A result from PCF theory

There is a very curious theorem by Shelah which states that:

If for every natural number $n$ we have $2^{\aleph_n} < \aleph_{\omega}$ then, $2^{\aleph_{\omega}} < \aleph_{\omega_4}$.

Helen Frankenthaler, The Human Edge, 1967

An other way to state this is this one: if $\aleph_{\omega}$ is a strong limit cardinal then: $$\prod_{n=0}^{\infty} \aleph_n < \aleph_{\omega_4} $$

Classification of dimensions for the non-existence of exotic spheres

It turns out that the only dimensions $n$ for which a sphere $S^n$ admits a unique differentiable structure are exactly $n=1, 2, 3, 5, 6, 12, 56, 61$.

In all other dimensions, except $n=4$, we get the existence of exotic spheres.

Vilhelm Hammershoi The farm, 1883

What happens at dimension $4$ is still largely unknown and it goes by the name of smooth Poincare conjecture, the only open case of generalized Poincare conjecture.

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χαῖρε, ξεῖνε, παρ᾿ ἄμμι φιλήσεαι

Odys, a, 123
Nicolas de Staël, Abstract Figure, 1954

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