( A formal proof for all finite sets would use the principle of mathematical induction to prove " for every natural number k, every family of k nonempty sets has a choice function.

* Metamath-a language for developing strictly formalized mathematical definitions and proofs accompanied by a proof checker for this language and a growing database of thousands of proved theorems ; while the Metamath language is not accompanied with an automated theorem prover, it can be regarded as important because the formal language behind it allows development of such a software ; as of March, 2012, there is no " widely " known such software, so it is not a subject of " automated theorem proving " ( it can become such a subject ), but it is a proof assistant.

Proofs in computability theory often invoke the Church – Turing thesis in an informal way to establish the computability of functions while avoiding the ( often very long ) details which would be involved in a rigorous, formal proof.

* Natural deduction, an approach to proof theory that attempts to provide a formal model of logical reasoning as it " naturally " occurs

The study of mathematical proof is particularly important in logic, and has applications to automated theorem proving and formal verification of software.

The Elements begins with plane geometry, still taught in secondary school as the first axiomatic system and the first examples of formal proof.

Gödel's ontological proof is a formal argument for God's existence by the mathematician Kurt Gödel.

It makes a close link between model theory that deals with what is true in different models, and proof theory that studies what can be formally proved in particular formal systems.

The completeness theorem says that if a formula is logically valid then there is a finite deduction ( a formal proof ) of the formula.

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 ).

It is deduced from the model existence theorem as follows: if there is no formal proof of a formula then adding its negation to the axioms gives a consisten theory, which has thus a model, so that the formula is not a semantic consequence of the initial theory.

James Margetson ( 2004 ) developed a computerized formal proof using the Isabelle theorem prover.

Depending on the particular formalism adopted for the calculus, it may be seen as a simple application of a " functional substitution " rule of inference, as in Gödel's paper, or it may be proved by considering the formal proof of, replacing in it all occurrences of Q by some other formula with the same free variables, and noting that all logical axioms in the formal proof remain logical axioms after the substitution, and all rules of inference still apply in the same way.

The unifying themes in mathematical logic include the study of the expressive power of formal systems and the deductive power of formal proof systems.

In the case of seL4, complete formal verification of the implementation has been achieved, i. e. a mathematical proof that the kernel's implementation is consistent with its formal specification.

In addition, from at least the time of Hilbert's program at the turn of the twentieth century to the proof of Gödel's incompleteness theorems and the development of the Church-Turing thesis in the early part of that century, true statements in mathematics were generally assumed to be those statements which are provable in a formal axiomatic system.

The same is true of proofs, which are often expressed as logically organized and clearly worded informal arguments, intended to convince readers of the truth of the statement of the theorem beyond any doubt, and from which arguments a formal symbolic proof can in principle be constructed.

The field of mathematics known as proof theory studies formal axiom systems and the proofs that can be performed within them.

Logic, especially in the field of proof theory, considers theorems as statements ( called formulas or well formed formulas ) of a formal language.

This incompleteness result is similar to Gödel's incompleteness theorem in that it shows that no consistent formal theory for arithmetic can be complete.

A deductive system is called complete if every logically valid formula is the conclusion of some formal deduction, and the completeness theorem for a particular deductive system is the theorem that it is complete in this sense.

Together with soundness ( whose verification is easy ), this theorem implies that a formula is logically valid if and only if it is the conclusion of a formal deduction.

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.

Gödel's incompleteness theorem, another celebrated result, shows that there are inherent limitations in what can be achieved with formal proofs in mathematics.

This is an immediate consequence of the completeness theorem, because only a finite number of axioms from Γ can be mentioned in a formal deduction of φ, and the soundness of the deduction system then implies φ is a logical consequence of this finite set.

To prove this theorem, Gödel developed a technique now known as Gödel numbering, which codes formal expressions as natural numbers.

During the summer of 1971, Colmerauer and Kowalski discovered that the clausal form of logic could be used to represent formal grammars and that resolution theorem provers could be used for parsing.

There is no formal distinction between a lemma and a theorem, only one of intention – see Theorem # Terminology.

Gödel's second incompleteness theorem ( 1931 ) shows that no formal system extending basic arithmetic can be used to prove its own consistency.

Gödel's theorem, informally stated, asserts that any formal theory expressive enough for elementary arithmetical facts to be expressed and strong enough for them to be proved is either inconsistent ( both a statement and its denial can be derived from its axioms ) or incomplete, in the sense that there is a true statement about natural numbers that can't be derived in the formal theory.

Theorems are derived deductively from objections according to a formal system of rules, sometimes as an end in itself and sometimes as a first step in testing or applying a theory in a concrete situation ; theorems are said to be true in the sense that the conclusions of a theorem are logical consequences of the objections.

In order to be proven, a theorem must be expressible as a precise, formal statement.

The most famous result is Gödel's incompleteness theorem ; by representing theorems about basic number theory as expressions in a formal language, and then representing this language within number theory itself, Gödel constructed examples of statements that are neither provable nor disprovable from axiomatizations of number theory.

Mathematical theorems, on the other hand, are purely abstract formal statements: the proof of a theorem cannot involve experiments or other empirical evidence in the same way such evidence is used to support scientific theories.

A theorem may be expressed in a formal language ( or " formalized ").

Logic / formal inference utilizes dependable processes that lead to a certain and firm conclusion in the fields in which it is applied.

The term " natural science " is used to distinguish the subject matter from the social sciences, which apply the scientific method to study human behavior and social patterns ; the humanities, which use a critical or analytical approach to study the human condition ; and the formal sciences such as mathematics and logic, which use an a priori, as opposed to factual methodology to study formal systems.

In computer programming, a precondition is a condition or predicate that must always be true just prior to the execution of some section of code or before an operation in a formal specification.

In computer programming, a postcondition is a condition or predicate that must always be true just after the execution of some section of code or after an operation in a formal specification.

As a consequence, for convergence of a sequence of elements of it then suffices that the coefficient of each power of Y converges to a formal power series in X, a weaker condition that stabilizing entirely ; for instance in the second example given here the coefficient of Y converges to, so the whole summation converges to.

Then R (( G )) is the ring of formal power series on G ; because of the condition that the indexing set be well-ordered the product is well-defined, and we of course assume that two elements which differ by zero are the same.

Expropriation is only allowed to serve the public interest and on the condition that prior formal assurance is given of ( full ) indemnity, meaning that some exact sum has to be determined.

In formal terms, a statement N is a necessary condition of a statement S if S implies N ( S N ).

In formal terms, a statement S is a sufficient condition of a statement N if S implies N ( S N ).

When the barrier field F is high enough for the CFE regime to be operating for metal emission at 0 K, then the condition k < sub > B </ sub > T < d < sub > F </ sub > provides a formal upper bound ( in temperature ) to the CFE emission regime.

In the theory of formal languages, the Myhill – Nerode theorem provides a necessary and sufficient condition for a language to be regular.

It also includes force used directly to preclude or impede the mission and / or duties of US forces, including the recovery of US personnel or vital US Government property .” A “ hostile force ” is defined as “ Any civilian, paramilitary, or military force or terrorist ( s ), with or without national designation, that have committed a hostile act, exhibited hostile intent, or have been declared hostile by appropriate US authority .” “ Armed forces ” are defined as “ The military forces of a nation or a group of nations .” Although the original Senate bill that produced the 1989 amendment intended to make the treatment of captives the operative qualifying condition for those held outside of formal armed conflict, this intent has only been sporadically enforced.

The formal garden which was designed by Lady Beaconsfield ( Queen Victoria created Mary Anne a Viscountess in her own right in 1868 ), has been restored to a similar condition to when occupied by the Disraelis.

In fact, if we consider these as formal generating functions, the existence of such a formal Euler product expansion is a necessary and sufficient condition that be multiplicative: this says exactly that is the product of the whenever factors as the product of the powers of distinct primes.

Although widely discussed, there had been scant information in the medical research literature until an article by Chalett and Nerenberg in Pediatrics 2000, which found little formal data regarding the condition but concluded that " The treatment is sexual release, or perhaps straining to move a very heavy object.

While such exclusive hiring halls do not, in a strictly formal sense, require union membership as a condition of employment, they do so in practical terms, in that an employee seeking to be dispatched to work through the union's hiring hall must either pay union dues or pay a roughly equivalent hiring hall fee.

The formal statement of the equilibrium condition of the Harris – Todaro model is as follows:

When Yingying's mother discovers what her daughter has done, she reluctantly consents to a formal marriage on one condition: Zhang must travel to the capital and pass the civil service examination.

As civilian non-combatants, according to Section XI, Article 6, of the 1907 Hague Conventions, merchant seamen "... are not made prisoners of war, on condition that they make a formal promise in writing, not to undertake, while hostilities last, any service connected with the operations of the war.

Flat feet ( also called pes planus or fallen arches ) is a formal reference to a medical condition in which the arch of the foot collapses, with the entire sole of the foot coming into complete or near-complete contact with the ground.

The last two conditions condition can be stated in less formal terms, that additively should be a free abelian group generated by a basis for A over.

The formal model underlying the hypothesis is the uncovered Interest Rate Parity condition which states that in absence of a risk premium, arbitrage will ensure that the depreciation or appreciation of a country's currency vis a vis another will be equal to the nominal interest rate differential between them.

He immediately stated that the condition of Ireland was topic of " all-absorbing importance " and went on to give a long speech in which all other matters were relegated to a formal mention at the end.