I cannot judge how serious these are, I just put Nelson predicative arithmetic in Google and came up with lots of stuff:

http://www.springerlink.com/content/gnvmapgw6cx2v40b/

"at the Nelson meeting in Vancouver in June 2004." http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.110.478 "This paper starts by discussing Nelson’s philosophy of mathematics, which is a blend of mathematical formalism and a radical constructivism. As such, it makes strong assertions about the foundations of mathematic and the reality of mathematical objects."

http://math.ucsd.edu/~sbuss/ResearchWeb/nelson/

http://www.illc.uva.nl/Publications/ResearchReports/X-1989-01.text.pdf

This one is skeptical: http://www.springerlink.com/content/v76473730365861x/

I cannot judge how serious these are, I just put Nelson predicative arithmetic in Google and came up with lots of stuff:

Link

"at the Nelson meeting in Vancouver in June 2004." http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.110.478 "This paper starts by discussing Nelson’s philosophy of mathematics, which is a blend of mathematical formalism and a radical constructivism. As such, it makes strong assertions about the foundations of mathematic and the reality of mathematical objects."

http://math.ucsd.edu/~sbuss/ResearchWeb/nelson/

http://www.illc.uva.nl/Publications/ResearchReports/X-1989-01.text.pdf

This one is skeptical: Link