Description:
In this note we prove that the Boolean
Prime Ideal Theorem (Form 14) is true in \(\cal N29\).
Content:
In this note we prove that the Boolean
prime ideal theorem (Form 14) is true in
\(\cal N29\).
We shall use Blass' theorem (Blass [1986], Theorem 2)
which characterizes Fraenkel-Mostowski models in which \(BPI\) is true.
According to this theorem \(BPI\) holds in a Fraenkel-Mostowski model if and
only if the model is determined by a filter of subgroups \(\Gamma\) with
the Ramsey property described below.
Definition
- A subgroup \(H\) of a group \(G\) is a Ramsey subgroup if for every finite \(F\subseteq G/H\) (the set of
left cosets of \(H\) in \(G\)) there is a finite \(Y\subseteq G/H\) such
that whenever \(Y\) is partitioned into two pieces there is a \(g\in G\)
such that \(gF\) is included in one of the pieces.
- A filter \(\Gamma\) of subgroups of \(G\) has the Ramsey
property if it has a basis consisting of Ramsey subgroups of \(G\).
The Boolean Prime Ideal Theorem (Form 14 (\(BPI\))) is true in
\(\cal N29\).
Let \(E\) be a support. By Blass' theorem it suffices to show that
fix\(_G(E)\) is a Ramsey subgroup of \(G\). We shall argue that this
follows from Rado's corollary which is given in Halpern [1964],
page 61. Rado's corollary is
(Rado's Corollary) Assume that \(n, p\in\omega\) and
that \(\langle m_i \rangle_{i\in n} \in \omega^n\). Then there is
a \(q\in\omega\) such that for all sequences \(\langle X_i\rangle_{i\in n}\)
of disjoint sets such that \(|X_i|\ge q\) for every \(i\in n\), if \(\Delta\)
is a partition of \(\{Z \subseteq \bigcup_{i\in n} X_i : (\forall i\in n)
(|Z\cap X_i| = m_i)\}\) into two pieces there then is a sequence
\(\langle Y_i\rangle_{i\in n}\) such that for all \(i\in n\),
\(Y_i \subset X_i\) and \(|Y_i| = p\) and
\(\{Z\subseteq \bigcup_{i\in n} Y_i : (\forall i\in n)
(|Z\cap Y_i| = m_i)\}\) is a subset of one of the pieces.
Let \(F = \{ \phi_1\circ \hbox{fix}_G(E),\ldots,
\phi_k\circ \hbox{fix}_G(E) \}\)
be a finite subset of \(G/\hbox{fix}_G(E)\). We will find a \(Y\)
satisfying the condition from definition 1 above. We first note that
it is easy to verify that
\begin{align}
(\forall \phi, \psi \in G)(\phi\circ \hbox{fix}_G(E) = \psi\circ
\hbox{fix}_G(E) \Leftrightarrow \phi(E) = \psi(E))\tag{1}
\end{align}
Let \(E' = \phi_1(E) \cup \cdots \cup \phi_k(E)\). \(E\) and \(E'\) are
finite and therefore there are natural numbers \(j_0,\ldots,j_{n-1}\)
such that \(E\cup E'\subseteq B_{j_0}\cup\cdots\cup B_{j_{n-1}}\).
For \(i\in n\), define \(E_i = B_{j_i}\cap E\), \(E'_i = B_{j_i}\cap E'\),
\(m_i = |E_i|\) and let \(p = \max\{|E'_i| :i\in n\}\). The integers \(n\),
\(p\) and the sequence \(\langle m_i \rangle_{i\in n} \in \omega^n\)
satisfy the hypothesis of Rado's corollary. Therefore there is
a \(q\in\omega\) satisfying the conclusion.
For \(i\in n\), choose a set \(X_i \subseteq B_{j_i}\) so that
\(|X_i| = q\). Let \(Y = \{\phi\circ \hbox{fix}_G(E) : \phi(E)
\subseteq \bigcup_{i\in n} X_i\}\). We now show that this \(Y\) satisfies
the condition of definition 1.
Let \(\Delta = \{\Delta_1,\Delta_2\}\) be a partition of \(Y\) into two
pieces. Let \(\Gamma_1 = \{\phi(E) : \phi\circ\hbox{fix}_G(E)\in
\Delta_1\}\) and \(\Gamma_2 = \{\phi(E) : \phi\circ\hbox{fix}_G(E)\in
\Delta_2\}\). By (1), \(\Gamma = \{\Gamma_1,\Gamma_2\}\) is a partition of
\(\{Z\subseteq \bigcup_{i\in n} X_i : (\forall i\in n)
|Z \cap X_i| = m_i\}\). By Rado's corollary, there is a sequence
\(\langle Y_i\rangle_{i\in n}\) such that \(|Y_i| = p\) for \(i\in n\) and
\(W = \{Z\subseteq \bigcup_{i\in n} Y_i : (\forall i\in n)
(|Z\cap Y_i| = m_i\}\) is either a subset of \(\Gamma_1\) or
a subset of \(\Gamma_2\).
We assume without loss of generality that \(W \subseteq \Gamma_1\).
Let \(\eta\in G\) be a permutation such that \(\eta(E'_i)
\subseteq Y_i\) for \(i\in n\). (This is possible since \(|Y_i| = p \ge
|E'_i|\).) Then (since \(\phi_t(E_i)\subseteq
E'_i\) for \(1\le t\le k\)) \(\eta\phi_t(E_i) \subseteq Y_i\) for \(i\in n\)
and \(1\le t\le k\). Therefore, for \(1\le t\le k\), \(\eta\phi_t(E)
\subseteq\bigcup_{i\in n} Y_i\). Using this and the fact that
\(|\eta\phi_t(E)\cap Y_i| = |\eta\phi_t(E_i)| =
|E_i| = m_i\) for \(1\le t\le k\) and \(i\in n\) we conclude that
\(\eta\phi_t(E)\in\Gamma_1\). Hence \(\eta\phi_t\circ\hbox{fix}_G(E) \in
\Delta_1\).
Howard-Rubin number:
136
Type:
proof of result
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