** If S is a set of sentences of first-order logic and B is a consistent subset of S, then B is included in a set that is maximal among consistent subsets of S. The special case where S is the set of all first-order sentences in a given signature is weaker, equivalent to the Boolean prime ideal theorem ; see the section " Weaker forms " below.

** Gödel's completeness theorem for first-order logic: every consistent set of first-order sentences has a completion.

The Löwenheim-Skolem theorems tell us that if we restrict ourselves to first-order logic, any axiom system for the reals admits other models, including both models that are smaller than the reals and models that are larger.

In first-order logic the axiom reads:

In contrast, other, more systematic algorithms achieved, at least theoretically, completeness for first-order logic.

A basic objective of the first normal form defined by Codd in 1970 was to permit data to be queried and manipulated using a " universal data sub-language " grounded in first-order logic.

Properties definable in first-order logic that an equivalence relation may or may not possess include:

* Tarski's axioms: Alfred Tarski ( 1902 – 1983 ) and his students defined elementary Euclidean geometry as the geometry that can be expressed in first-order logic and does not depend on set theory for its logical basis, in contrast to Hilbert's axioms, which involve point sets.

The asks for an algorithm that takes as input a statement of a first-order logic ( possibly with a finite number of axioms beyond the usual axioms of first-order logic ) and answers " Yes " or " No " according to whether the statement is universally valid, i. e., valid in every structure satisfying the axioms.

By the completeness theorem of first-order logic, a statement is universally valid if and only if it can be deduced from the axioms, so the can also be viewed as asking for an algorithm to decide whether a given statement is provable from the axioms using the rules of logic.

Alphabets can also be infinite ; e. g. first-order logic is often expressed using an alphabet which, besides symbols such as ∧, ¬, ∀ and parentheses, contains infinitely many elements x < sub > 0 </ sub >, x < sub > 1 </ sub >, x < sub > 2 </ sub >, … that play the role of variables.

It is also known as first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic ( a less precise term ).

A theory about some topic is usually first-order logic together with: a specified domain of discourse over which the quantified variables range, finitely many functions which map from that domain into it, finitely many predicates defined on that domain, and a recursive set of axioms which are believed to hold for those things.

Sometimes " theory " is understood in a more formal sense, which is just a set of sentences in first-order logic.

The adjective " first-order " distinguishes first-order logic from higher-order logic in which there are predicates having predicates or functions as arguments, or in which one or both of predicate quantifiers or function quantifiers are permitted.

There are many deductive systems for first-order logic that are sound ( all provable statements are true ) and complete ( all true statements are provable ).

The general first-order theory of the natural numbers expressed in Peano's axioms cannot be decided with such an algorithm, however.

No first-order theory, however, has the strength to describe fully and categorically structures with an infinite domain, such as the natural numbers or the real line.

While the first-order experiments could be explained by a modified stationary aether, more precise second-order experiments were expected to give positive results, however, no such results could be found.

These are restricted, however, in that all terms that they form must be either first-order terms ( which can be substituted for a first-order variable ) or second-order terms ( which can be substituted for a second-order variable of an appropriate sort ).

It has the same quantifier complexity as the definition of uniform continuity in terms of sequences in standard calculus, which however is not expressible in the first-order language of the real numbers.

That is, every consistent set of first-order sentences can be extended to a maximal consistent set.

The kinds of logical sentences that obey this restriction on quantification are referred to as statements in first-order logic.

However, it is well known that sentences of this kind cannot be interpreted in first-order logic, where individual variables stand for individual things.

In mathematical logic, the compactness theorem states that a set of first-order sentences has a model if and only if every finite subset of it has a model.

Horn clauses are named after the logician Alfred Horn, who investigated the mathematical properties of similar sentences in the non-clausal form of first-order logic.

Then sentences that were second-order become first-order, with the formerly second-order quantifiers ranging over the second sort instead.

That theorem implies that there is some countably infinite subset of the real numbers, whose members we will call internal numbers, and some countably infinite collection of sets of internal numbers, whose members we will call " internal sets ", such that the domain consisting of internal numbers and internal sets satisfies exactly the same first-order sentences satisfied as the domain of real numbers and sets of real numbers.

A first-order theory consists of a fixed signature and a fixed set of sentences ( formulas with no free variables ) in that signature.

A substructure of a σ-structure M is obtained by taking a subset N of M which is closed under the interpretations of all the function symbols in σ ( hence includes the interpretations of all constant symbols in σ ), and then restricting the interpretations of the relation symbols to N. An elementary substructure is a very special case of this ; in particular an elementary substructure satisfies exactly the same first-order sentences as the original structure ( its elementary extension ).

Note that an individual instance of the sentence, such as " Alice, Bob and Carol admire only one another ", need not involve sets and is equivalent to the conjunction of the following first-order sentences:

Let T denote the set of L-sentences true in N, and T * the set of Gödel numbers of the sentences in T. The following theorem answers the question: Can T * be defined by a formula of first-order arithmetic?

Nonfirstorderizable sentences are sometimes presented as evidence that first-order logic is not adequate to capture the nuances of meaning in natural language.

" Boolos argued that such sentences call for second-order symbolization, which can be interpreted as plural quantification over the same domain as first-order quantifiers use, without postulation of distinct " second-order objects " ( properties, sets, etc.

* Diagonal lemma, also known as the fixed-point lemma, for producing self-referential sentences of first-order logic

However, the following criteria can be coded as first-order sentences in the language of fields, and are equivalent to the above definition.

Logic forms are simple, first-order logic knowledge representations of natural language sentences formed by the conjunction of concept predicates related through shared arguments.

If the system is suitably complex, like first-order arithmetic, then the set T of Gödel numbers of true sentences in the system will be a productive set, which means that whenever W is a recursively enumerable set of true sentences, there is at least one true sentence that is not in W. This can be used to give a rigorous proof of Gödel's first incompleteness theorem, because no recursively enumerable set is productive.

In the second-order theory, one of the normal stress differences can be calculated from the first-order stress relaxation function.

The rate of a first-order reaction depends only on the concentration and the properties of the involved substance, and the reaction itself can be described with the characteristic half-life.

Such a definition can be formulated in terms of equivalence classes of smooth functions on M. Informally, we will say that two smooth functions f and g are equivalent at a point x if they have the same first-order behavior near x.

For example, if we can enumerate all such definable numbers by the Gödel numbers of their defining formulas then we can use Cantor's diagonal argument to find a particular real that is not first-order definable in the same language.

Rather, we say that a real a is definable in the language of arithmetic ( or arithmetical ) if its Dedekind cut can be defined as a predicate in that language ; that is, if there is a first-order formula φ in the language of arithmetic, with two free variables, such that

The question whether a given Turing machine halts or not can be formulated as a first-order statement, which would then be susceptible to the decision algorithm.

Many common axiomatic systems, such as first-order Peano arithmetic and axiomatic set theory, including the canonical Zermelo – Fraenkel set theory ( ZF ), can be formalized as first-order theories.

Unlike natural languages, such as English, the language of first-order logic is completely formal, so that it can be mechanically determined whether a given expression is legal.

If some specific deductive system of first-order logic is sound and complete, then is it " perfect " ( a formula is provable iff it is a semantic consequence of the axioms ), thus equivalent to any other deductive system with the same quality ( any proof in one system can be converted into the other ).

Precisely, we can systematically define a model of any consistent effective first-order theory T in PA by interpreting each symbol of T by an arithmetical formula whose free variables are the arguments of the symbol.

Its first-order Fresnel lens can be seen for about 19 miles ( 30 kilometers ), in good conditions.

* November 1 – The current Cape Lookout, North Carolina, lighthouse is lighted for the first time ( its first-order Fresnel lens can be seen for 19 miles ).

Precise expressions for the transition probability, based on first-order perturbation Hamiltonians, can be found in Thompson and Baker.

This can be used as a very rough first-order approximation for SHF ( microwave ) communication links ;

The first formal investigation of unification can be attributed to John Alan Robinson, who used first-order unification as a basic building block of his resolution procedure for first-order logic, a great step forward in automated reasoning technology, as it eliminated one source of combinatorial explosion: searching for instantiation of terms.

Other all-pole second-order filters may roll off at different rates initially depending on their Q factor, but approach the same final rate of 12 dB per octave ; as with the first-order filters, zeroes in the transfer function can change the high-frequency asymptote.