**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?



Is there a set $S\subseteq [0,1]$ with $|S|=2^{\aleph_0}$ and distinct pairwise distances?
Consider $\Bbb R$ as a vector space over $\Bbb Q$, choose (Zorn) a basis $S$ consisting (after rescaling with rational scalars) of elements in $(0,1)$. Then we have no relation of the shape $\pm(s\_1-s\_2)=(s\_3-s\_4)$ for $s\_1,s\_2,s\_3,s\_4\in S$. The cardinality of the basis is also the required one.


