Re:A Bad Trip to Infinity
by btilly on Saturday March 25, 2023 @05:44PM
(#35307753)
You can construct a mapping where an infinite number in the one are mapped to an infinite number in the other.
You can't always construct a mapping where all of one maps to all of the other.
Classical mathematics concludes from this that one set truly has more things than the other does. These presentations always assume that classical mathematics is right. But it ACTUALLY depends on philosophical assumptions that are both unprovable, and questionable.
In particular, classical mathematics assumes that it makes sense to talk about whether a statement is absolutely true or false. And to build constructions that require a series of decisions based on the absolute truth value of the statement. This despite the fact that we do not know whether it is true, have no procedures to determine it, and in some cases the statement is independent of our axioms.
Attempts to create finite parallels to this type of reasoning inevitably run into self-referential paradoxes and contradictions. It appears that classical mathematics avoids such paradoxes from the simple fact that nobody can actually carry out these impossible procedures. If we could, then we would certainly find similar contradictions.
People have attempted to figure out what mathematics would look like if we limited ourselves to things we can prove true and false, instead of making statements about the truth value of things that we have (and may never have) any proof of. The results go by names such as "constructivism" and "intuitionism". In those systems some infinite sets have more self-referential structure, but none has "more" elements than any other.
There is no logical reason to choose classical mathematics over these alternatives. Only arguments about philosophy and convenience help us choose. I wish that this fact was more often acknowledged.
You can't always construct a mapping where all of one maps to all of the other.
Classical mathematics concludes from this that one set truly has more things than the other does. These presentations always assume that classical mathematics is right. But it ACTUALLY depends on philosophical assumptions that are both unprovable, and questionable.
In particular, classical mathematics assumes that it makes sense to talk about whether a statement is absolutely true or false. And to build constructions that require a series of decisions based on the absolute truth value of the statement. This despite the fact that we do not know whether it is true, have no procedures to determine it, and in some cases the statement is independent of our axioms.
Attempts to create finite parallels to this type of reasoning inevitably run into self-referential paradoxes and contradictions. It appears that classical mathematics avoids such paradoxes from the simple fact that nobody can actually carry out these impossible procedures. If we could, then we would certainly find similar contradictions.
People have attempted to figure out what mathematics would look like if we limited ourselves to things we can prove true and false, instead of making statements about the truth value of things that we have (and may never have) any proof of. The results go by names such as "constructivism" and "intuitionism". In those systems some infinite sets have more self-referential structure, but none has "more" elements than any other.
There is no logical reason to choose classical mathematics over these alternatives. Only arguments about philosophy and convenience help us choose. I wish that this fact was more often acknowledged.