The Inglorious One wrote:Just note, Scott, that AXIOM 4 implies an AXIOM 5: there are no true axioms.
I was only giving an example and would likely require sitting down to consider them in more depth. I don't know if I agree or not with your own addition if only because by one interpretation, this might still defeat any following proof regardless. Your interpretation that it follows AXIOM 4, however, is not clear. To me, regular "contradiction" applies within one argument given a particular 'dimension'. But this should not be interpreted as an end to further reasoning. Rather, it should act as a 'function' to require finding a 'place' or "
dimension" that allows both to exist in a larger universal (logic universe).
Example,
Given a number line, if it should be found in some argument that something exists at point X
AND non-existing at X at the same time, this implies a 'perpendicular' dimension at that point of contradiction that contains its own number line where they meet at X. This means that all other points in the perpendicular line from that point are 'true' with respect to the additional dimension. For instance, if one were to say that their coordinates are at (x,y) = (0,3), with regards to the x-axis, x = 0 and somehow also x does not = 0. To someone stuck in only the first x-axis line, the point (0,3) is perceived as (x) = (0 AND -0) [at zero and not at zero]. Do you follow?
EDIT as addition: That is, assuming one demonstrate an apparent contradiction from some given or understood dimension or set of dimensions, no matter how they are defined or map to reality, a
solution to contradiction is to find a place where they
do exist. In the above example, if you are trapped in a world that only perceives that something cannot both be x and not-x, label a place where they CAN exist, even if untrue, such that it defines another line, the y-axis, where an infinite possible solutions exist except at the point of intersection of the x-axis.
Given a world with only a line, if it can be found that somewhere on that line a real contraction exists, this is resolved by the process of creating dimensions.