Fundamental Examples    I asked this question over mathoverflow
and it yielded around 100 examples. They are not equally
fundamental and they are not equally suitable to be regarded as
“examples,” but overall it is a very good list.  If you see some
important example missing please, please add it.  Here are the
examples classified to areas. (Of course, sometimes, the same
example may fit several areas.)  Logic and foundations:
\aleph_\omega (~1890),  Russell’s paradox (1901), Halting
problem (1936), Goedel constructible universe L (1938), McKinsey
formula in modal logic (~1941), 3SAT (*1970), The theory of
Algebraically closed fields (ACF) (?),    Brachistochrone
problem (1696), Ising model (1925), The harmonic oscillator (?),
Dirac’s delta function (1927), Feynman path integral (1948),
Real and Complex Analysis:   Harmonic series (14th Cen.) and
Riemann zeta function (1859), li(x), The elliptic integral that
launched Riemann surfaces (*1854?), Chebyshev polynomials
(?1854) punctured open set in C^n</a> (Ha