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Math Help - fixed point theorem

  1. #1
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    fixed point theorem

    Prove that for every function f: \omega_1 \rightarrow \omega_1, there is a \beta \in \omega_1 such that f[\beta] \subseteq \beta.

    Hint: A fixed point theorem may be useful. We may us the Countable Principle of Choice, but not the Axiom of Choice.
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  2. #2
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    Are you referring to Knaster–Tarski fixpoint theorem? Since it requires a monotonic function, maybe one can consider g(\beta)=\bigcup_{\alpha\le\beta}f(\alpha). Then g is monotonic, so it has a fixpoint \beta_0: g(\beta_0)=\beta_0, which implies f(\beta_0)\subseteq\beta_0. And the Countable Principle of Choice is used to show that \omega_1 is a complete lattice. I am not sure about this, but it may be a start...
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