First-order logic, ZFC axioms, ordinals, cardinality, the axiom of choice, completeness and Godel theorems.
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- What is a first-order language?
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A collection of logical symbols (variables, connectives, quantifiers) and non-logical symbols (predicates, functions, constants).- What is a formula in first-order logic?
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A finite sequence of symbols from the language's alphabet that follows the rules of syntax for well-formed expressions.- What does it mean for a formula to be satisfiable?
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There exists a model and an interpretation of variables under which the formula evaluates to true.- Define a model in first-order logic.
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A structure assigning meaning to the language's symbols: a domain of objects and interpretations of predicates and functions over that domain.- What is the difference between satisfiability and validity in first-order logic?
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A formula is satisfiable if true in some model, and valid if true in every model.- State the Soundness Theorem for first-order logic.
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If a formula can be proven from axioms using valid inference rules, then it is valid (true in all models).- What does logical consequence mean?
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Formula B is a logical consequence of set A if B is true in every model where all formulas in A are true.- State the Axiom of Extensionality.
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Two sets are equal if and only if they have exactly the same elements.- State the Axiom of Foundation.
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Every nonempty set contains an element with which it has no common elements (no infinite descending membership chains).- State the Axiom of Pairing.
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For any two sets x and y, there exists a set containing exactly x and y as its elements.- State the Axiom of Union.
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For any set S, there exists a set U containing all elements of all sets in S.- State the Axiom of Power Set.
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For any set S, there exists a set P whose elements are exactly the subsets of S.- State the Axiom of Infinity.
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There exists a set that contains the empty set and is closed under the operation of taking successors.- State the Axiom of Replacement.
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If a property P(x, y) associates each x in set A with a unique y, then there is a set B containing all such y.- State the Axiom of Separation.
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For any set S and property P(x), there exists a set T whose elements are exactly those elements of S for which P(x) is true.
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