A structure consists of a domain together with interpretations of the symbols of a formal language. In the usual finitary setting it interprets constant, function, and relation symbols of their declared arities. A first-order structure is such an interpretation for first-order logic.
Overspill is the principle that an internal subset of a nonstandard extension containing every standard natural number must contain some nonstandard natural number. The overspill lemma is its definable-set instance in nonstandard arithmetic.
Craig's theorem says that a theory with a computably enumerable axiomatization has an equivalent decidable axiomatization. The Craig trick gives a constructive proof by encoding an enumeration index in repeated conjunctions of the enumerated axiom.
A theory is a set of sentences in a formal language, sometimes understood to be closed under deduction. A first-order theory uses a first-order language; consistency and completeness are separate properties of a theory.
A sentence is a closed formula, with no free variable occurrences. Its truth does not depend on a variable assignment. A first-order sentence uses a first-order formal language.
A variable occurrence is bound when it falls within the scope of a binding operation that binds that occurrence; otherwise it is free. Quantifiers and lambda abstractions are common binders. In first-order formulas, a free variable occurrence is outside the scope of a corresponding quantifier.
A proof system is sound with respect to its semantics when every derivable conclusion is true in every interpretation satisfying its premises. Both soundness theorem for propositional logic and soundness theorem for first-order logic establish this property for particular languages and semantics.
A theory has the disjunction property when provability of a closed disjunction implies provability of one disjunct. It has the existence property when a provable existential sentence has a witness whose substitution is provably valid. These are distinct constructive properties; disjunction property of intuitionistic propositional logic is one instance of the former.
Kripke semantics interprets logical formulas at worlds related by an accessibility relation or an information order. Modal logics use accessibility to interpret modal operators; intuitionistic logics use persistent valuations and quantify over stronger worlds for implication.
A Priestley space is a compact topological partially ordered set that is totally order-disconnected: if , a clopen upward-closed set contains and omits . A Priestley dual space is a space of this kind constructed from the prime filters of a bounded distributive lattice.
Infinitary logic permits infinitary syntactic operations, such as infinite conjunctions, together with an interpretation of its formulas. The infinitary language specifies the permitted conjunction and quantifier lengths.
A mathematical definition fixes the meaning of a term or symbol using previously specified concepts. A definition by a universal property can still require a mathematical proof that an object with that property exists. For example, the definition of a product in a category specifies its mapping behavior, while existence is a further question about that category.
An axiom is a sentence admitted as a starting assumption of a formal theory. An axiom schema specifies a collection of such sentences, usually one for each formula of an allowed form. A model of a first-order theory satisfies all its axioms; a relative consistency construction can show that the other axioms do not imply a particular omitted one.
Proof theory studies formal proofs and the deductive systems that produce them. It examines which conclusions follow from axioms, how proofs can be transformed, and what those transformations reveal about the chosen logic. Natural deduction organizes a proof through introduction and elimination rules for logical connectives and quantifiers.
Mean-variance optimization balances expected linear return against a positive-definite quadratic variance penalty.
Modern portfolio theory constructs portfolios by comparing expected return with variance risk and the effects of diversification.
Implied volatility is the volatility input of an option-pricing model that reproduces an observed option price. Black-Scholes implied volatility specializes this inversion to the Black-Scholes model.
The Greeks are sensitivities of a financial claim value to model inputs, expressed through partial derivatives. The option delta measures sensitivity to the underlying price; a delta hedge uses that sensitivity to cancel first-order price exposure.
The binomial options pricing model values a contingent claim on a two-outcome asset-price tree using risk-neutral probability and backward option pricing. Its underlying discrete-time binomial market distinguishes the market assumptions from the pricing procedure.
A risk-neutral measure is a probability measure equivalent to the physical probability measure , meaning they agree on which events have zero probability. For assets without intermediate payments, prices expressed in the chosen numéraire must be martingales under . For a positive bank account and a stock without dividends, this requires to be a martingale; with dividends, use discounted dividend gains. In an unrestricted finite-state one-period financial market, existence of such a measure is equivalent to absence of arbitrage, by the fundamental theorem of asset pricing.
Physical probability describes the modeled frequency or belief about an up move. The risk-neutral probability is constrained by traded prices. In the worked binomial market, even if or . Calling risk-neutral does not assume that investors have no risk aversion; it changes the weights used for risk-neutral pricing.
A restriction that the initial cost of all allocated resources equals the available budget. In a portfolio with risky holdings and bank holdings , the displayed equation is the budget identity. Barring borrowing makes , equivalently ; banning the bank account instead imposes equality.
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