Robdemanc wrote:

I am not asking here about what is good or bad. I am talking about our mindset that things have a true or false value.

The reason I ask is because I saw a documentary about infinity. It stated that if we had two infinte sets, can we say they are both equal?

Example:

A = {The set of all numbers}

B = {The set of all even numbers}

Can we say that A > B? If both sets are infinite how can we say one is greater than the other? Perhaps we could say that the statement A > B is neither true nor false, or is both.

The point here is that you do not define what ">" is. When you compare 2 numbers, the meaning of > is clear, the standard greater-than relation. For sets of arbitrary size, however, there are multiple ways to interpret >.

Some other posters in this thread have already mentioned comparing both sets on their "cardinality" (a technical term for size, which is extended for infinite sets). If you look at cardinality, then A = B. If you interpret the > sign as denoting "is a superset of", then A > B is true.

In mathematics, a statement is always either true or false. Not both or neither. But that does require that the statement is very detailed and that all terms are well defined. There is a small branch of math called "fuzzy logic", where a statement can also take on truth-values between the two extremes of true and false (if 0 is false and 1 is true, fuzzy logic allows the truthfulness of a statement to be any real in [0,1], allowing a statement to be "somewhat true"), but this is a sidetrack of mainstream math.

In real life, the question of binary truth is obscured by the fact that many statements are not factual and are polluted with subjective elements. "The weather is nice today" can be both true and false depending on who you ask.