LYCOS RETRIEVER 
First-Order Logic: Set
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The ordered conjecture states that least fixed-point logic LFP is strictly more expressive than first-order logic FO on every infinite class of ordered finite structures. It has been established that either way of settling this conjecture would resolve open problems in complexity theory. In fact, this holds true even for the particular instance of the ordered conjecture on the class of BIT-structures, that is, ordered finite structures with a built-in BIT predicate. Using a well known isomorphism from the natural numbers to the hereditarily finite sets that maps BIT to the membership relation between sets, the ordered conjecture on BIT-structures can be translated to the problem of comparing the expressive power of FO and LFP in the context of finite set theory. The advantage of this approach is that we can use set-theoretic concepts and methods to identify certain fragments of LFP for which the restriction of the ordered conjecture is already hard to settle, as well as other restricted fragments of LFP that actually collapse to FO. These results advance the state of knowledge about the ordered conjecture on BIT-structures and contribute to the delineation of the boundary where this conjecture becomes hard to settle.
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[O]rder logic ... has the operator, =, expressing that two expressions refer to the same object. For example, Father(Richard) = Henry, means Henry and the object returned by Father(Richard) are the same. It can also be used to assert that an object is a member of a set, e.g. King(Richard) = True, Richard is a member of the set of Kings.
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[F]irst-order logic is strong enough to formalize all of set theory and thereby virtually all of mathematics. It is the classical logical theory underlying mathematics. It is a stronger theory than sentential logic, but a weaker theory than arithmetic, set theory, or Second-order logic.
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A first order logic is given by a set of function symbols and a set of predicate symbols. Each function or predicate symbol comes with an arity, which is natural number. Function symbols of arity 0 are known as constant symbols. Now terms are recursively defined by
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The predicate calculus is an extension of the propositional calculus that defines which statements of first order logic are provable. It is a formal system used to describe mathematical theories. If the propositional calculus is defined with a suitable set of axioms and the single rule of inference modus ponens (this can be done in many different ways), then the predicate calculus can be defined by appending some additional axioms and the additional inference rule "universal generalization". More precisely, as axioms for the predicate calculus we take:
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The concept of general semantics for second-order logic avoids any pretense that the power-set operation is a fixed well-understood resource. Instead, the range of the quantifier ∀X must be directly specified.
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