It is not so much the question of what *additional* axioms are   
   needed, but which ones need to be removed. Physical quantities are   
   not numbers, so that not all the infrastructure of a field (or more   
   generally: of the number system) has physical meaning with them;   
   both rather only a substructure thereof.   
      
   The dimensions associated with the quantities are connected with   
   constraints on which ways they may be combined; namely that for   
   quantities of different physical dimensions, addition and subtraction   
   are not meaningful operations.   
      
   On Saturday, July 15, 2017 at 3:54:00 PM UTC-5, J. J. Lodder wrote:   
   > Before going on it is necessary to understand   
   > that dimensions have no physical reality.   
      
   Quite the contrary: they are deeply connected with the operational   
   meaning of the various quantities involved. The most notable example   
   being E and B whose "variability" of dimensions "in different   
   systems" is grounded in a confusion of concepts; namely the confusion   
   of E and B with D and H.   
      
   This is particularly important when the fields are non-abelian   
   because in that case, E and B aren't even the same KIND of quantity   
   as D and H (in particular, the former pair are Lie-vector valued   
   while the latter are Lie-CO-vector valued!) So the conflating of   
   them not only worsens the confusion but Sapir-Worfs matters of great   
   physical relevance (which later get put back in "by hand" under the   
   guise of moduli and coupling constants).   
      
   This is something that gets almost universally ignored in contemporary   
   liteature -- for which reason, the literature itself is also subject   
   to the same criticism. Your standpoint and its fallacious nature   
   is basically just an echo-chambering the contemporary literature,   
   which is just as wrong, so this needs to be addressed to them as   
   well.   
      
   E and B arise from the Lorentz force law; and as such are 2-form   
   valued:   
     F = (E_x dx + E_y dy + E_z dz) ^ dt   
         + B^x dy ^ dz + B^y dz ^ dx + B^z dx ^ dy.   
   This is actually how Maxwell originally presented the relevant   
   quantities (apart from the joining up of the E 1-form with dt, that   
   is). Because of their involvement in the force law, the 2-form must   
   have dimensions conjugate to that of electric charge. Calling the   
   dimensions of charge Q, and the dimensions of action H, (and using   
   L for length and T for time) that gives (E_x,E_y,E_z) the dimensions   
   of H/(QLT) and (B^x,B^y,B^z) the dimensions of H/(QL^T).   
      
   For both Maxwell and non-abelian fields, the field is derived from   
   the potential 1-form A = (A_x dx + A_y dy + A_z dz) - phi dt (and   
   again, Maxwell himself used this form originally for the 3-vector   
   A before using vector notation). That makes the dimensions of   
   (A_x,A_y,A_z): H/(QL) and of phi: H/(QT).   
      
   All of the foregoing is universally the case for ALL field theories   
   that can be cast in Lagrangian form, not just electromagnetism or   
   gauge theory. Corresponding to (A,phi) are the field configuration   
   variables; corresponding to (E,B) are the field gradients or   
   combinations thereof; together comprising the field kinematics.   
      
   They are always conjugate to the quantities that comprise the field   
   dynamics -- which (in a Lagrangian theory) are connected to the   
   respective derivatives of the Lagrangian density.   
      
   So, here, the Lagrangian (as a 4-form) would be   
     S = L dt ^ dx ^ dy ^ dz;   
   and the and would have the units of action H. Out of the total   
   differential: dS = dA ^ J - dF ^ G comes the conjugate field --   
   which comprise the field dynamics.   
      
   The respective forms are what give us the (D,H) fields:   
     G = D^x dy ^ dz + D^y dz ^ dx + D^z dx ^ dy   
         - (H_x dx + H_y dy + H_z dz) ^ dt;   
   while J gives us the source terms   
     J = rho dx ^ dy ^ dz - (J^x dy ^ dz + J^y dz ^ dx + J^z dx ^ dy) ^ dt;   
   where rho is the charge density and J the current density.   
      
   The units of G and Q are those of electric charge, so one has the   
   dimensions (H_x, H_y, H_z): Q/(LT), (D^x, D^y, D^z): Q/L^2;   
   (J^x,J^y,J^z): Q/(L^2 T) and rho: Q/L^3   
      
   All systems for (E,B,D,H) have these dimensions. Appearances otherwise   
   are rectified by bringing back out the various quantities that the   
   respective system has Sapir-Worfed away (and making explicit again   
   the *relevant* Physics it omitted by doing so!)   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)