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Sagot :
George Canter showed that, for any set a, we have |a|<|p(a)|. This is a well-known theorem of canter, known as the 'Canter's Theorem'.
Proving this theorem:
Proof: At the first instance, we need to show that, A bar ≤P(A) bar, we will define the injection by f:A→P(A) by: f(a)={a}. Let g be the contradiction.
We need to prove that there is no bijection g:A→P(A).
Now, Let: S={a∈A:a∉g(a)}⊆A.}
In the present condition, there are two possibilities, first, x∈S and second, x∉S.
Therefore,
1. If x∈S, then x∉g(x)=S, i.e., x∉S, a contradiction.
2. If x∉S, then x∈g(x)=S, i.e., x∈S, a contradiction.
Thus, such a bijection, is not possible.
This theorem, coined by Canter, basically implies that there is no largest cardinal number present. There are 'n' number of cardinal numbers present, i.e., there are infinite number of cardinal numbers.
Now, suppose that, A is a set of all sets, this was proved by the continuum hypothesis. But, the continuum hypothesis can't be proved. It can't be disproved also.
To know more about the Cantor's Theorem:
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