Let L be a first order language. Let A be any set of sentences of L. We extend L0 (=L) to L1 by adding denumerably many constants c1, …,cn,… to L. We enumerate the existential formulas of L. We add Henkin axioms to L by taking each formula in the enumeration and making it the antecedant of a conditional whose consequent is an instantiation of the existential variable in the antecedent to the next earliest constant in the enumeration c1, …,cn,…(adsbygoogle = window.adsbygoogle || []).push({});

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.

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# Not Understanding a Move in Henkin Completeness Proof

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