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Diagonal Argument

1 The Argument

  • Theorem 1.1 (Diagonal Argument). In a well-pointed topos, \(X \ncong \Omega ^X\).

  • Proof. Suppose it were isomorphic, via an isomorphism \(f\).

    (-tikz- diagram) (-tikz- diagram)

    Then, we can consider \(f_x\), the composite in the diagram above. There is an element of \(X\), \(x\) corresponding to \(f_x\), and if we chase \(x\) around the diagram, we get a contradiction.