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If ZF has a standard model there is a least ordinal $\sigma$ such that $L_{\sigma}$ is a model of ZFC. What is $\sigma$ called?

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It does not have any particular name or notation: it is just ''the height of the minimal model of set theory''

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I thought as much. Wikipedia misleadingly calls it an $unprovable \ ordinal $. Perhaps one should start calling it $ the \ Cohen \ ordinal $. –  Frode Bjørdal Aug 3 '13 at 18:01
    
``The Cohen ordinal'' would be a misattribution: John Shepherdson also identified the ordinal more than a decade earlier in one of three JSL articles in the late forties (I forget which one) analysing the Goedel model. I doubt he was alone. Mostowski? –  Philip Welch Aug 8 '13 at 18:56
    
I knew about Shepherdson, who I thought wrote in the early fifties. If I remember correctly, though, Cohen made some improvements in describing the minimal model. Nevertheless, a $Shepherdson \ ordinal$ would be fine. A reference to Mostowski on this would be welcome. –  Frode Bjørdal Aug 10 '13 at 1:03

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