For strong arithmetizable theories T there exists a primitive recursive selector that proves every instance of the consistency schema, yielding uniformity that cannot be internalized as the single sentence Con(T).
Detlefsen, What does G\"odel's second theorem say?, Philosophia Mathematica 9 (2001), no
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Uniformity of Consistency in Arithmetic and G\"odel's Second Incompleteness Theorem: Ein M\"archen
For strong arithmetizable theories T there exists a primitive recursive selector that proves every instance of the consistency schema, yielding uniformity that cannot be internalized as the single sentence Con(T).