The Alfven spectrum of resistive magnetohyclroclynamics (RMHD) consists of discrete. exponentially clamped modes. and the corresponding eigenvalues lie on specific cnrves in the complex plane with spacing of O(h1/2), where h denotes the resistivity. These curves are independent of resistivity for small resistivity. The ·Alfven Paradox· is that as resistivity decreases. the exponential damping is independent of the resistivity. and the discrete eigenmodes do not converge to the generalized eigennwcles of the ideal Alfven continumn. To resolve the paradox. the e­pseuclospectrum of the RMHD operator, L h is considered. l is in the e-pseuclospectrum if there exists a function, u, such that ||Lhu - lu|| e and ||u|| = 1. It is proven that for any e, the e­pseudospectrum contains the Alfvcn continuum for sufficiently small resistivity. Formal e­pseudoeigenmodes are constructed using the formal Wentzel-Kramers-Brillouin-Jeffreys solutions. and it is shown that the entire stable half-annulus of complex frequencies with r|w|2 = |k · B (x)| is resonant to order e, i.e. belongs to the e­pseudospectrum. In this frequency half-annulus. the 1 norm of the frequency response Green's function is proportional to exp( RM1/2). where RM is the magnetic Reynolds number.