What question did this study set out to answer?

The aim is to introduce a metric-free definition of convergence known as indistinctio per gradus (IPG).

May 9, 2026Open Access

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.

Abstract

This paper proposes a new definition of convergence — indistinctio per gradus(IPG) — grounded in the structure of the generalized Cantor space C. The ε-δconditions, the Cauchy completion, and the Lebesgue integral are instances ofconvergence IPG. The Dominated Convergence Theorem illustrates the simplifi-cation this brings. The equivalence theorems in Part II confirm that every metricnotion of convergence is subsumed exactly. This is a constructive and inclusivegeneralization.

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