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The magmatic universe revisited: we define ordered pairs, relations, numbers and a special form of Separation

Athanassios Tzouvaras

Abstract

This is a companion article to \cite{Tz24}. We address the following two questions: 1) Can we define in the magmatic universe $M$ of \cite{Tz24} counterparts, or just analogues, of some very basic set-theoretic objects which are missing from $M$, specifically ordered pairs, binary relations, especially functions, as well as natural and ordinal numbers? 2) Are there restricted forms of the Separation and, perhaps, Replacement schemes that hold in $M$? We show the following: 1) Magmatic analogues of ordered pairs can indeed be defined by means of certain magmas called ``magmatic pairs''. However when we use them to generate relations and especially functions, some unsurmountable problems come up. These problems are due to the peculiarity of the elements of magmas to be distinguished into ``intended'' and ``collateral'' ones, a distinction due to their inherent relation of dependence. So magmatic functions are defined under very special conditions. 2) A certain class of formulas, called ``magmatic formulas'' is isolated, and the scheme of Separation restricted to these formulas, called ``Magmatic Separation Scheme'' (MSS), is proven to hold in $M$. On the other hand Replacement fails badly, and this is due to its functional form.

The magmatic universe revisited: we define ordered pairs, relations, numbers and a special form of Separation

Abstract

This is a companion article to \cite{Tz24}. We address the following two questions: 1) Can we define in the magmatic universe of \cite{Tz24} counterparts, or just analogues, of some very basic set-theoretic objects which are missing from , specifically ordered pairs, binary relations, especially functions, as well as natural and ordinal numbers? 2) Are there restricted forms of the Separation and, perhaps, Replacement schemes that hold in ? We show the following: 1) Magmatic analogues of ordered pairs can indeed be defined by means of certain magmas called ``magmatic pairs''. However when we use them to generate relations and especially functions, some unsurmountable problems come up. These problems are due to the peculiarity of the elements of magmas to be distinguished into ``intended'' and ``collateral'' ones, a distinction due to their inherent relation of dependence. So magmatic functions are defined under very special conditions. 2) A certain class of formulas, called ``magmatic formulas'' is isolated, and the scheme of Separation restricted to these formulas, called ``Magmatic Separation Scheme'' (MSS), is proven to hold in . On the other hand Replacement fails badly, and this is due to its functional form.
Paper Structure (10 sections, 17 theorems, 43 equations)

This paper contains 10 sections, 17 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. The content of the paper
  3. A brief overview of Tz24
  4. The consequences of dependence on the behavior of magmas
  5. Some preliminary technical facts about magmas
  6. Magmatic ordered pairs
  7. Magmatic products. Intended vs collateral objects
  8. Magmatic relations and functions in $\widehat{M}$
  9. Magmatic natural and ordinal numbers in $\widehat{M}$
  10. The question about Separation and Replacement in $M$

Key Result

Proposition 1.1

(Tz24) (i) For any $x,y\in {\cal P}(A)$, $y\subseteq x \Rightarrow y\preccurlyeq^+ x$, so ${\cal P}(x)\subseteq pr^+(x)$. (ii) If $x\in LO(A,\preccurlyeq)$, then the converse of (i) holds, i.e., for every $y\in {\cal P}(A)$, $y\preccurlyeq^+ x \Rightarrow y\subseteq x$, so $pr^+(x)\subseteq {\cal

Theorems & Definitions (25)

  • Proposition 1.1
  • Proposition 1.2
  • Proposition 2.1
  • Corollary 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Definition 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Theorem 3.4
  • ...and 15 more