The Gödelian insight is all that we need
It's not necessary to see the consistency of an entire formal system. The ability to pass from one formal system to the Gödel sentence of that formal system is enough.
This kind of Gödelian insight, which is not captured by formal rules, is characteristic of mathematical insight and is non-algorithmic.

Roger Penrose (1990).
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Artificial Intelligence »Artificial Intelligence
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Are thinking computers mathematically possible? [7] »Are thinking computers mathematically possible? [7]
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No: computers are limited by Gödel's theorems »No: computers are limited by Gödel's theorems
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The problem of consistency »The problem of consistency
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We don't know that mathematics is consistent »We don't know that mathematics is consistent
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The Gödelian insight is all that we need
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The Gödelian insight has already been formalised »The Gödelian insight has already been formalised
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Roger Penrose »Roger Penrose
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