Description:
Results and consequences of von Rimscha [1982]
Content:
In von Rimscha [1982] three transitivity conditions are considered:
| \(Tr\): | \(\forall x\exists u\exists f\) such that \(u\) is transitive and \(f\) is a function from \(f\) one to one and onto \(u\). This is Form 175. |
| \(Tr'\): | \(\forall x \exists u \exists f\) such that \(u\) is transitive and \(u \subseteq TC(x)\) and \(f\) is a one to one function from \(x\) onto \(u\). |
| \(Tr''\): | \(\forall x\exists u\exists f\) such that \(u\) is transitive and \(f\) is a one to one function from \(x\) onto \(u\) and \(\forall s\in x\), \(f(s)\in TC(s)\). |
It is however possible to prove in \(ZF^0\) that \(Tr''\) (Form 174) implies Dedekind finite = finite (Form 9). We proceed as follows: Assume \(Tr''\). We first show that every infinite subset of the atoms \(A\) is Dedekind infinite. Assume \(B \subseteq A\) and that\(B\) is infinite. We will use \(Tr''\) to get a choice function for \({\cal P}(B)\) from which it follows that \(B\) has a denumerable subset. Let \(\kappa\) be the least well ordered cardinal which is not \(\le|{\cal P}(B)|\). For each \(a \in B\) and ordinal \(\alpha\), define\(a_{\alpha}\) by induction: \(a_{0} = a\), \(a_{\lambda} =\bigcup^{}_{\alpha <\lambda } a_{\alpha }\) for limit ordinals \(\lambda\) and \(a_{\alpha +1} = a_{\alpha} \cup \{a_{\alpha }\}\). (\(a_{\alpha }\) is the ordinal \(\alpha \) with every occurrence of \(\emptyset\) replaced by \(a\).) Then for each \(\alpha\), \(|a_{\alpha}| = |\alpha|\). Let \(B' = \{a_{\kappa}: a\in B \}\) and let \(x = {\cal P}(B')\).By \(Tr''\) there is a transitive set \(u\) and a function \(f : x\rightarrow u\) which is one to one and onto and such that for each \(s \in x\), \(f(s)\in TC(s)\). Suppose that \(C \subseteq B\) and let \(s = \{a_{\kappa}:a\in C\}\in x\). Then \[TC(s)=\{s \}\cup s\cup\left(\bigcup^{}_{a\in C}a_{\kappa}\right)\] since \(|TC(s)| \ge \kappa\), \(s \not\in u\).\((|u| = |{\cal P}(B)|\) which is not \(\ge\kappa\).). So \(f(s) \in \bigcup^{}_{a\in C} a_{\kappa }\). Since \(a\neq b\) implies\(a_{\kappa } \cap b_{\kappa } = \emptyset\), \(f(s)\) determines a unique element of \(C\).
Now let \(x\) be any infinite set. By \(Tr''\) there is a transitive \(u\) and a one to one function \(f\) from \(x\) onto \(u\). If \(u\) contains only finitely many atoms, then the von Rimscha proof can be modified to show that \(x\) has a denumerable subset. If \(u\) contains an infinite set of atoms \(B\), then by the first argument \(B\) and therefore\(x\) contains a denumerable subset.
(from von-Rimscha-1982): Are any of the implications \(Tr \rightarrow Tr' \rightarrow Tr'' \rightarrow AC\) provable in \(ZF\)?
Howard-Rubin number: 49
Type: Proofs and statements
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