  We investigate integration of classes of real-valued continuous functions on
(0,1]. Of course difficulties arise if there is a non-$L^1$ element in the
class, and the Hadamard finite part integral ({\em p.f.}) does not apply. Such
singular integrals arise naturally in many contexts including PDEs and singular
ODEs.
  The Lebesgue integral as well as $p.f.$, starting at zero, obey two
fundamental conditions: (i) they act as antiderivatives and, (ii) if $f =g$ on
$(0,a)$, then their integrals from $0$ to $x$ coincide for any $x\in (0,a)$.
  We find that integrals from zero with the essential properties of $p.f.$,
plus positivity, exist by virtue of the Axiom of Choice (AC) on all functions
on $(0,1]$ which are $L^1((\epsilon,1])$ for all $\epsilon>0$. However, this
existence proof does not provide a satisfactory construction. Without some
regularity at $0$, the existence of general antiderivatives which satisfy only
(i) and (ii) above on classes with a non-$L^1$ element is independent of ZF
(the usual ZFC axioms for mathematics without AC), and even of ZFDC (ZF with
the Axiom of Dependent Choice). Moreover we show that there is no mathematical
description that can be proved (within ZFC or even extensions of ZFC with large
cardinal hypotheses) to uniquely define such an antiderivative operator.
  Such results are precisely formulated for a variety of sets of functions, and
proved using methods from mathematical logic, descriptive set theory and
analysis. We also analyze $p.f.$ on analytic functions in the punctured unit
disk, and make the connection to singular initial value problems.
