Random stop and probability

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Suppose you have a particle moving on a staight line at speed $v_0$. The particle will stop when a totally random event happens (for example a radioactive nucleus decay). So the stop point will be a random point $x_0$. Because the Cantor - Dedekind axiom, the calculated probability to stop exatly in that point is $P(x_0)=0$ that means there is no possibility for the particle to stop there. Despite this, the particle will stop in $x_0$. Is this a paradox or there is a simple explanation I miss? Thanks in advance.