Start by calling P the set of all prime numbers. Since there are an infinite number of primes, but P $\subseteq$ N, $|\text{P}|={\aleph }_0$. If we now call $A=\{n\in \text{<mstyle fontweight="bold">N</mstyle>}:\text{n has no repeated prime factors}\}\cup \{1\}$, then every element in A except 1 is expressible as a unique combination of prime factors, i.e. non-repeated elements of P. Every element of A can therefore be identified with a unique subset of P and vice versa, identifying $1\in \text{A}$ with $\varnothing \subseteq \text{P}$.

There exists an isomorphism between A and the power set of P, $P^{*}$ say. $|A|=|P^{*}|$. However, since A $\subseteq$ N, $|A|\le |\text{<mstyle fontweight="bold">N</mstyle>}|$, so $|\text{<mstyle fontweight="bold">N</mstyle>}|\ge |P*|$. Cantor showed that $|P|<|P*|$ (as a strict inequality, even with infinite cardinals) , and so $|P*|\ge {\aleph }_1$, and I think that this reusult is independent of the truth of the continuum hypothesis. This means that $|\text{<mstyle fontweight="bold">N</mstyle>}|\ge {\aleph }_1$, i.e. is uncountable! As this is clearly not the case, the above argument is paradoxical. However, I cannot find the flaw. Might it be the case that it is in some way linked more deeply than my comment on the penultimate step suggests to the fact that the continuum hypothesis is undecidable?

Many thanks,

Alex.