On Sun, Jun 29, 2014 at 6:05 PM, Alexander Solla <alex.solla@gmail.com> wrote:

Is x == x being false really any more surprising than x + 1 == x being true?

Yes, a lot more. Equality is a symmetric relation. So "x equals x" is true for every x. A logic/language in which that is not true is unsound.

x + 1 == x is just an equation. Coming up with a theory where it holds is straightforward. For example, addition followed by the "fractional part" operation, on the set [0,1].

Your theory is wrong. This is the same problem you run into with "x == x" being False: you create theories from assumptions about how these things behave based on your experience with abstract mathematical objects, and there are values for which those assumptions do not hold. Surprise results when you run into those cases.

IEEE 754 floats under + and * are *not* ℝ. Or even ℚ. They just play them in the computer. There are simple large subsets of possible operands for which addition (and hence the other three operators) doesn't work like the mathematical realities they model.