A Question Regarding Reverse Mathematics and V=L Can all of "ordinary mathematics" ("ordinary mathematics" as understood by practitioners of reverse mathematics) be formulated in ZF[0]+V=L (ZF[0]is simply ZF without the Power Set Axiom)?  It is my understanding that second order arithmetic interprets ZF[0]+V=L (this from Colin Mc'Larty's paper 'Interpreting Set theory in Higher Order Arithmetic", ArXiv:1207.6357 [math.LO] section 3.2). 
 A: Apart from a few exceptions, all of the theorems that we study in reverse mathematics are provable in ZF+V=L. Indeed they are provable in "full second order arithmetic", $\mathsf{Z}_2$.  So ZF+V=L is much stronger than is needed to prove the theorems that are usually considered. 
When Simpson says "ordinary mathematics" in his book Subsystems of second-order arithmetic, he refers to his provisional definition on page 1:

We identify as ordinary or non-set-theoretic that body of mathematics which is prior to or independent of the introduction of abstract set-theoretic concepts. We have in mind such branches as geometry, number theory, calculus, differential equations, real and complex analysis, countable algebra, the topology of complete separable metric spaces, mathematical logic, and computability theory. 

This type of theorem will have a "countable version" in which we restrict discrete objects to be countable and continuous objects (e.g. complete metric spaces) to be separable. 
The hierarchy of subsystems comes by looking at which countable versions of these theorems imply which (countable versions of) others relative to a much weaker base system. 

Regarding "ordinary", there are really three aspects of these theorems, from my perspective:


*

*"ordinary" theorems do not depend on abstract set theory for their statements or proofs, but can be proved in the "usual" way. For example, a result in dynamics that uses the technique of enveloping semigroups is "extraordinary" in this sense. (Theorems that are ordinary in this sense would likely survive the discovery of an inconsistency in ZFC.) 

*"ordinary" theorems tend to relate to countable or separable objects. For example, Ramsey's theorem for $\mathbb{N}$ is an ordinary theorem, but the Erdős-Rado theorem that generalizes it to arbitrary cardinalities  is "extraordinary" in this sense.

*"ordinary" results tend to be found in undergraduate texts, and form the well-known core of mathematics. More esoteric theorems that are only found at the graduate level, or which are cooked up only to serve as examples in mathematical logic, are "extraordinary" in this sense.
Theorems are are "ordinary" in all three senses are particularly important for foundations of mathematics, and they are the theorems that traditional Reverse Mathematics is most interested in studying. 
A: You can reconstruct all "ordinary mathematics" in $\mathsf{ACA_0}$, or so the story goes. And $\mathsf{ACA_0}$ is a heck of a lot weaker than $\mathsf{ZF}$. So you can certainly reformulate all that "ordinary mathematics" in $\mathsf{ZF}$ + whatever. 
The 'F' bit of $\mathsf{ZF}$, i.e. the Axiom Schema of Replacement, in particular is humungously strong -- and it is certainly arguable that is overkill, much more than you  need even for non-ordinary mathematics like big chunks of descriptive set theory. Or so I'm told by local heroes.
