For every set exists another stronger set

Mathematics Asked by 45465 on November 28, 2020

Is the following claim true?

For every set exists another stronger set.

By "if $$A$$ is stronger than $$B$$", we mean that $$B$$ is equinumerous to a subset of $$A$$, and $$A$$ is equinumerous to none of the subsets of $$B$$.

The question had been asked in a class and the professor used Cantor’s theorem to prove the claim.

I think the statement can be wittern formally as:

$$forall A,exists B: text{card}(A) < text{Card}(B)$$

One may use Cantor’s theorem as the professor did, but I think the claim is true as long as the universal set does not exist because if it does and is denoted by $$U$$ then from the definition we know that every other set is contained in $$U$$ and so the negation of the claim does hold.

I like to know whether I’m right and it’s interesting to know more about the claim, thanks in advance.

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