Second-order variables: elements of domain only without quantifiers?

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SUMMARY

Second-order variables are defined in two primary ways: one definition encompasses all elements of Pk(M), where P represents the power set of M, and k denotes natural numbers. The alternative definition focuses on first-order relations and predicates. A critical point of discussion is whether "predicates" includes first-order sentences that contain quantifiers and variables or is limited to those with only constant symbols. The clarification of these definitions is essential for understanding the implications in model theory.

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Logicians, mathematicians, and computer scientists interested in advanced topics in model theory and the nuances of second-order variables.

nomadreid
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On one side one can define second-order variables as ranging over all elements of Pk(M) for all natural numbers k (P=power set of M, M is the universe of the model, superscript being iteration). On the other side it is sometimes defined as ranging over all first-order relations and predicates. In this latter definition, does "predicates" include first-order sentences with quantifiers and variables, or only first-order sentences with only constant symbols?
Thanks.
 
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er, sorry, I meant P(Mk), where the exponent refers to the Cartesian product
 

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