Short version of question. Is there a set $S\subseteq [0,1]$ with $|S|=2^{\aleph_0}$ such that all points of $S$ have distinct pairwise distances?

Formal version of question. If $X$ is a set, let $[X]^2=\big\{\{x,y\}:x\neq y\in X\big\}$. Let $(X,d)$ be a metric space. Let $d_{\text{set}}:[X]^2\to \mathbb{R}$ be defined by $$\{x,y\}\in [X]^2\mapsto d(x,y)=d(y,x).$$ We say $S\subseteq X$ has the distinct pairwise distance property (dpdp) if the restriction of $d_{\text{set}}$ to $S$ is injective. Is there a set $S\subseteq [0,1]$ with (dpdp) and $|S|=2^{\aleph_0}$, where $[0,1]$ is endowed with the Euclidean metric?