Wednesday, June 8, 2011

Some thought on the Axiom of Choice and Peano Axioms.

  1. A set is a "Peano set", i.e. a set satisfying Peano Axioms, if and only if it is a well ordered infinite set, with all finite initial chains (this may actually be a possible equivalent simpler alternative to Peano Axioms); 
  2. it does exist at least one infinite well ordered set if and only if does exist at least one Peano set;
  3. for any cardinality equivalence class k = { sets with the same cardinality }, we may define a specialized Axiom of Choice, AC(k): "for any non empty family of subsets of an element X of k it does exist a choice function";
  4. AC(k) is true if and only if any element X of k may be well ordered and hence it is "inductive" in this well order.

To my eyes these properties seem all true and they seem to point out that assuming the existence of at least one Peano set, i.e. of natural integers, imply assuming AC(Aleph0), in other words, assuming "Mathematical Induction" imply assuming AC for the countable cardinality.