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  2. Mathematical proof - Wikipedia

    en.wikipedia.org/wiki/Mathematical_proof

    Modern proof theory treats proofs as inductively defined data structures, not requiring an assumption that axioms are "true" in any sense. This allows parallel mathematical theories as formal models of a given intuitive concept, based on alternate sets of axioms, for example axiomatic set theory and non-Euclidean geometry.

  3. Conjecture - Wikipedia

    en.wikipedia.org/wiki/Conjecture

    Sometimes, a conjecture is called a hypothesis when it is used frequently and repeatedly as an assumption in proofs of other results. For example, the Riemann hypothesis is a conjecture from number theory that — amongst other things — makes predictions about the distribution of prime numbers. Few number theorists doubt that the Riemann ...

  4. Axiom - Wikipedia

    en.wikipedia.org/wiki/Axiom

    An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word ἀξίωμα ( axíōma ), meaning 'that which is thought worthy or fit' or 'that which commends itself as evident'.

  5. List of axioms - Wikipedia

    en.wikipedia.org/wiki/List_of_axioms

    This is a list of axioms as that term is understood in mathematics. In epistemology, the word axiom is understood differently; see axiom and self-evidence. Individual axioms are almost always part of a larger axiomatic system.

  6. Without loss of generality - Wikipedia

    en.wikipedia.org/wiki/Without_loss_of_generality

    For example, if some property P(x,y) of real numbers is known to be symmetric in x and y, namely that P(x,y) is equivalent to P(y,x), then in proving that P(x,y) holds for every x and y, one may assume "without loss of generality" that x ≤ y.

  7. List of paradoxes - Wikipedia

    en.wikipedia.org/wiki/List_of_paradoxes

    For example, some unicellular organisms have genomes much larger than that of humans. Cole's paradox: Even a tiny fecundity advantage of one additional offspring would favor the evolution of semelparity. Gray's paradox: Despite their relatively small muscle mass, dolphins can swim at high speeds and obtain large accelerations.

  8. Proof by contradiction - Wikipedia

    en.wikipedia.org/wiki/Proof_by_contradiction

    A typical example is the proof of the proposition "there is no smallest positive rational number": assume there is a smallest positive rational number q and derive a contradiction by observing that ⁠ q / 2 ⁠ is even smaller than q and still positive.

  9. Mathematical induction - Wikipedia

    en.wikipedia.org/wiki/Mathematical_induction

    Mathematical induction can be informally illustrated by reference to the sequential effect of falling dominoes. [1] [2]Mathematical induction is a method for proving that a statement () is true for every natural number, that is, that the infinitely many cases (), (), (), (), … all hold.