In the same way that we work with delta functions as though they were ordinary functions, we can work with lexicographic preferences as though they were ordinary utility-function preferences.
There’s a pattern which comes up in math sometimes where some rules hold for all the functions in some set, and the rules also apply “in the limit” to things “on the boundary” of the set, even if the “boundary” is kinda weird—e.g. a boundary at infinity. Delta functions are a prototypical example: we usually define them as a limit of ordinary functions, and then work with them as though they were ordinary functions. (Though this is not necessarily the best way to think of delta functions—the operator formalism is cleaner is some ways.)
Lexicographic preferences fit this pattern: they are a limiting case of ordinary utility functions. Specifically, given the lexicographic utility sequence (u1,u2,...) and some real number a, we can construct a single utility function ua:=∑ia−iui. In the limit as a→∞, this converges to the same preferences as the lexicographic utility. So, just like we can work with delta functions like ordinary functions and then take the limit later if we have to (or, more often, just leave the limit implicit everywhere), we can work with lexicographic preferences like ordinary utilities and then take the limit later if we have to (or, more often, just leave the limit implicit everywhere).
To answer your exact question: lexicographic utilities are consistent with the coherence theorems, as long as we drop smoothness/boundedness assumptions. They shouldn’t be excluded by coherence considerations.
In the same way that we work with delta functions as though they were ordinary functions, we can work with lexicographic preferences as though they were ordinary utility-function preferences.
There’s a pattern which comes up in math sometimes where some rules hold for all the functions in some set, and the rules also apply “in the limit” to things “on the boundary” of the set, even if the “boundary” is kinda weird—e.g. a boundary at infinity. Delta functions are a prototypical example: we usually define them as a limit of ordinary functions, and then work with them as though they were ordinary functions. (Though this is not necessarily the best way to think of delta functions—the operator formalism is cleaner is some ways.)
Lexicographic preferences fit this pattern: they are a limiting case of ordinary utility functions. Specifically, given the lexicographic utility sequence (u1,u2,...) and some real number a, we can construct a single utility function ua:=∑ia−iui. In the limit as a→∞, this converges to the same preferences as the lexicographic utility. So, just like we can work with delta functions like ordinary functions and then take the limit later if we have to (or, more often, just leave the limit implicit everywhere), we can work with lexicographic preferences like ordinary utilities and then take the limit later if we have to (or, more often, just leave the limit implicit everywhere).
To answer your exact question: lexicographic utilities are consistent with the coherence theorems, as long as we drop smoothness/boundedness assumptions. They shouldn’t be excluded by coherence considerations.