Indistinctio Per Gradus: A Metric-Free Definition of Convergence with Equivalence to Classical Convergence in Metric

Key Points

• The aim is to introduce a metric-free definition of convergence known as indistinctio per gradus (IPG).
• Proposes a definition based on the structure of the generalized Cantor space C.
• Explores examples including ε-δ conditions, the Cauchy completion, and the Lebesgue integral.
• Illustrates the simplification brought by the Dominated Convergence Theorem.
• Demonstrates that the new definition of convergence subsumes all classical metric notions of convergence.
• Confirms conversely that classical convergence notions are completely captured by IPG.