A recalculation of set theory underlying Categories.

oorahb120206 February 8, 2018Continuum of Mathematics

1 Sets and Propositions

We will assume the simple tool of a arrow, and the intuition that our mind can associate one object of our imagination with a imagined slight; surely this is not to much of a stretch upon which to build a theory. A proposition P is a statement, a propositional function P( $\mathbf{x}$ ) is a statement equipped with a variable $\mathbf{x}$ allowed to vary over a domain of definition (a intuitive notion of those objects we wish to prove in contrast to our proposition.) of our choice. A arrow $\longrightarrow$ is a association of a object $\mathbf{a}$ (called the source of the arrow) with a logic value $\langle\textbf{T},\textbf{F}\rangle$ ,(called the target of the arrow) such that T is true and F is false. For a object $\mathbf{a}$ of the domain of definition we call $\mathbf{P(a)}$ the image of $\mathbf{a}$ under P, and write $\mathbf{P(a)}=\mathbf{T}$ or $\mathbf{P(a)}=\mathbf{F}$ . The satisfactory data that is generated by the propositional function is called the Range of P. The Range is partitioned into two $ur-sets$ (a collection of logical values with respect to some proposition). We denote the two ur-sets as T(x) the collection of all truth values, and F(x) as the collection of all false values. We denote the preimage of the ur-sets as $\mathbf{T^{-1}(x)}$ and $\mathbf{F^{-1}(x)}$ and call them sets.

Definition.

Set
Let S be a set, P be a proposition, and $\mathbf{a}$ an object, then it follows,

\mathbf{S}=\{\mathbf{a}:\mathbf{P(a)}\}

this is read as, the set S equals the collection of those $\mathbf{a}$ such that $\mathbf{P(a)}=\mathbf{T}$ We call the set $\mathbf{F^{-1}(x)}$ the relative complement of S denoted $\lnot\mathbf{S}$ , such that

\lnot\mathbf{S}=\{\mathbf{a}:\lnot\mathbf{P(a)}\}

In particular $\lnot\mathbf{S}=\mathbf{F^{-1}(x)}$ . Let U denote the domain of definition, then we write P: U $\longrightarrow\langle\textbf{T},\textbf{F}\rangle$ . This brings us to our first axiom.

Definition.

Axiom of Extenionality
Two sets are equal if and only it they are compiled of the same data. ¹¹I refrain from using the word contain since this in my mind suggest that the elements of a set are allocated to a specific space, and this is not the definition that I wish to portray. A set is a symbol denoting a satisfaction, that is the objects satisfying some proposition, even though the objects may be scattered through our space.

If we allow the variable in the propositional function to vary over sets then the ur-set $\mathbf{T^{-1}(x)}$ is called a Class. And from this we may derive all the tools we need to accomplish what is called mathematics using the objects in Category Theory. There is the matter of the $\in$ relation, to use this notation as a suggestive argument that the object $\mathbf{a}$ is associated with a truth logic value, is not a improper abuse of notation I would suppose. So if we wrote $\mathbf{a}\in\mathbf{S}$ to suggest that $\mathbf{P(a)}=\mathbf{T}$ this too is not a improper abuse of notation I would suppose.

Title	A recalculation of set theory underlying Categories.
Canonical name	ARecalculationOfSetTheoryUnderlyingCategories
Date of creation	2013-11-04 21:38:12
Last modified on	2013-11-04 21:38:12
Owner	joseph120206 (1000115)
Last modified by	joseph120206 (1000115)
Numerical id	1
Author	joseph120206 (1000115)
Entry type	Definition