The Implied Axioms of Science

When a child and first time heard of "logic", I was so amazed, and somehow amused, by the way they use only 2 values TRUE/FALSE; and later by the 2 values ZERO/ONE in computing, too. "How can it be?", I wondered. Why is any thing must either true or false? Why can't they be somehow-true-somehow-false, like "30% true"? Why don't we do computation with real numbers like 1.2315, but only with 0 and 1? When studied about the Theory of Computation and first met the Diangonal Argument, I was surprised about its strength, but soon get more surprised with the way Cantor choosed an element out of an arbitrary set! I thought that if we donot know the underlying structure, or if we cannot assign any structure to a set, how can we "pick something up" from that set, while a set is simply a "bag" and no more.

➠ Axiom of Choice

Cannot be satisfied with the simple explaination "Because we defined it, we can choose something from it!", I've searched for the definition of set. The rationale of my dissatisfaction is that if we can choose, then other actions like sorting must be able, but no one allows that, they just give a special allowance to the "choosing" action! Then, at last I've found that the "right to choose" is no more than an axiom! People accept it just because they feel it axiomatic!

➟ Axiom of Separation

Even though there is no axiomatic system for the whole science, but for science to be "clear" and "precise", all scientists presume that "A" and "not A" are always distinct, i.e. we can always separate things into one side and the other side. It sounds self-evident, but if we kick the word "of course" out of our mind, we can see that this separation is no more than an axiom.