Sure. For example, if P is the Continuum Hypothesis, eg “There exists a set with a cardinality greater than that of the integers but less than that of the real numbers”.
In this case we know that ZF cannot prove this statement true and also cannot prove it false. In this case, I’d be tempted to say that it really is not true and not false.
(I’m told that this statement would have to be either true or false in any “model of ZF”, but I don’t understand model theory well enough to have a good grasp of what that means. It might be the case that what I’m really saying is that I’m unwilling to grant “truth status” to model A but not model B if both models satisfy the axioms of ZF. I’ll probably have to to both more learning and more thinking before I can clarify my thoughts on this any further, but anyone reading this should feel free to prod me with thought experiments, cutting questions, etc.)
Sure. For example, if P is the Continuum Hypothesis, eg “There exists a set with a cardinality greater than that of the integers but less than that of the real numbers”.
In this case we know that ZF cannot prove this statement true and also cannot prove it false. In this case, I’d be tempted to say that it really is not true and not false.
(I’m told that this statement would have to be either true or false in any “model of ZF”, but I don’t understand model theory well enough to have a good grasp of what that means. It might be the case that what I’m really saying is that I’m unwilling to grant “truth status” to model A but not model B if both models satisfy the axioms of ZF. I’ll probably have to to both more learning and more thinking before I can clarify my thoughts on this any further, but anyone reading this should feel free to prod me with thought experiments, cutting questions, etc.)