I have been considering the possibility of an approach to set theory that represents the following property of the free variable x, in the terms of parameters h and k: ((x = h) or (x = k))) by means of a set existence schema that uses something other than the approach that Frege used before he was informed of Russell's paradox.
It has come to my attention that in some quarters this may be received as being a sign of obsessive dedication to my own pet theory, given that Frege's approach is still being used in ZFC.
How important is it, if we are seeking true foundations for mathematics, to ensure that the property of x that restricts x to being equal to one of two named variables be represented exactly as Frege represented it?
To explain my train of thought, I am responding to the following comment:
"it's not clear what the motivation for the ZFC pairing axiom would be" Seriously? In what universe do you not want to be able to form pairs? Why does your pet principle take precedence over how sets are actually used? That's absolutely an example of something "going wrong" - what you have to throw out to accommodate it is far too much. –
The above comment was posted in response to the question ...