If induction only licenses perception of constructibility without proving the infinite generalization, then transfinite arithmetic and completed infinities lose any rigorous foundation.

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Completed infinities(as what Cantor's work addressed): The idea of infinity as a finished, whole thing you can study as a complete object, rather than just an endless process that keeps going.
Constructibility(as used in mathematics and computer science): The quality of being able to build or compute something using a defined set of tools or rules.
Infinite generalization(as used in logic and mathematics): A conclusion that applies to an endless or unlimited set of things, rather than just a few examples—like saying 'all numbers follow this pattern' rather than 'these five numbers do.'
Rigorous foundation(as used in philosophy of mathematics and epistemology): A set of basic principles and proof methods so solid and logically airtight that everything built on top of them is absolutely reliable and justified.

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