has ``more information''
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than $q$.
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A filter $g \subseteq \mathbb{P}$ is $\mathbb{P}$-generic over $M$
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iff $g \cap D \neq \emptyset$ for all dense $D \subseteq \mathbb{P}$,
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$D \in M$.
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Next steps:
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\begin{enumerate}[(1)]
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\item Define the \vocab{forcing extension} $M[g]$.
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\item Show that $M[g] \models \ZFC$.
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\item Determine other facts about (the theory of) $M[g]$.
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This depends on the partial order $\mathbb{P}$ we chose
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in the beginning (and maybe $M$).
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\end{enumerate}
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\end{idea}
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\begin{example}[Prototypical example]
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Let $\bP = 2^{< \omega}, p \le q \mathop{:\iff} p \supseteq q$ be Cohen forcing,
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often denoted $\bC$.
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Let $M$ be a countable transitive model of $\ZFC$.
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Since the definition of $\bC$ is simple enough,
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$\bC \in M$.
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Let $g$ be $\bC$-generic over $M$.
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\begin{claim}
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\label{ex:cohen:c1}
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For each $n \in \omega$,
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the set $D_n \coloneqq \{ p \in \bC : n \in \dom(p)\}$
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is dense.
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\end{claim}
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\begin{subproof}
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This is trivial.
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\end{subproof}
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\begin{claim}
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\label{ex:cohen:c2}
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$D_n \in M$.
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\end{claim}
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\begin{subproof}
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The definition of $D_n$ is absolute.
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\end{subproof}
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\begin{claim}
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\label{ex:cohen:c3}
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If $p,q \in g \cap D_n$,
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then $p(n) = q(n)$.
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\end{claim}
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\begin{subproof}
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$g$ is a filter, so $p$ and $q$ are compatible.
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$p,q \in D_n$ makes sure that $p(n)$ and $q(n)$ are defined.
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\end{subproof}
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Let $x = \bigcup g$.
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By \yaref{ex:cohen:c3}, $x \in 2^{\le \omega}$.
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By \yaref{ex:cohen:c1} and \yaref{ex:cohen:c2},
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we have $g \cap D_n \neq \emptyset$ for all $n < \omega$,
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hence $n \in \dom(x)$ for all $n < \omega$.
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So $x \in 2^{\omega}$.
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\begin{claim}
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Let $z \in 2^{\omega}$, $z \in M$.
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Then $D^z = \{p \in \bC : \exists n \in \dom(p) .~p(n) \neq z(n)\} $
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is dense.
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\end{claim}
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\begin{subproof}
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Trivial.
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\end{subproof}
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\begin{claim}
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$D^z \in M$ for all $z \in 2^{< \omega}$ with $z \in M$.
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Therefore, $g \cap D^z \neq \emptyset$ for all $z \in M$,
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$z\colon 2^{<\omega}$.
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Hence $x \neq z$ for all $z \in M$, $z \in 2^{< \omega}$.
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In other words $x \not\in M$.
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\end{claim}
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The new real $x$ does not do too much damage to $M$
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when adding it.\footnote{We still need to make this precise.}
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(Some reals would completely kill the model.)
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Now let $\alpha$ be an ordinal in $M$.
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Let
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\begin{IEEEeqnarray*}{rCll}
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\bC(\alpha) &\coloneqq& \{p \colon &\text{$p$ is a function with domain $\alpha$,}\\
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&&&\text{$p(\xi) \in \bC$ for all $\xi < \alpha$,}\\
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&&&\text{$\{\xi < \alpha : p(\xi) \neq \emptyset\}$ is finite}\}
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\end{IEEEeqnarray*}
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($\alpha$ many copies of $\bC$ with \vocab{finite support}).
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For $p, q \in \bC(\alpha)$ define $p \le q :\iff \forall \xi < \alpha .~p(\xi) \supseteq q(\xi)$.
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We have $\bC(\alpha) \in M$
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Let $g$ be $\bC(\alpha)$-generic over $M$.
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Let $x_\xi = \bigcup \{p(\xi) : p \in g\}$
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for $\xi < \alpha$.
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$x_\xi \in 2^{ \omega}$:
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For each $n < \omega$ and $\xi < \alpha$,
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\[
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D_{n,\xi} \coloneqq \{ p \in \bC(\alpha) : n \in \dom(p(\xi))\} \in M
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\]
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and $D_{n,\xi}$ is dense.
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\begin{claim}
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For all $\xi, \eta < \alpha$, $\xi \neq \eta$,
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\[
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D^{\xi, \eta} \coloneqq \{ p \in \bC(\alpha) : \exists n \in \dom(p(\xi)) \cap \dom(p(\eta)) .~
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p(\xi)(n) \neq p(\eta)(n)\}
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\]
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we have that $D^{\xi, \eta} \in M$
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and is $D^{\xi, \eta}$ dense.
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\end{claim}
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Therefore if $\xi \neq \eta$, $x_\xi \neq x_\eta$.
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Currently this is not very exciting,
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since we only showed that for a countable transitive model $M$,
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there is a countable set of reals not contained in $M$.
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The interesting point will be, that we can actually add these reals
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to $M$.
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\end{example}
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Next we want to define $M[g]$.
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