Z cardinals
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From: zuhair on 31 Jan 2010 21:48 Here is another presentation of Z cardinals. Define H(b,d): H(b,d) <-> For all y ( y e b <-> for all z ( z e {y} -> z strictly subnumerous to d) ). so H(b,d) is a formula (in first order logic with identity) in two free variables b and d. so there is no intention here to stipulate that for every set d there exist a set b : H(b,d) this is not the case. Define(base-set) b is a base-set iff Exist d ( d is ordinal & H(b,d) ) also there is not intention here to stipulate that for every ordinal d there exist a set b such that H(b,d) this is not the case at all. In ZF- it can be proven that for some ordinal d there exist a set b such that H(b,d), this is obvious since when d=0 would would have a set that is 0 such that H(0,0) also when d=1, we would have the set 1 such that H(1,1), however in ZF- we cannot prove that for ordinals bigger than 1, on the other hand ZF- do not prove that we don't have a set b such that H(b,d) for ordinals bigger than d, so some models of ZF- can have these sets, some don't. Define (Pi(b)) by recursion: for every base-set b, Pi(b) = b if i=0 Pi(b) = P(Pi-1(b)) for every successor ordinal i Pi(b) = Union (j<i) Pj(b) for every limit ordinal i Define ( minimal for x ) for every set x , for every base-set b, for every ordinal i Pi(b) is minimal for x iff x subnumerous to Pi(b) & For every ordinal j (x subnumerous to Pj(b) -> i subset of j). So when Pi(b) is minimal for x, then Pi(b) is said to be the "minimal for x iterative power of the base-set b". Define (the nearest base-set to x): For every set x, for every b b is the nearest base-set to x iff b is a base-set & for all d (H(b,d) -> Exist an ordinal i ( Pi(b) is minimal for x & For every base set c, for every k, for every ordinal j ((H(c,k) &Pj(c) is minimal for x) -> (i subset of j & d subset of k)))). The Cardinality of a set x can be defined as: Define(Card(x)): Card(x)=A iff for all y ( y e A iff ( y equinumerous to x & for some base-set b, for some ordinal i (b is the nearest base-set to x & Pi(b) is minimal for x & y subset of Pi(b) ) ) ). In words: ------------------------------------------------------------------- Card(x) is the set of all sets Equinumerous to x, that are subsets of the minimal for x iterative power of the nearest base-set to x. ------------------------------------------------------------------- This approach albeit more complex than the one that I presented at http://www.cs.nyu.edu/pipermail/fom/2010-January/014327.html yet it is more appropriate to use with ZF-. The first approach lead to the confusion that I am defining H(x) for every x, which is not the case, so here in this approach I defined H as a formula in two free variables, which might help to avoid this confusion. Zuhair |