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Pentashagon comments on How to cheat Löb's Theorem: my second try - Less Wrong
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First, assume the existence of up(n) that replaces K with K+1 in the godel number of a sentence.
Let C="#[up(&'If K>0, and PPT.2 |- #[&C], then #[down(C)]')]" so the axiom "If K>0, and PPT.2 |- #[&C], then #[down(C)]" implies #[down(#[up(&'If K>0, and PPT.2 |- #[&C], then #[down(C)]')])] and cancelling the ups and downs shows we've just proven our own consistency.
I probably butchered the syntax, but couldn't something similar to this be used put Löb's theory back in business?