Depends what you mean by ‘large’ I suppose. A non-well founded model of ZFC is ‘larger’ than the well-founded submodel it contains (in the sense that it properly contains its well-founded submodel), but it certainly isn’t “standard”.
By Gödel’s constructive set theory are you talking about set theory plus the axiom of constructibility (V=L)? V=L is hardly ‘dismissed as an aberration’ any more than the field axioms are an ‘aberration’ but just as there’s more scope for a ‘theory of rings’ than a ‘theory of fields’, so adding V=L as an axiom (and making a methodological decision to refrain from exploring universes where it fails) has the effect of truncating the hierarchy of large cardinals. Everything above zero-sharp becomes inconsistent.
Furthermore, the picture of L sitting inside V that emerges from the study of zero-sharp is so stark and clear that set theorists will never want to let it go. (“No one will drive us from the paradise which Jack Silver has created for us”.)
Depends what you mean by ‘large’ I suppose. A non-well founded model of ZFC is ‘larger’ than the well-founded submodel it contains (in the sense that it properly contains its well-founded submodel), but it certainly isn’t “standard”.
By Gödel’s constructive set theory are you talking about set theory plus the axiom of constructibility (V=L)? V=L is hardly ‘dismissed as an aberration’ any more than the field axioms are an ‘aberration’ but just as there’s more scope for a ‘theory of rings’ than a ‘theory of fields’, so adding V=L as an axiom (and making a methodological decision to refrain from exploring universes where it fails) has the effect of truncating the hierarchy of large cardinals. Everything above zero-sharp becomes inconsistent.
Furthermore, the picture of L sitting inside V that emerges from the study of zero-sharp is so stark and clear that set theorists will never want to let it go. (“No one will drive us from the paradise which Jack Silver has created for us”.)