We show that the so-called weak Markov's principle (WMP) which
states that every pseudo-positive real number is positive is underivable in
![${\cal T}^{\omega}:=$](Abs/img1.gif)
E-HA
![$^{\omega}+$](Abs/img2.gif)
AC. Since
![${\cal T}^{\omega}$](Abs/img3.gif)
allows to
formalize (at least large parts of) Bishop's constructive mathematics this
makes it unlikely that WMP can be proved within the framework of Bishop-style
mathematics (which has been open for about 20 years). The underivability even
holds if the ineffective schema of full comprehension (in all types) for
negated formulas (in particular for
![$\exists$](Abs/img4.gif)
-free formulas) is added which
allows to derive the law of excluded middle for such formulas.