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Hand in 3 problems; do as many as you can.
Any problem may use the previous problems as established fact.
\(
\def\Ucal{\mathscr{U}}
\def\Vbf{\mathbf{V}}
\def\Ult{\mathrm{Ult}}
\def\Mbf{\mathbf{M}}
\def\jbf{\mathbf{j}}
\def\Lbf{\mathbf{L}}
\def\Pcal{\mathscr{P}}
\)
A \(\Sigma_1\)-formula is a formula in the language of set theory of the form
\(\exists u \phi(u,\bar v)\) where \(\phi\) is quantifier free.
A \(\Sigma^1_2\)-formula is a formula of the form
\[
\exists \bar u \in \Pcal(\omega)^m \forall \bar v \in \Pcal(\omega)^n \phi(\bar u,\bar v,\bar w)
\]
where all quantification in \(\phi\) ranges over \(\omega\).
-
Prove that if \(\phi (\bar v)\) is a \(\Sigma_1\)-formula, then there is a \(\Sigma^1_2\)-formula \(\psi(\bar v)\)
such that whenever \(\bar x\) is a tuple of subsets of \(\omega\) of the appropriate length, \(\psi(\bar x)\) is equivalent to
\(H_{\omega_1} \models \phi(\bar x)\).
Let \(\mu\) be the Borel measure on \(\Pcal(\omega)\) determined by declaring \(\mu(\{A \subseteq \omega : n \in A\}) = 1/2\) for each \(n \in \omega\).
-
Show that membership to \(\Pcal(\omega) \cap \Lbf\) and
\(<_\Lbf \restriction \Pcal(\omega) \cap \Lbf\) are expressible by a \(\Sigma^1_2\)-formulas.
Show that if \(\Pcal(\omega) \cap \Lbf\) is not measure 0, then \(<_\Lbf \restriction \Pcal(\omega) \cap \Lbf\) is a
nonmeasurable \(\Sigma^1_2\)-set.
Hint: use Fubini's Theorem.
Remark: The map \(A \mapsto \sum_{n \in A} 2^{-n}\) is a measure preserving map from \((\Pcal(\omega),\mu)\) to
\(([0,1],\lambda)\) where \(\lambda\) is Lebesgue measure.
Moreover this map is one-to-one except on a countable (hence measure 0) set.
If \(\Ucal\) is an ultrafilter on a set \(I\), define a class relations \(\in_{\Ucal}\) and \(=_\Ucal\)
on all functions with \(I\) by
\[
f =_\Ucal g \textrm{ if and only if } \{i \in I : f(i) = g(i) \} \in \Ucal
\]
\[
f \in_\Ucal g \textrm{ if and only if } \{i \in I : f(i) \in g(i) \} \in \Ucal
\]
Let \(\operatorname{Ult}(\mathbf{V},\Ucal)\) denote the set of \(=_\Ucal\) equivalence classes of elements
of \(\mathbf{V}^I\) equipped with the binary relation which \(\in_\Ucal\) induces on \(=_\Ucal\)-classes.
Los's Theorem implies that the map from \(\mathbf{V}\) into \(\operatorname{Ult}(\mathbf{V},\Ucal)\)
which sends \(x\) to (the equivalence class of) the function which is constantly \(x\) is an elementary embedding –
it preserves the truth of first order formulas.
In particular \(\operatorname{Ult}(\mathbf{V},\Ucal)\) is a model of ZFC.
-
Show that \(\Ucal\) is closed under countable intersections if and only if \(\in_\Ucal\) is well-founded.
Conclude that if there is an ultrafilter \(\Ucal\) is closed under countable intersections and is nonprinciple, then there is a transitive class \(\Mbf\)
which contains the ordinals
and an elementary embedding \(\jbf:\Vbf \to \Mbf\) which is not the identity (here \(\jbf\) is a class function).
-
Show that if \(\jbf:\Vbf \to \Mbf\) is a nontrivial elementary embedding into a transitive class \(\Mbf\),
then there is an ordinal \(\kappa\) such that \(\jbf(\kappa) \ne \kappa\).
This ordinal is called the critical point of \(\jbf\).
-
Show that if \(\jbf:\Vbf \to \Mbf\) is an elementary embedding with critical point \(\kappa\), then
\(\{U \subseteq \kappa : \kappa \in \jbf(U)\}\) is a countably complete ultrafilter on \(\kappa\).
Moreover, show that if \(U \in \Ucal\) and \(r:U \to \kappa\) is such that \(r(\alpha) < \alpha\) for all
\(\alpha \in U\),
then \(r\) is constant on a set in \(\Ucal\).
An ultrafilter with this property is said to be normal.
-
Show that the critical point \(\kappa\) of an elementary embedding \(\jbf :\Vbf \to \Mbf\)
is a strongly inaccessible cardinal.
Show moreover that the set of all \(\lambda \in \kappa\) such that \(\lambda\) is a strongly inaccessible
cardinal is stationary in \(\kappa\).
A cardinal which is the critical point of an elementary embedding \(\jbf : \Vbf \to \Mbf\) is called a
measurable cardinal.
-
Show that \(\Lbf\) satisfies ``there are no measurable cardinals.''