[bmacho]

"Carrier" is much more a minority slang than a well-known mathematical term. "Carrier set" is sometimes mentioned as an alternative, but the common term for that is "underlying set".

I have no idea what TFA tries to say, it seems to argue about aesthetics, I am not really into that. If it is very clear, and the authors enthusiasm about their favourite is sticky then sure, but TFA is unclear to me.

[qazxcvbnm]

It seems to me that the author is attempting to put together some analysis on the essential meaning, purposes, and distinctions of type theory and set theory. While what on offer may not satisfy you or me, it seems a task worthy of doing.