Axiom of Induction???

Hi all,

In a theory that have V ( the proper class of all sets) in it, like
NBG and Morse-Kelley, Triclass theory? why not the following axiom
instead of infinity.


Axiom of Induction:


AfAm( (meV & f:V->V) -> Ec(mec & (Ax(xec -> f(x)ec) ) ) ).


read as: for every function f , for every class m such that m is a
set
and f is defined from V to V then their exist a set c such that m in
c
and Avery x in c implies that f(x) in c.



From this infinity can be proved.


Let f:V->V ,f(x)=xU{x} and let m=0 then we have
Ec(0ec & ( Ax(xec ->xU{x}ec))). Infinity Proved.

Also this axiom can prove the existence of {x} for every x.
By simplly letting f to be the identity function from V to V.


According to this axiom the set of all powers of a set is a set.


so for example lets take w={0,1,2,3,4,...}


w is a set ( the above axiom )


then P(w) is a set , and so is P(P(w)), etc... (Axiom of Power).


then P:V->V.


Then we have the set of w and its powers.


M = { w , P(w) , P(P(w)), P(P(P(w))),......... }


It is clear that this is a set in Triclass theory since every member
in M is a set and the whole M is equinumerous with w, which is a set
and thus sumbnumerous to V, thus M is subnumerous to V and thus M is
a
set.


Is their something wrong with this? Am I missing something?


Zuhair

.

Relevant Pages

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