For each member of ℘(T), there is a further truth specifying whether a given truth t1 is or is not a member of that subset. So there are at least as many truths as there are members of ℘(T).
power set(Used in Cantor's theorem: every set has cardinality strictly less than its power set.)
The set of all subsets of a given set.
℘(T)(in logic and set theory)
A mathematical symbol meaning 'the power set of T'—basically, a collection of all possible subsets (smaller groups) that can be made from a larger set T.
Another recent concern is whether it really is possible to know all truths. Grim (1988) has objected to the possibility of omniscience on the basis of an argument that concludes that there is no set of all truths. The argument (by reductio) that there is no set \(\mathbf{T}\) of all truths goes by way of Cantor’s Theorem. Suppose there were such a set. Then consider its power set, \(\wp(\mathbf{T})\), that is, the set of all subsets of \(\mathbf{T}\). Now take some truth \(t_1\). For each member of \(\wp(\mathbf{T})\), either \(t_1\) is a member of that set or it is not. There will thus corres...