Description:
In this note we prove that in \(\cal N41\),
Form 9 (Dedekind finite and finite are equivalent) is true and
\(C(WO,< \aleph_0)\) is true.
Content:
In this note we prove that in \(\cal N41\),
Form 9 (Dedekind finite and finite are equivalent) is true and
\(C(WO,< \aleph_0)\) is true.
Assume that \(\psi\) is an order automorphism of
\(({\Bbb Q},\le)\) with the property that \(\forall a\in {\Bbb Q}\),
\(\lim_{n\to\infty}\psi^n(a) = \infty\). (That is, \((\forall a\in {\Bbb Q})
(\forall b\in {\Bbb Q})(\exists n \in \omega)(\psi^n(a) \ge b)\).)
Then there is an order automorphism \(\eta\) of \(({\Bbb Q},\le)\) such that
for all \(a\in\Bbb Q\), \(\eta\psi\eta^{-1}(a) = a+1\).
It follows from the hypotheses that \(\psi^n(0) <
\psi^{n+1}(0)\) for each \(n\in\omega\)
and that \({\Bbb Q}\) is the disjoint union \({\Bbb Q} = \bigcup_{n\in\
{\Bbb Z}}[\psi^n(0),\psi^{n+1}(0))\).
Let \(\sigma\) be an order isomorphism from the interval \([0,\psi(0))\)
onto the interval \([0,1)\). We define the function \(\eta\) on \(\Bbb Q\)
as follows: Assume \(a \in [\psi^n(0),\psi^{n+1}(0))\), then
\(\eta(a) = \sigma(\psi^{-n}(a))+n\).
A straightforward argument shows that \(\eta\mid [\psi^n(0),\psi^{n+1}(0))\)
is an order automorphism from \([\psi^n(0),\psi^{n+1}(0))\) onto \([n,n+1)\)
for all \(n\in \Bbb Z\) and that for \(b\in [n,n+1)\), \(\eta^{-1}(b) =
\psi^n\sigma^{-1}(b-n)\). It follows from the first assertion that
\(\eta\) is an order automorphism of \(\Bbb Q\) and from the second we
can calculate for \(b\in[n,n+1)\):
\begin{eqnarray*}
\eta\psi\eta^{-1}(b) & = & \eta\psi(\psi^n\sigma^{-1}(b-n))\\
& = & \eta(\psi^{n+1}\sigma^{-1}(b-n)) \\
& = & \sigma(\psi^{-(n+1)}(\psi^{n+1}\sigma^{-1}(b-n)))+(n+1) \\
& & (\mathrm{since\;} \psi^{n+1}(\sigma^{-1}(b-n)) \in [\psi^{n+1}(0),\psi^{n+2}(0)))\\
& = & b+1
\end{eqnarray*}
This completes the proof of the lemma.
Any order automorphism \(\psi\) of \((\Bbb Q,\le)\)
satisfying
\[(\forall a\in \Bbb Q)(\forall b\in \Bbb Q)(\exists n\in\omega)(b \le \psi^n(a))\]
has an \(n^{th}\) root for every \(n\in\omega-\{0\}.\)
That is, there is an order automorphism \(\psi'\) of \((\Bbb Q,\le)\)
such that \((\psi')^n = \psi\).
With \(\eta\) as in the lemma, \(\psi = \eta^{-1}\pi
\eta\) where \(\pi(a) = a + 1\). Assume \(n\in\omega-\{0\}\) and
let \(\pi_n(a) = a + \frac{1}{n}\). If we let \(\psi' = \eta^{-1}\pi_n\eta\),
then \((\psi')^n = \psi\).
Any order automorphism \(\psi\) of \((\Bbb Q,\le)\)
which satisfies \((\forall a\in {\Bbb Q})(\forall b\in {\Bbb Q})
(\exists n \in \omega)(\psi^n(a) \le b)\) has an \(n^{th}\) root for all
\(n \in \omega-\{0\}\).
Every order automorphism of \((\Bbb Q,\le)\)
has an \(n^{th}\) root for all \(n\in\omega-\{0\}\).
Let \(\phi\) be an order automorphism of
\((\Bbb Q,\le)\) and for each \(a\in \Bbb Q\) let \([a] =
\{ b\in \Bbb Q: \exists n \in \Bbb Z\) such that \(b\) is between
\(a\) and \(\phi^n(a)\}\). If \(a\) is a fixed point of \(\phi\) then \([a]
=\{ a \}\) otherwise \([a]\) is an open interval in the sense that if
\(c < d < e\) and \(c\) and \(e\) are in \([a]\), then \(d \in [a]\) and \([a]\)
has no first or last element. Further the set \(\{[a]: a\in {\Bbb
Q}\}\) is a partition of \(\Bbb Q\).
If \(a\) is not a fixed point of \(\phi\), then it is also the case that
\(\phi\mid [a]\) is an order automorphism of \([a]\) and
either \((\forall b \in [a])(\forall c \in [a])(\exists n\in\omega)
(c \le \phi^n(b))\) (if \(\phi(a) > a\)) or
\((\forall b \in [a])(\forall c \in [a])(\exists n\in\omega)
(\phi^n(b) \le c)\) (if \(\phi(a) < a\)). Therefore, by the previous two
corollaries \(\phi\mid [a]\) has an \(n^{th}\) root which we denote by
\((\phi\mid [a])^{\frac{1}{n}}\). Then \(\psi'\) defined by
\(\psi'(a) = (\phi \mid [a])^{\frac{1}{n}}(a)\) is an \(n^{th}\) root
of \(\psi\).
It will follow from the lemma below that Form 9 (\(DF = F\)) and
Form 122 (\(C(WO,<\aleph_0)\)) are true in \(\cal N41\).
If \(x \in \cal N41\) has support \(E = B_0 \cup \cdots
\cup B_k\) and there is some \(\phi \in G\) which fixes \(E\) pointwise but
does not fix \(x\) pointwise, then \(x\) has a well orderable, infinite
subset.
Assume the hypotheses. Then there is some
\(y\in x\) and \(\phi\in G\) such that \(\phi\) fixes \(E\) pointwise and
\(\phi(y) \ne y\). If we assume that \(y\) has support
\(E' = (B_0\cup\cdots B_k)\cup(B_{k+1}\cup \cdots \cup B_r)\) then we
may also assume without loss of generality that for some \(i\) with
\(k+1 \le i \le r\), \(\phi\) fixes \(A-B_i\) pointwise.
We note that if \(z\) is any element of \(\cal N41\) of the form
\(z = \psi(y)\) where \(\psi \in G\) and \(\psi\) fixes \(A -B_i\) pointwise,
then \(z\) has support \(E'\) (because \(\psi(E') = E'\)) and \(z\in X\) (since
\(\psi\) fixes \(E\) pointwise).
We intend to show that there are infinitely many such elements \(z\).
Assume not. Let \(W\) be the set of all such \(z\) and assume \(|W| = q\in
\omega\). By the last corollary, the permutation \(\phi\) has an
\(n^{th}\) root for every positive integer \(n\) and we choose an \(n\)
which is divisible by all of the integers \(1, 2, \ldots, q+1\). Let
\(\eta\in G\) satisfy \(\eta^n = \phi\) (and we may assume that \(\eta\)
fixes \(A - B_i\) pointwise). Clearly, for any \(j\), \(1\le j
\le q+1\), \(\eta^j(y) \ne y\) (otherwise \(\eta^n(y) = \phi(y) = y\)).
It follows that for \(1\le j< m \le q+1\), \(\eta^j(y) \ne \eta^m(y)\)
because equality would imply that \(y = \eta^{m-j}(y)\). Therefore,
\(|W|\ge q+1\) a contradiction and \(W\) is infinite.
Since each \(z\in W\) has support \(E'\), \(W\) can be well ordered.
This complete the proof of the lemma.
To show that \(C(WO,< \aleph_0)\) is true in \(\cal N41\) let \(Y\) be a well
ordered set of finite sets in \(\cal N41\). Suppose that \(E\) is a
support of a well ordering of \(Y\). Then for each \(x\) in \(Y\), \(E\) is
a support of \(x\). By the lemma, since \(x\) is finite, \(E\) is a
support of every element of \(x\). It follows that if \(f\) is any
choice function for \(Y\) (in the ground model) then \(E\) is a
support of \(f\).
Howard-Rubin number:
112
Type:
proofs of results
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