Tarski’s Undefinability Theorem and the Diagonal Lemma

The Diagonal Lemma Fails in Aristotelian Logic

Gödel's theorem does not need an introduction. But it is generally not acknowledged that all instances of Gödel's sentence are vacuous (see below). On the contrary Aristotelian logic permits only sentences that are not vacuous. The consequences of these observations are surprising. What does the article title mean? First of all by " Aristotelian logic " we will understand the logic of the tradi...

Fatou's Lemma and the Lebesgue's Convergence Theorem

For simplicity, we adopt the following rules: X denotes a non empty set, F denotes a sequence of partial functions from X into R with the same dom, s1, s2, s3 denote sequences of extended reals, x denotes an element of X, a, r denote extended real numbers, and n, m, k denote natural numbers. We now state several propositions: (1) If for every natural number n holds s2(n) ≤ s3(n), then inf rng s...

The diagonal lemma as the formalized Grelling paradox 1

György Serény Gödel's diagonal lemma (which is often referred to as fix-point or self-referential lemma) summarizes very succinctly the ability of first-order arithmetic to 'talk about itself', a crucial property of this system that plays a key role in the proof of Gödel's incompleteness theorem and in those of the theorems of Tarski and Church on the undefinability of truth and undecidability ...

The LSB Theorem Implies the KKM Lemma

Let S be the unit d-sphere, the set of all points of unit Euclidean distance from the origin in R . Any pair of points in S of the form x,−x is a pair of antipodes in S. Let ∆ be the d-simplex formed by the convex hull of the standard unit vectors in R. Equivalently, ∆ = {(x1, ..., xd+1) : ∑ i xi = 1, xi ≥ 0}. The following are two classical results about closed covers of these topological spaces:

A TRANSLATION THEOREM 1 Lemma