[FOM] Why ZF as foundation of mathematics?

[Timothy Y. Chow]

Zuhair Abdul Ghafoor Al-Johar wrote:
> So why ZF is mentioned for that purpose if much weak class\set theories 
> can do the job of formalizing mainstream mathematics in set theory?

This question presupposes that weakness is a strength.

For most anyone not actively working in foundations, an existence proof of 
a foundation suffices, just as most people don't care to know anything 
about non-measurable sets other than that they exist.  It's useful to have 
a standard answer that you can cite, but for non-specialist purposes it 
doesn't matter too much what exactly the standard is.

If, of course, there is some motivation for coming up with a weaker 
foundation, then people can, and have, come up with alternatives to ZF. 
But I don't see the point of trying to change what people consider to the 
be the standard.  "The nice thing about standards is that there are so 
many to choose from."

Tim