He uses this as a proof of the existence of atoms:

In this case, if a proof uses this statement, researchers will often look for a new proof that doesn't require the hypothesis ( in the same way that it is desirable that statements in Euclidean geometry be proved using only the axioms of neutral geometry, i. e. no parallel postulate.

Empire magazine praised the film saying " the gaudily gory, virtuoso, hyper-kinetic horror sequel / remake uses every trick in the cinematic book " and confirms that " Bruce Campbell and Raimi are gods " and Caryn James of The New York Times called it " genuine, if bizarre, proof of Sam Raimi's talent and developing skill.

The proof uses Euclid's lemma ( Elements VII, 30 ): if a prime p divides the product of two natural numbers a and b, then p divides a or p divides b ( or perhaps both ).

The proof of Gödel's completeness theorem given by Kurt Gödel in his doctoral dissertation of 1929 ( and a rewritten version of the dissertation, published as an article in 1930 ) is not easy to read today ; it uses concepts and formalism that are outdated and terminology that is often obscure.

George Boolos has since sketched an alternative proof of the first incompleteness theorem that uses Berry's paradox rather than the liar paradox to construct a true but unprovable formula.

The proof of this statement uses the linearity of classical probability, and has exactly the same structure as the proof of the quantum no-cloning theorem.

Newman's proof is arguably the simplest known proof of the theorem, although it is non-elementary in the sense that it uses Cauchy's integral theorem from complex analysis.

Perelman's proof uses a modified version of a Ricci flow program developed by Richard Hamilton.

The proof uses the classification of finite simple groups.

Although BB84 does not use entanglement, Ekert's protocol uses the violation of a Bell's inequality as a proof of security.

The proof of Euler's identity uses only the formula for the geometric series and the fundamental theorem of arithmetic.

A sig block is easily forged, whereas a digital signature uses cryptographic techniques to provide verifiable proof of authorship.

* Centipede game, the Nash equilibrium of which uses a similar mechanism as its proof.

The alternative is formal software verification, which uses mathematical proof techniques to show the absence of bugs.

An alternative proof uses Prüfer sequences.

While the proof that is a Noetherian ring uses the order structure of, typical proofs in ring theory in general do not assume such additional structure on the ring.

One proof of the impossibility of finding a planar embedding of K < sub > 3, 3 </ sub > uses a case analysis involving the Jordan curve theorem, in which one examines different possibilities for the locations of the vertices with respect to the 4-cycles of the graph and shows that they are all inconsistent with a planar embedding.

The original proof uses hypergeometric functions and innovative tools from the theory of Hilbert spaces of entire functions, largely developed by de Branges.

In logic, a formal proof is not written in a natural language, but instead uses a formal language consisting of certain strings of symbols from a fixed alphabet.

This proof uses definition of even integers, as well as distribution law.

He also made important contributions to proof theory by clarifying the connections between classical logic, intuitionistic logic, and modal logic.

Gödel's incompleteness theorem marks not only a milestone in recursion theory and proof theory, but has also led to Löb's theorem in modal logic.

( 4 ) A formal modal logic proof of the necessity of identity.

The mathematical structure of modal logic, namely Boolean algebras augmented with unary operations ( often called modal algebras ), began to emerge with J. C. C. McKinsey's 1941 proof that S2 and S4 are decidable, and reached full flower in the work of Alfred Tarski and his student Bjarni Jonsson ( Jonsson and Tarski 1951 – 52 ).

Kurt Gödel wrote the first paper on provability logic, which applies modal logic — the logic of necessity and possibility — to the theory of mathematical proof, but Gödel never developed the subject to any significant extent.

A benchmark in Royce ’ s career and thought occurred when he returned to California to speak to the Philosophical Union at Berkeley, and ostensibly to defend his concept of God from the criticisms of George Holmes Howison, Joseph Le Conte, and Sidney Mezes, a meeting the New York Times called “ a battle of the giants .” There Royce offered a new modal version of his proof for the reality of God based upon ignorance rather than error, based upon the fragmentariness of individual existence rather than its epistemological uncertainty.

It is the most popular proof procedure for modal logics ( Girle 2000 ).

Other than the knowability thesis, his proof makes only modest assumptions on the modal nature of knowledge and of possibility.

For the frequent case of propositional logic, the problem is decidable but Co-NP-complete, and hence only exponential-time algorithms are believed to exist for general proof tasks.

* Conditional proof, in logic: a proof that asserts a conditional, and proves that the antecedent leads to the consequent.

As an example of a conditional proof in symbolic logic, suppose we want to prove A → C ( if A, then C ) from the first two premises below:

Hilbert is known as one of the founders of proof theory and mathematical logic, as well as for being among the first to distinguish between mathematics and metamathematics.

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

Concepts such as infinite proof trees or infinite derivation trees have also been studied, e. g. infinitary logic.

In propositional logic, disjunction elimination ( sometimes named proof by cases or case analysis ), is the valid argument form and rule of inference that allows one to eliminate a disjunctive statement from a logical proof.

Euclid frequently used the method of proof by contradiction, and therefore the traditional presentation of Euclidean geometry assumes classical logic, in which every proposition is either true or false, i. e., for any proposition P, the proposition " P or not P " is automatically true.

First-order logic also satisfies several metalogical theorems that make it amenable to analysis in proof theory, such as the Löwenheim – Skolem theorem and the compactness theorem.

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 ).

In modern logic texts, Gödel's completeness theorem is usually proved with Henkin's proof, rather than with Gödel's original proof.

The version given below attempts to represent all the steps in the proof and all the important ideas faithfully, while restating the proof in the modern language of mathematical logic.

As is true of all axioms of logic, the law of non-contradiction is alleged to be neither verifiable nor falsifiable, on the grounds that any proof or disproof must use the law itself prior to reaching the conclusion.

Although it was based on the proof methods of logic, Planner, developed at MIT, was the first language to emerge within this proceduralist paradigm 1969.

While it is one of the most commonly used concepts in logic it must not be mistaken for a logical law ; rather, it is one of the accepted mechanisms for the construction of deductive proofs that includes the " rule of definition " and the " rule of substitution " Modus ponens allows one to eliminate a conditional statement from a logical proof or argument ( the antecedents ) and thereby not carry these antecedents forward in an ever-lengthening string of symbols ; for this reason modus ponens is sometimes called the rule of detachment.

Intuitionistic logic substitutes constructability for abstract truth and is associated with a transition from the proof to model theory of abstract truth in modern mathematics.

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