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

In addition, a consistent formal theory that contains the first-order theory of the natural numbers

In contrast to the frequency domain analysis of the classical control theory, modern control theory utilizes the time-domain state space representation, a mathematical model of a physical system as a set of input, output and state variables related by first-order differential equations.

A real number a is first-order definable in the language of set theory, without parameters, if there is a formula φ in the language of set theory, with one free variable, such that a is the unique real number such that φ ( a ) holds in the standard model of set theory ( see Kunen 1980: 153 ).

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

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.

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.

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.

It says that for any first-order theory T with a well-orderable language, and any sentence S in the language of the theory, there is a formal proof of S in T if and only if S is satisfied by every model of T ( S is a semantic consequence of T ).

( Here, Peano Arithmetic ( PA ) is understood as the first-order theory of arithmetic with symbols of addition and multiplication, and schema of recursion.

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.

An important consequence of the completeness theorem is that it is possible to recursively enumerate the semantic consequences of any effective first-order theory, by enumerating all the possible formal deductions from the axioms of the theory, and use this to produce an enumeration of their conclusions.

Henkin's proof directly constructs a term model for any consistent first-order theory.

Under such conditions every formula of the form, where ( T ) is a string of quantifiers containing all quantifiers in ( S ) and ( S ') interleaved among themselves in any fashion, but maintaining the relative order inside ( S ) and ( S '), will be equivalent to the original formula Φ '( this is yet another basic result in first-order predicate calculus that we rely on ).

In the first-order phase transition the range of (< i > P, V, T </ i >), where the liquid phase and the gas phase are in equilibrium, it does not exhibit the empirical fact that < var > p </ var > is constant as a function of < i > V </ i > for a given temperature: although this behavior can be easily inserted into the van der Waals model ( see Maxwell's correction below ), the result is no longer a simple analytical model, and others ( such as those based on the principle of corresponding states ) achieve a better fit with roughly the same work.

Let T be a first-order theory in the language of arithmetic and capable of representing all computable functions.

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?

" It is possible to define a formula True ( n ) whose extension is T *, but only by drawing on a metalanguage whose expressive power goes beyond that of L. For example, a truth predicate for first-order arithmetic can be defined in second-order arithmetic.

The thought " John is tall " is clearly composed of two sub-parts, the concept of John and the concept of tallness, combined in a manner that may be expressed in first-order predicate calculus as a predicate ' T ' (" is tall ") that holds of the entity ' j ' ( John ).

T. Considering the first-order terms in the expansion for small λ, we must have M * T

When a speaker addresses somebody using the V form instead of the T form, they index ( via first-order indexicality ) their understanding of the need for deference to the addressee.

Silverstein comments that while exhibiting a basic level of first-order indexicality, the T / V system also employs second-order indexicality vis-à-vis ' enregistered honorification '.

Therefore, people will use T / V deference entailment in 1 ) a first-order indexical sense that distinguishes between speaker / addressee interpersonal values of ' power ' and ' solidarity ' and 2 ) a second-order indexical sense that indexes an interlocutor's inherent " honor " or social merit in employing V forms over T forms in public contexts.

A sentence σ is independent of a given first-order theory T if T neither proves nor refutes σ ; that is, it is impossible to prove σ from T, and it is also impossible to prove from T that σ is false.

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

Although the logical consequence relation is only semidecidable, much progress has been made in automated theorem proving in first-order logic.

In Gödel's paper, he uses a version of first-order predicate calculus which has no function or constant symbols to begin with.

Prolog has its roots in first-order logic, a formal logic, and unlike many other programming languages, Prolog is declarative: the program logic is expressed in terms of relations, represented as facts and rules.

Each real set, function, and relation has its natural hyperreal extension, satisfying the same first-order properties.

It has also been shown by L. Berman in 1980 that the problem of verifying / falsifying any first-order statement about real numbers that involves only addition and comparison ( but no multiplication ) is in EXPSPACE.

If the transfer function of a first-order low-pass filter has a zero as well as a pole, the Bode plot will flatten out again, at some maximum attenuation of high frequencies ; such an effect is caused for example by a little bit of the input leaking around the one-pole filter ; this one-pole – one-zero filter is still a first-order low-pass.

A first-order lens has a focal length of 920 mm ( 36 in ) and an optical area 2590 mm ( 8. 5 ft ) high.

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.

Proof: Fix a first-order language L, and let Σ be a collection of L-sentences such that every finite subcollection of L-sentences, i ⊆ Σ of it has a model.

# A real closed field has all the first-order properties of the real number system, regardless of whether they are usually taken as axiomatic, for statements involving the basic ordered-field relations +, *, and ≤.

For an instrument whose payment stream is described by f ( t ), the value V ( t ) satisfies the inhomogeneous first-order ODE (" inhomogeneous " is because one has f rather than 0, and " first-order " is because one has first derivatives but no higher derivatives ) – this encodes the fact that when any cash flow occurs, the value of the instrument changes by the value of the cash flow ( if you receive a $ 10 coupon, the remaining value decreases by exactly $ 10 ).

It is more expressive than propositional logic but has more efficient decision problems than first-order predicate logic.

For example, while it was once assumed that catagenetic processes were first-order reactions, some research has shown that this may not be the case.

There are no first-order administrative divisions as defined by the United States Government, but Puerto Rico has 78 municipalities or " municipios " at the secondary order.

A second-order low-pass filter with a very low quality factor has a nearly first-order step response ; the system's output responds to a step input by slowly rising toward an asymptote.

The first-order change in the n-th energy eigenket has a contribution from each of the energy eigenstates k ≠ n. Each term is proportional to the matrix element, which is a measure of how much the perturbation mixes eigenstate n with eigenstate k ; it is also inversely proportional to the energy difference between eigenstates k and n, which means that the perturbation deforms the eigenstate to a greater extent if there are more eigenstates at nearby energies.

Second-and third-order winning percentage has been shown to predict future actual team winning percentage better than both actual winning percentage and first-order winning percentage.