The Σ₁ lattice consists of Σ₁
arithmetical sentences modulo provable equivalence in
Peano Arithmetic.
We give an overview of the structure of this lattice,
compare it to the lattice of recursively enumerable sets,
discuss the latest progress in the study of the lattice
including extensions of lattice embeddings, its 1st order
theory, and, if time allows, the ordertypes of maximal paths through the
Priestley space of the lattice.
We point out connections to recursion theory, model theory
of arithmetic and self-reference,
and conclude with some open questions.
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