On Dec 20, 7:11 am, Zuhair <zaljo...@gmail.com> wrote: > Some people are trying to prove from definability that only countably > many sets can exist. The idea is that a set is an object extension of > a predicate that is definable after a parameter free finite formula, > and since we have countably many such formulas and since under > Extensionality each parameter free formula can define only ONE set, > then there is One-One correspondence between sets and their defining > formulas and accordingly we can only have countably many sets. > > In symbols: > > X is a parameter free definable set <-> [Exist Phi. for all y. y in X > <-> Phi(y)] > > where Phi(y) is a formula having y as the sole free variable. > > Now is the above rationale correct? > > The answer is NO! > > If one insists that EVERY set must be parameter free definable, then > we will end up having UNCOUNTABLY many parameter free definable sets! > > Proof: > Lets assume we have countably may parameter free definable sets. > Then the set of all parameter free definable reals must be countable. > Accordingly there exist a bijection between the set N of all naturals > and the set R* of all parameter free definable reals. > Now since EVERY set is parameter free definable, then this bijection > is definable! > By The diagonal argument of Cantor, then the diagonal defined after > that bijection would be PARAMETER FREE definable real that is NOT in > the set of ALL parameter free definable reals. A CONTRADICTION!