On 2/27/2014 12:18 PM, firstname.lastname@example.org wrote: > > Why have just these axioms of set theory been chosen? Any idea? >
I could answer that question.
You, however, would have neither the temperament nor patience.
Most of the axioms had been chosen to represent arithmetic in some form or another. The original intent -- and the priorities of the time -- had been an arithmetization capable of investigating transfinite arithmetic.
Union and pairing, for example, express the structure of a directed set.
But, it is more complex because the set notion derives from Cantor's topological notions. The part relation may be compared to the notion of magnitudes from Euclid. They are not intrinsically discrete.
Now, if you want angels on the head of a pin, find someone who respects Russell and the stench of philosophy that trailed off like dogshit dripping out of the ass end of "Principia Mathematica".