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Such a proof is again a refutation 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.
The following famous example of a nonconstructive proof shows that there exist two irrational numbers a and b such that is a rational number. This proof uses that 2 {\displaystyle {\sqrt {2}}} is irrational (an easy proof is known since Euclid ), but not that 2 2 {\displaystyle {\sqrt {2}}^{\sqrt {2}}} is irrational (this is true, but the proof ...
Reductio ad absurdum, painting by John Pettie exhibited at the Royal Academy in 1884. In logic, reductio ad absurdum (Latin for "reduction to absurdity"), also known as argumentum ad absurdum (Latin for "argument to absurdity") or apagogical arguments, is the form of argument that attempts to establish a claim by showing that the opposite scenario would lead to absurdity or contradiction.
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. This is done by first proving a simple case, then also showing that if we ...
Logicism. Appearance. In the philosophy of mathematics, logicism is a programme comprising one or more of the theses that – for some coherent meaning of ' logic ' – mathematics is an extension of logic, some or all of mathematics is reducible to logic, or some or all of mathematics may be modelled in logic. [ 1 ]
Completeness is a property of the real numbers that, intuitively, implies that there are no "gaps" (in Dedekind's terminology) or "missing points" in the real number line. This contrasts with the rational numbers, whose corresponding number line has a "gap" at each irrational value. In the decimal number system, completeness is equivalent to ...
It is a proposition that is unconditionally false (i.e., a self-contradictory proposition). [2][3]This can be generalized to a collection of propositions, which is then said to "contain" a contradiction. History. [edit] By creation of a paradox, Plato's Euthydemusdialogue demonstrates the need for the notion of contradiction.
Kant's antinomies are four: two "mathematical" and two "dynamical". They are connected with (1) the limitation of the universe in respect of space and time, (2) the theory that the whole consists of indivisible atoms (whereas, in fact, none such exist), (3) the problem of free will in relation to universal causality, and (4) the existence of a necessary being.