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() is a statement equipped with a variable 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 is a association of a object (called the source of the arrow) with a logic value ,(called the target of the arrow) such that T is true and F is false. For a object of the domain of definition we call the image of under P, and write or . The satisfactory data that is generated by the propositional function is called the Range of P. The Range is partitioned into two (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 and and call them sets.
Let S be a set, P be a proposition, and an object, then it follows,
this is read as, the set S equals the collection of those such that We call the set the relative complement of S denoted , such that
In particular . Let U denote the domain of definition, then we write P: U . This brings us to our first axiom.
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 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 relation, to use this notation as a suggestive argument that the object is associated with a truth logic value, is not a improper abuse of notation I would suppose. So if we wrote to suggest that this too is not a improper abuse of notation I would suppose.
|Title||A recalculation of set theory underlying Categories.|
|Date of creation||2013-11-04 21:38:12|
|Last modified on||2013-11-04 21:38:12|
|Last modified by||joseph120206 (1000115)|