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Up to here I think I get it. I have an extended system such that any existential claims entailed by the formulas in A, together with the appropriate Henkin axioms will yield instances of the existential claims via what I have seen called "witnessing constants". What I'm failing to see is why I need to go on? Once I have L1, why not just construct my model of L1? I am not understanding the move to add another set of constants and Henkin axioms, and then another, and so on to get L2,L3,etc.?

I'm sure I'm missing something obvious.