Peter Krautzberger on the web

Red workbook, p14

Source

red workbook, p14
Red Workbook, p.14

Transcript

  • Beweis: (a) => (b): \mathfrak{B} \subseteq q \Rightarrow \mathfrak{D} = { D_e : e \in [ \mathfrak{B} ]^{< \omega} } \subseteq q
    • Nimm g_e \in [S^{< \omega}] mit \beta S \cdot q \subseteq \bigcup_{x \in g_e} \widehat{ x^{-1} D_e} = \widehat{C_e}
    • \Rightarrow \beta S \cdot q \subseteq \bigcap_{C \in \mathfrak{C}} \widehat{C_e} = Y_e \stackrel{3.3}{\Rightarrow} Beh.
  • (b) => (a): seien g_e, C_e, \mathfrak{C} wie in (b).
    • Nimm p \in \beta S mit L := \beta S \cdot p \subseteq Y_{\mathfrak{C}} oBdA p\in K(\beta S) .
    • \Rightarrow p\in L .
    • Fuer e \in [\mathfrak{B}]^{<\omega}: p \in \widehat{C_e} = \bigcup_{x\in g_e} \widehat{x^{-1} D_e}
    • \Rightarrow \exists x_e \in g_e: p \in \widehat{ x_e^{-1}D_e} .
  • ((?) wieso eDe?) Nimm w\in \beta S mit S_e := { x_f : f \supseteq e, f \in [ \mathfrak{B}]^{<\omega} } \underset{?}{\in} \omega (\forall e \in \beta S)
  • Setze q = w \cdot p \Rightarrow q \in \beta S \cdot p \subseteq K(\beta S) und
  • es ist \mathfrak{B} \subseteq q [ B\in \mathfrak{B} zeige: B\in q = w\cdot p . Aber e:{B} \in [\mathfrak{B}]^{\omega}
    • D_e = B, S_e \in \omega ; Fuer alle f\supseteq e: x_f \cdot p \in \widehat{D_f}
    • \widehat{D_f} \subseteq \widehat{D_e} = \widehat{B} \Rightarrow S_e \cdot p \subseteq \widehat{B}
    • \Rightarrow w \cdot p \in \widehat{B} ]
  • Also folgt die Behauptung.□

partial Translation

  • Proof:
  • (a) => (b):
    • \mathfrak{B} \subseteq q \Rightarrow \mathfrak{D} = { D_e : e \in [ \mathfrak{B} ]^{< \omega} } \subseteq q
    • Take g_e \in [S^{< \omega}] with \beta S \cdot q \subseteq \bigcup_{x \in g_e} \widehat{ x^{-1} D_e} = \widehat{C_e}
    • \Rightarrow \beta S \cdot q \subseteq \bigcap_{C \in \mathfrak{C}} \widehat{C_e} = Y_e \stackrel{3.3}{\Rightarrow} the claim.
  • (b) => (a): let g_e, C_e, \mathfrak{C} as in (b).
    • Then take p \in \beta S mit L := \beta S \cdot p \subseteq Y_{\mathfrak{C}} ; without loss p\in K(\beta S) .
    • \Rightarrow p\in L .
    • For e \in [\mathfrak{B}]^{<\omega}: p \in \widehat{C_e} = \bigcup_{x\in g_e} \widehat{x^{-1} D_e}
    • \Rightarrow \exists x_e \in g_e: p \in \widehat{ x_e^{-1}D_e} .
    • Take w\in \beta S with S_e := { x_f : f \supseteq e, f \in [ \mathfrak{B}]^{<\omega} } \in \omega (\forall e \in \beta S)
    • Now define q = w \cdot p \Rightarrow q \in \beta S \cdot p \subseteq K(\beta S) and
    • since \mathfrak{B} \subseteq q
      • [ B\in \mathfrak{B} show: B\in q = w\cdot p . But e:{B} \in [\mathfrak{B}]^{\omega}
      • D_e = B, S_e \in \omega ; For all f\supseteq e: x_f \cdot p \in \widehat{D_f}
      • \widehat{D_f} \subseteq \widehat{D_e} = \widehat{B} \Rightarrow S_e \cdot p \subseteq \widehat{B}
      • \Rightarrow w \cdot p \in \widehat{B} ]
    • The claim follows.

Notes

This page contains the proof of Theorem 3.4 of the previous part (I guess I should’ve included that yesterday). I can’t really make much of it. It’s the dull of writing up a new notion. But if you look closer, you might stumble over a few details (as I did when I took these notes). Writing this up just now I find the choice of w quite striking.