[FOM] What do you lose if you ditch Powerset?

[Timothy Y. Chow]

Sorry if this is a dumb question with a well-known answer, but...

What sorts of theorems are lost if we delete from (say) ZFC the assumption
that every infinite set has a powerset?

Perhaps more precisely, what sorts of theorems of "classical" mathematics
are there that (1) do not, on the surface at least, require the existence
of powersets of infinite sets for one to make sense of their statements,
but (2) cannot be proved without the powerset axiom or something similar?

This question was posed to me by a friend who is not a mathematician but
enjoys mathematics and who has some constructivist intuitions.  Curiously,
he does not seem to be too bothered by Infinity or Choice, but does not
like the notion of an arbitrary set of integers or an arbitrary decimal
expansion.  This reminds me a little of Bill Richter's article on
sci.math.research some years ago.

   http://www.google.com/groups?selm=va684t4cf2.fsf%40conley.math.nwu.edu

Tim