Does having publications on a completely separate field helps my graduate school application?
Should I ultimately decide to apply for PhD under the set theory/logic
department in the future?

It's up to you. Nothing you have said indicates that you should not pursue your dream.

How much would my past research experiences in other fields help in my
application?

A lot.
When I read the title, I thought it was going to be that you're applying to mathematics but published in biology, or something like that.  After reading the question I've learned that you've published a number theory paper in a small local journal, and may publish a paper heavy in functional analysis, and you want to apply for a PhD in set theory/logic. These aren't &quot;completely&quot; different fields, they are all mathematics subjects and there is always some overlap.
Having a publication, no matter what the field, does show at least a bit of aptitude for research, and that you have experience in academic research and publishing. Not all undergraduate students that get into prestigious PhD programs have publications (though it certainly would help). Even fewer will have a publication in precisely the same field as their PhD.