Kreisel's analysis shows undecidability results like the halting problem can be established relative only to the formal system itself, without invoking Church's Thesis as a bridge principle to informal effectivity.

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Bridge principle(describes how Church's Thesis was being used): An assumption or rule that connects two different domains—in this case, connecting what's theoretically computable to what's actually doable in practice.
Church's Thesis(Also called the Church-Turing Thesis; surveyed in Section 1.6 of the source text.): The claim that the class REC coincides with the class of effectively computable functions.
Formal system(as used in logic and mathematics): A set of rules and symbols (like mathematical axioms) that you use to prove whether statements are true or false, similar to how a chess game has specific rules that determine what moves are legal.
Informal effectivity(what Church's Thesis bridges toward): The idea that something can be done in practice by a human using intuition and common sense, without needing a formal set of rules.

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