If there is a ''set'' W in V which is a standard model of ZF, and the ordinal κ is the set of ordinals which occur in W, then Lκ is the L of W. If there is a set which is a standard model of ZF, then the smallest such set is such a Lκ. This set is called the minimal model of ZFC. Using the downward Löwenheim–Skolem theorem, one can show that the minimal model (if it exists) is a countable set.
Of course, any consistent theory must have a model, so even within the minimal model of set theory there are sets which are models of ZF (assuming ZF is consistent). However, those set models are non-standard. In particular, they do not use the normal element relation and they are not well founded. (http://en.wikipedia.org/wiki/Constructible_universe)
I don't get it. OK, so in this case, one assumes that there exists some standard model that is not a class, but a set. Then one constructs a constructible universe inside such set which is also a set. Then one uses Lowenheim-Skolem to show that the smallest such model is countable.
My question is, it seems that the model is externally not standard (viewed from a bigger model) - so why call it standard first place? Or am I being confused, and is the model really standard? And the quote in the later part seems to say that the model is not standard, which confuses me much.
And I do not get the second paragraph - why would any consistent theory having at least one model relates to the minimal model containing models of ZF? I am not sure how Godel's completeness theorem is being matched here.