  Recent measurements at the LHC of the p-p total cross section have reduced
the uncertainty in simulations of cosmic ray air showers. In particular of the
depth of shower maximum, called $X_{max}$. However, uncertainties of other
important parameters, in particular the multiplicity and elasticity of high
energy interactions, have not improved, and there is a remaining uncertainty
due to the total cross section. Uncertainties due to extrapolations from
accelerator data, at a maximum energy of $\sim$ one TeV in the p-p center of
mass, to 250 TeV ($3\times10^{19}$ eV in a cosmic ray proton's lab frame)
introduce significant uncertainties in predictions of $<X_{max}>$. In this
paper we estimate a lower limit on these uncertainties. The result is that the
uncertainty in $<X_{max}>$ is larger than the difference among the modern
models being used in the field. At the full energy of the LHC, which is
equivalent to $\sim 1\times10^{17}$ eV in the cosmic ray lab frame, the
extrapolation is not as extreme, and the uncertainty is approximately equal to
the difference among the models.
