Reordered Computable Numbers

Abstract A real number is called left-computable if there exists a computable increasing sequence of rational numbers converging to it. In this article we are investigating a proper subset of the left-computable numbers. We say that a real number x is reordered computable if there exist a computable function $$f :\mathbb {N} \rightarrow \mathbb {N}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>:</mml:mo> <mml:mi>N</mml:mi> <mml:mo>→</mml:mo> <mml:mi>N</mml:mi> </mml:mrow> </mml:math> with $$\sum _{k=0}^{\infty } 2^{-f(k)} = x$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msubsup> <mml:mo>∑</mml:mo> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:mi>∞</mml:mi> </mml:msubsup> <mml:msup> <mml:mn>2</mml:mn> <mml:mrow> <mml:mo>-</mml:mo> <mml:mi>f</mml:mi> <mml:mo>(</mml:mo> <mml:mi>k</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:msup> <mml:mo>=</mml:mo> <mml:mi>x</mml:mi> </mml:mrow> </mml:math> and a bijective function $$\sigma :\mathbb {N} \rightarrow \mathbb {N}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>σ</mml:mi> <mml:mo>:</mml:mo> <mml:mi>N</mml:mi> <mml:mo>→</mml:mo> <mml:mi>N</mml:mi> </mml:mrow> </mml:math> such that the rearranged series $$\sum _{k=0}^{\infty } 2^{-f(\sigma (k))}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msubsup> <mml:mo>∑</mml:mo> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:mi>∞</mml:mi> </mml:msubsup> <mml:msup> <mml:mn>2</mml:mn> <mml:mrow> <mml:mo>-</mml:mo> <mml:mi>f</mml:mi> <mml:mo>(</mml:mo> <mml:mi>σ</mml:mi> <mml:mo>(</mml:mo> <mml:mi>k</mml:mi> <mml:mo>)</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> converges computably. In this article we will give some examples and counterexamples for reordered computable numbers and we will show that these numbers are closed under addition, multiplication and the Solovay reduction. Finally, we will also present a density theorem for reordered computable numbers.