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

TL;DR

This paper explores the magmatic universe, defining analogues of set-theoretic objects like ordered pairs and numbers, and investigates which set-theoretic schemes such as Separation and Replacement hold within it.

Contribution

It introduces magmatic analogues of ordered pairs and relations, and establishes a restricted Separation scheme, highlighting limitations of Replacement in the magmatic universe.

Findings

Magmatic analogues of ordered pairs are definable.

Magmatic Separation scheme holds for certain formulas.

Replacement scheme fails in the magmatic universe.

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.