I started writing some rules for an alternative cardinal math. So, tell me what your problem is with them. So, I just wrote them, here are some ideas.
Ross
Rules for my cardinal math
1. Assume a set-theoretical universe where the underlying logic is ZF, or ZF with anti-foundation instead of foundation axiom, and inverse axiom. Anti-foundation axiom says "no sets of all sets that do not contain themselves."
2. Some sets are infinite, there are more than finite elements in that set. As set is infinite if either another infinite set has an injection into it or the set surjects onto any finite set.
3. Any pair of infinite sets may have a bijection, injection, and surjection between them, and infinite quantities of them. There is no upper bound on the number of mappings from one infinite set to another.
4. A finite set is the same size as another finite set if a bijection exists between them, a bijection is always an injection and a surjection.
5. An infinite set is larger than any finite set.
6. Any set has a number of distinguished orderings. The number of distinguished orderings is the total number of possible permutations of the set. For any set, a distinguished ordering may be selected into a class oforderings.
7. Given two orderings, the orderings can be compared on a synchronous basis such that the actual relative size of the two orderings can be determined.
8. Important in that process is the determination of construction rules or generating functions that when applied to finite or infinite quantities yield other finite or infinite quantities.
For example, the set of natural numbers is the set containing first zero and then for any n, n+1. The construction of the powerset is the set of all subsets of a set. The set of reals on [0,m] is the result of a scalar applied to the set [0,1].
Any set that is a subset of another is smaller than the superset, thus injectable at least once into the superset.
For example, there are more rationals than integers, and complex nubmers than reals.
9. So then, to determine the existence of an immediate injection of one infinite set to another, it is then necessary to compare the construction or generation rules to see which sequence "completes" first.
For example, the even integers immediately inject into the integers, the integers immediately inject into the powerset of the integers, etc. The construction rules of the set guarantee that one set based on another set and some construction rules that specify or determinately modify set size will have more, the same, or less elements than the input set.
10. Under these rules, the cardinal unit is the size of the natural numbers, omega or aleph_zero. These cardinal numbers exist, aleph_n+1= 2 ^ aleph_n.
11. Another unit that is related to |N| by a constant is the size of the unit real interval R[0,1]. For some cardinal coefficient x, 0 < x <1, |N|= x |R[0,1]|.
