  Given a Banach space $X$, we say that a sequence $\{x_n\}$ in the unit ball
of $X$ is $L$-orthogonal if $\Vert x+x_n\Vert\rightarrow 1+\Vert x\Vert$ for
every $x\in X$. On the other hand, an element $x^{**}$ in the bidual sphere is
said to be $L$-orthogonal (to $X$) if $\|x+x^{**}\|= 1+\Vert x\Vert$ for every
$x\in X$. A result of V. Kadets, V. Shepelska and D. Werner asserts that a
Banach space contains an isomorphic copy of $\ell_1$ if and only if there
exists an equivalent renorming with an $L$-orthogonal sequence, whereas a
result of G. Godefroy claims that containing an isomorphic copy of $\ell_1$ is
equivalent to the existence of an equivalent renorming with $L$-orthogonals in
the bidual. The aim of this paper is to clarify the relation between
$L$-orthogonal sequences and $L$-orthogonal elements. Namely, we study whether
every $L$-orthogonal sequence contains $L$-orthogonal elements in its
weak*-closure. We provide an affirmative answer whenever the ambient space has
small density character. Nevertheless, we show that, surprisingly, the general
answer is independent of the usual axioms of set theory. We also prove that,
even though the set of $L$-orthogonals is not a vector space, this set contains
infinite-dimensional Banach spaces when the surrounding space is separable.
