From Cantor's diagonal argument and Russell's antinomy to Gödel's incompleteness and Turing's undecidability–classical results about infinity are usually read as limitations to be endured within an infinitary foundation. This paper proposes a different diagnosis. We show that the standard derivations uniformly instantiate reductio patterns once one isolates the specific infinitary postulate each result requires. We organise the landscape into three normal forms by premise: (AT) absolute totalities (``the set of all sets/ordinals''), (IR) a single actually infinite registry (global listings of formulas/proofs or machines/inputs), and (AC\ (infty\) ) unrestricted Choice over uncountable families. In each case, baseline coherence assumptions (consistency, and where relevant completeness or basic measure) combined with the corresponding infinitary premise yield contradiction or impossibility. We recast Gödel I as an inconsistent triad (Consistency + Completeness + IR), so that rejecting IR–rather than completeness–is a coherent resolution. Crucially, the finite/periodic foundation is taken as ontologically primary: it recovers the practical functionality often attributed to infinitary methods (global choice becomes definable; periodic/equivariant choice handles repeated families; AC-dependent pathologies cannot arise; and diagonal ``escape'' fails without a global infinite registry). The infinitary framework is retained only as an idealisation, a convenient approximation language layered on top of the finite/periodic reality. The philosophical payoff is a conservative, practice-preserving foundation in which mathematics is finite, relational, and paradox-free.
Yosef Akhtman (Tue,) studied this question.