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Everything posted by wtf

  1. I hope nobody minds my resurrecting this thread. I wanted to mention that if we work in ZFC, the standard axiom system for modern math, there are sets that satisfy neither of those properties. Consider the set of real numbers; and if x and y are real numbers, define the relation x ~ y if it happens to be the case that x - y is a rational number. For example, (1/3 + pi) ~ pi because (1/3 + pi) - pi = 1/3 is rational. This is readily seen to be an equivalence relation. For all real numbers x, we have x ~ x; also for all x and y we have x ~ y implies y ~ x; and for x, y, and z we have x ~ y and y ~ z implies x ~ z. This equivalence relation partitions the real numbers into a set of mutually disjoint equivalence classes. By the Axiom of Choice (that's the C in ZFC) there is a set, call it V, that consists of exactly one representative from each of the equivalence classes. We know from the rules of set theory that V exists. We can not name any of its specific members; nor is there any property, other than membership in V, that's shared by the elements of V. We can't look at a real number and say whether that number is in V. If you ask, "Is 1/3 in V?" I answer that I don't know. I do know that V contains exactly one rational (why?) but I have no idea which one it is. V is a set whose members are not known, whose members can not be recognized as belonging or not belonging to V, and share no property in common. I should mention that this kind of construction is exactly why some people don't like the Axiom of Choice. However if you reject Choice, you get other unpleasant anomolies. So mathematicians just accept it and get used to using it. What that means metaphysically is a personal choice.
  2. New here. Amateur math fan. I'll take the other side of that one. I dispute the truth of CH. CH is known to be independent of the standard axioms of set theory (ZFC). The only way to prove CH true or false is to cook up a nonstandard model of the axioms and show that in that model, CH is either true or false. That's what Gödel and Cohen did, respectively. It may be the case that CH is true (or false) about the universe of sets, and that we just haven't found the right axiomatization. Another view is that the matter has no truth or falsity, since sets are just a formal abstractions with ultimately no meaning. Or it may be the case that there's some other resolution. But you can't say that CH is true without supplying further context. By the way, both Gödel and Cohen were of the opinion that CH is false. A heuristic argument for that position is by analogy with finite sets. Between the powerset of a 3 element set and the powerset of a 4 element set are sets of many intermediate cardinalities. Why shouldn't the same be true in the infinite case? Of course that's not even remotely any kind of proof; but it does give some understanding of why people think CH may well be false. The Wiki page for Cohen outlines his argument against CH along the same lines. He feels that the powerset operation is so powerful that it must make a great leap in cardinalities, not a one-unit step. He thought that the cardinality of [math]\mathbb{R}[/math] must be an extremely large Aleph.
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