In his lecture notes he seems to put in a great amount of effort to prove results without utility functions.
Why does Ariel Rubinstein hate utility functions

I don’t know if it’s intentional, but the way in which this post makes no sense is actually the substantive reason to make sure you can phrase your definitions in terms of preferences rather than utility.
u(x)<u(y) means something (and can be stated in terms of induced preferences), but u(x)<0 doesn’t (and can’t).
His utility function assigns negative value to utility functions.

I don’t know if it’s intentional, but the way in which this post makes no sense is actually the substantive reason to make sure you can phrase your definitions in terms of preferences rather than utility.
u(x)<u(y) means something (and can be stated in terms of induced preferences), but u(x)<0 doesn’t (and can’t).His utility function assigns negative value to utility functions.
\[U_{A}(x) =  U_{B}(x)\]

I don’t know if it’s intentional, but the way in which this post makes no sense is actually the substantive reason to make sure you can phrase your definitions in terms of preferences rather than utility.
u(x)<u(y) means something (and can be stated in terms of induced preferences), but u(x)<0 doesn’t (and can’t).His utility function assigns negative value to utility functions.
Consider the negative exponential utility function.

Consider 1+ the negative exponential function, which describes identical preferences.
I don’t know if it’s intentional, but the way in which this post makes no sense is actually the substantive reason to make sure you can phrase your definitions in terms of preferences rather than utility.
u(x)<u(y) means something (and can be stated in terms of induced preferences), but u(x)<0 doesn’t (and can’t).His utility function assigns negative value to utility functions.
Consider the negative exponential utility function.

dudes, you are embarrassing urselves by taking these seriously,
Consider 1+ the negative exponential function, which describes identical preferences.
I don’t know if it’s intentional, but the way in which this post makes no sense is actually the substantive reason to make sure you can phrase your definitions in terms of preferences rather than utility.
u(x)<u(y) means something (and can be stated in terms of induced preferences), but u(x)<0 doesn’t (and can’t).His utility function assigns negative value to utility functions.
Consider the negative exponential utility function.

In his proofs, you see him use utility functions sometimes, but not others.
I think the point of view is that utility functions are representations which can be technically useful, but preferences are the fundamental objects of interest. So, for some results utility functions are employed and additional assumptions are employed and for other results, they are more general and these technical assumptions can be avoided.