If an axiomatic system is consistent it means that if x logically follows, then “not x” does not logically follow.

While this might seem overly pedantic (hardly something to complain about in this discussion), I'd like to point out that this definition only matches the usual one if you also accept the law of non-contradiction. More precisely, a system is consistent if it does not contain a contradiction.

Also, your definitions of Mathematics and "logically follows" don't seem very... good. Did you make them up?

EDIT: changed "excluded middle" to "non-contradiction". duh.

ETA: Yeah, my point was stupid. Nevermind.