Is forcing force on us

[Timothy Y. Chow]

On Fri, 21 Aug 2020, John Baldwin wrote:
> What is the distinction between arithmetic and set theory that makes 
> forcing possible?

I should clarify that I had in mind potentially "exotic" weakenings of ZF 
that may not have been studied much.  In other words, I'm imagining 
tinkering with the ZF axioms specifically to see what is forcing us to use 
forcing (as opposed to tinkering with the axioms to create systems of 
intrinsic interest).

But as for your question here, I tried asking something similar on 
MathOverflow a while ago.

https://mathoverflow.net/q/100792

Tim