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\begin{document}

\begin{flushright}
  A.Miller\\
  M771 \\
  Spring 2000\\
  http://www.math.wisc.edu/$\sim$miller/m771/logfile.html\\
\end{flushright}


\prob  \thm (Zermelo) The axiom of choice AC is equivalent to
every set can be well-ordered WO.

\prob  \thm (Shroeder,Bernstein) If there are one-to-one maps
taking $X$ to $Y$ and $Y$ to $X$, then there is a bijection taking $X$ to $Y$.

\prob  \thm $V_{\om+\om}\models$Z.

\prob  \thm  $\om$ is an ordinal.

\prob  \thm Every well-ordering is isomorphic to a unique ordinal.

\prob  \thm For any set $X$ its transitive closure exists.

\prob  \thm There exists a Bernstein set.

\prob  (Given $G:V\to V$ any class function)
\par\noindent  \thm There exists a unique
$F:V\to V$ such that $F(\al)=G(F\res\al)$ for every ordinal $\al$.

\prob  \thm Axiom of Foundations is equivalent to every
set $x$ is in some $V_\al$.

\prob  \thm $cof(\ka^+)=\ka^+$.

\prob  \thm (K\"onig) $\ka^{cof(\ka)}>\ka$.

\prob  \thm The Downward going Lowenheim-Skolem-Tarski theorem.

\prob  \thm (Vaught) Reflection Schema.

\prob  \thm  $(X,E)$ is a well-founded
relation iff there exists a rank function $$r:X\to ORD$$

\prob  \thm (Mostowski Collapse) If $(X,E)$ is a well-founded
extensional relation, then there exists a transitive set $M$ and
an isomorphism $j:(X,E)\to(M,\in)$.

\prob \thm $\Delta_0$ formulas are absolute for transitive models.

\prob \thm $\Delta_1$ formulas are absolute for transitive models
of ZFC.

\prob Lemma. $N=Def(M)$ is $\Delta_1$ - ZF.

\prob \thm (G\"odel 1938)  $L\models$ZF+AC+GCH.
Con(ZF)$\to$Con(ZFC+GCH)

\prob  (For any large enough finite fragment of ZFC, ZFC$_n$,)
\par\noindent Lemma. If $M$ is a standard transitive model of ZFC$_n$+V=L,
then $M=L_\de$ for some ordinal $\delta$.

\prob \thm  ZF has a well-founded model iff it has a countable
transitive standard model.  If ZF has a model, it has one in which there
is no transitive standard model.

\prob  \thm (Kurepa) there exists a Souslin line iff there
exists a Souslin tree.

\prob  Lemma. If $I_\al$ for $\al<\om_1$ are intervals constructed
in a Souslin line so that $I_\al$ avoids all the endpoints of
$I_\be$ for $\be<\al$, then reverse inclusion on these intervals
makes them a Souslin tree.

\prob  Lemma. If there exists a Souslin tree, then there exists
a nice Souslin subtree of $\qq^{<\om_1}$.

\prob  Lemma. The lexicographical order on the maximal chains
of a nice tree is a Souslin line.

\prob \thm (Tarski) Elementary Chains of substructures.

\prob \thm (Fodor) Push down lemma.

\prob \thm (Jenson) $\Diamond_{\om_1}$ implies there exists a nice
Souslin tree.

\prob \thm (Jenson) V=L implies $\Diamond_{\om_1}$.

\prob \thm For every $p\in\pp$ there exists $G$ which is
$\pp$-generic over $M$ and $b\in\pp$.

\prob \thm $M[G]$ is transitive, $M\su M[G]$, $G\in M[G]$,
rank($M[G]$)=rank($M$).

\prob \thm Given the Truth and Definability Lemma, $M[G]$ is
a model of ZFC.

\prob \thm  ccc forcing preserves cofinalities and cardinals.

\prob \thm $\Delta$-systems lemma.

\prob \thm FIN$(\ka,2)$ has ccc. Forcing with it gives
model of notCH.


\prob \thm $G$ $\pp$-generic over $M$ iff it is a $\pp$-filter which
meets every maximal antichain in $M$.

\prob \thm Generic filters are maximal compatible sets.

\prob How to remove the well-foundedness assumption from forcing
proofs, e.g., reflection, forcing over nonstandard models, boolean
value models, syntatical methods.

\prob (MA$_{\om_1}$) No Souslin trees, notCH, for any
$\{f_\al:\al<\om_1\}\su \om^\om$
there exists $f\in\om$ such that $f_\al\leq^* f$ for all $\al$.

\prob (MA$_{\om_1}$) implies $2^\om=2^{\om_1}$. (almost disjoint families)

\prob (MA$_{\om_1}$) additivity of measure, category.  ccc is productive,
property K.  Aronszahn trees are special.


\prob Con(ZFC) $\to$ Con(ZFC+MA$_{\om_1}$).

\end{document}