$$Let \; R = \{ x | x \notin x \}, then \; R \in R \iff R \notin R$$

Russell shared this paradox with Frege in a famous letter that undermined his work: http://www.thatmarcusfamily.org/philosophy/Course_Websites/Readings/Russell%20-%20Letter%20to%20Frege.pdf

The barber’s paradox is an applied version:

The barber is the “one who shaves all those, and those only, who do not shave themselves”. The question is, does the barber shave himself?

See wikipedia: https://en.wikipedia.org/wiki/Russell%27s_paradox