Transfinite approximation of Hindman’s theorem

Some Transfinite Generalisations of Gödel's Incompleteness Theorem

Transfinite Equations in Transfinite Strings

.1 Kronecker's Approximation Theorem

Theorem 2 (Kronecker Approximation Theorem (1884)) ∃x ∈ Z s.t. Ax = b if and only if ∀y, y>b is an integer whenever y>A is an integral vector. Proof: To prove the forward implication, take an integral solution x∗. For any y, since y>Ax∗ = y>b, if y>A is integral then y>b must be an integer too. We next prove the converse. First, suppose that Ax = b does not have a solution (over the reals). The...

The Block Approximation Theorem

The block approximation theorem is an extensive generalization of both the well known weak approximation theorem from valuation theory and the density property of global fields in their henselizations. It guarantees the existence of rational points of smooth affine varieties that solve approximation problems of local-global type (see e.g. [HJP07]). The theorem holds for pseudo real closed field...

Analytic Approximation Theorem