Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

$ω$-consistency for Different Arrays of Quantifiers

Paulo Guilherme Santos

Abstract

We study the formalized v statement by allowing the occurrence of different arrays of quantifiers in it. We prove that for some specific arrays of quantifiers we get consistency statements that are S-equivalent to the original $ω$-consistency statement (S denotes the basis theory to develop metamathematics). We end our paper by creating a theory of truth that proves each $ω$-consistency-statement.

$ω$-consistency for Different Arrays of Quantifiers

Abstract

We study the formalized v statement by allowing the occurrence of different arrays of quantifiers in it. We prove that for some specific arrays of quantifiers we get consistency statements that are S-equivalent to the original -consistency statement (S denotes the basis theory to develop metamathematics). We end our paper by creating a theory of truth that proves each -consistency-statement.

Paper Structure

This paper contains 5 sections, 11 theorems, 15 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Our results
  4. Conclusions
  5. Acknowledgement

Key Result

Proposition 1

$S\vdash \omega\text{-}\textup{Con}_T\leftrightarrow\omega\text{-}\textup{Con}^{\forall}_T$.

Figures (1)

  • Figure 1: $S$-implications and $S$-equivalences for the general notion of consistency $\omega\text{-}\textup{Con}^{\vec{Q}}_T$ depending on the array of quantifiers; $S$-implications are read from the top to the bottom, id est the upper part implies the lower part; all the formulas are provable in $\textup{Tr}(S)_T$. This image relies on our Main Theorem and Theorem \ref{['theo55']}.

Theorems & Definitions (29)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Definition 5
  • Theorem 1
  • ...and 19 more