I originally asked this on MSE as math.stackexchange.com/questions/3674921/… . Asaf Karagila conjectured that $cf (|v\Bbb R$ \ $\Bbb R |)\ne \omega.$

(Add to my previous comment): Every uncountable closed set of reals has the cardinal $c$ of the reals. So if $|Y|<c$ then $Y$ has inner measure $0.$

This would require that every $Y\subset X$ with $|Y|=\aleph_1$ be a Lebesgue-null set and that every $Y\subset X$ with $|Y|=\aleph_2$ be non-measurable, with inner measure 0 and positive outer measure. I dk whether this is consistent.

There are more gems in Dandelin's figure but I need words : Let circles $k_1,\; k_2$ lie in planes $J_1.\; J_2 $ respectively. Let the ellipse be in plane $J_E$. Then $J_1\cap J_E ,\; J_2\cap J_E$ are the directrices.. Let line $l_P$ be the tangent to the ellipse at $P,$ with $l_P\subset J_E.$ Let $l_P$ meet $J_1,\;J_2$ at $Q_1\;,Q_2$ respectively. For $ i=1,2 , $ triangles $Q_iPP_i ,\; Q_iPF_i$ are congruent (as $Q_iP_i.\; Q_iF_i$ are tangent to sphere $G_i$), So angles $Q_iPP_i = Q_iPF_i$ . But $Q_1PQ_2$ and $P_1PP_2$ are lines. So angles $Q_2PF_2=Q_2PP_2=Q_1PP_1=Q_1PF_1.$

i recall that Mathematics Magazine, a more elementary cousin of mathematical Monthly (and also published by A.M.A.) used to have a fairly regular feature titled Proofs Without Words.... with some surprising pictures.

Looking forward to it.

@PaulLarson. i had thought that the def'n of $S$ unbounded in $X=[k]^{\omega}$ was $\forall t\in X\; \exists s\in S\; (t\subset s).$

Yes! Thanks. # 1644446857 has now been marked Duplicate but it was the one that I was looking for.