theorem$82791$ - translation to ελληνικό
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theorem$82791$ - translation to ελληνικό

THEOREM
Lob's Theorem; Löb's Theorem; Lob's theorem; Loeb's theorem; Loeb theorem; Lob theorem; Löb theorem; Loeb's Theorem

theorem      
n. θεώρημα
inscribed angle         
  • Animated gif of proof of the inscribed angle theorem. The large triangle that is inscribed in the circle gets subdivided into three smaller triangles, all of which are isosceles because their upper two sides are radii of the circle. Inside each isosceles triangle the pair of base angles are equal to each other, and are half of 180° minus the apex angle at the circle's center. Adding up these isosceles base angles yields the theorem, namely that the inscribed angle, <math> \psi </math>, is half the central angle, <math> \theta </math>.
  • For fixed points ''A'' and ''B'', the set of points ''M'' in the plane for which the angle ''AMB'' is equal to&nbsp;''&alpha;'' is an arc of a circle. The measure of ∠ ''AOB'', where ''O'' is the center of the circle, is&nbsp;2''&alpha;''.
  • Case: One chord is a diameter
  • Case: Center interior to angle
ANGLE FORMED IN THE INTERIOR OF A CIRCLE
Inscribed angle theorem
εγγεγραμένη γωνία
absolute convergence         
A PROPERTY OF INFINITE SERIES
Absolut convergence; Absolutely convergent; Absolute Convergence; Absolutely convergent series; Absolutely convergent improper integral; Absolute summability; Converges absolutely; Unconditional summability; Absolute convergence theorem; Absolutely summable
απόλυτη σύγκριση

Ορισμός

theorem
n.
Proposition (to be demonstrated), position, dictum, thesis.

Βικιπαίδεια

Löb's theorem

In mathematical logic, Löb's theorem states that in Peano arithmetic (PA) (or any formal system including PA), for any formula P, if it is provable in PA that "if P is provable in PA then P is true", then P is provable in PA. If Prov(P) means that the formula P is provable, we may express this more formally as

If
P A P r o v ( P ) P {\displaystyle PA\,\vdash \,{{\rm {Prov}}(P)\rightarrow P}}
then
P A P {\displaystyle PA\,\vdash \,P}

An immediate corollary (the contrapositive) of Löb's theorem is that, if P is not provable in PA, then "if P is provable in PA, then P is true" is not provable in PA. For example, "If 1 + 1 = 3 {\displaystyle 1+1=3} is provable in PA, then 1 + 1 = 3 {\displaystyle 1+1=3} " is not provable in PA.

Löb's theorem is named for Martin Hugo Löb, who formulated it in 1955. It is related to Curry's paradox.