You can encode halting of a program P as a comparison of a computable real number Q with a fixed number R by defining Q as 0.111..1 where each step of P adds on

You can encode halting of a program P as a comparison of a computable real number Q with a fixed number R by defining Q as 0.111..1 where each step of P adds one digit of 1 to Q’s expansion. P will halt iff Q is less than R=0.111… Any subset of reals is ordered as it inherits the usual order from reals. The existence of well ordering (related to AOC) is difficult issue). But the trichotomy of A>=<B does fail for a different but useful logic - remove the law of contradiction. There is a number e which is neither equal nor not equal to 0, with e^2=0. This leads to simplifications of concepts and proofs - you can define derivatives without limits for instance. This topic is studied in synthetic differential geometry. But the real response to the comment ‘everything is just a word game’ is ‘just’ is not apt. You are free to fix rules of the game, once done you face questions which are possibly beyond your ability to answer. A person could run a program checking id Fermat equations had solutions in 1950’s. Only In 1990’s we know after great advances (like discovering a route between mountain ranges) that this program wont halt (or ZFC is inconsistent which would be even more surprising).