> truth being relative only to ZFC is the standard mathematical foundation
I think this is at odds with:
> most mathematicians have a sense of truth that is not bound to any axiom system
Re.:
> You literally just said that working in ZFC is "standard"
Nowhere did I say that considering ZFC to be the arbiter of mathematical truth is standard, in fact I’m claiming the opposite.
Deciding to adopt a particular set of axioms as standard just means that I’ll accept a proof from those axioms without question; it doesn’t mean that I believe mathematical truth is precisely that which is a syntactic consequence of ZFC/TG/whatever.
Perhaps I’m misinterpreting your claim? At the moment I’m reading it as: “The belief of the majority of working research mathematicians is that mathematical truth is defined relative to ZFC.”
I don't think so. Mathenatical statements are almost always framed in the context of ZFC. This does not contradict that mathematicians think ZFC is not an absolute truth.
>Perhaps I’m misinterpreting your claim? At the moment I’m reading it as: “The belief of the majority of working research mathematicians is that mathematical truth is defined relative to ZFC.”
All I am saying is that mathematicians are framing their results in the context of ZFC and that this makes it the "standard" theory. I think that statement is absolutely not controversial, even alternative theories are framed in opposition to ZFC.
I absolutely do not think that mathematicians believe that ZFC is "true", as in it is the one and only perfect set of axioms.
My initial argument (and I am sorry if that was unclear) was that IF you believe that mathematical truth is about formal derivations from axioms (ZFC would be such a theory, same as ZF or any variation) then either you have to say that there is one perfect system and all truth is relative to it alone or that there are multiple equally true, but incompatible, theories.
The "IF" is of course important and I don't think many mathematicians actually agree with the IF clause. I actually completely agree with: "Deciding to adopt a particular set of axioms as standard just means that I’ll accept a proof from those axioms without question; it doesn’t mean that I believe mathematical truth is precisely that which is a syntactic consequence of ZFC/TG/whatever." and I am sorry if I wasn't clear. I actually do not think there is any disagreement here.
> truth being relative only to ZFC is the standard mathematical foundation
I think this is at odds with:
> most mathematicians have a sense of truth that is not bound to any axiom system
Re.:
> You literally just said that working in ZFC is "standard"
Nowhere did I say that considering ZFC to be the arbiter of mathematical truth is standard, in fact I’m claiming the opposite.
Deciding to adopt a particular set of axioms as standard just means that I’ll accept a proof from those axioms without question; it doesn’t mean that I believe mathematical truth is precisely that which is a syntactic consequence of ZFC/TG/whatever.
Perhaps I’m misinterpreting your claim? At the moment I’m reading it as: “The belief of the majority of working research mathematicians is that mathematical truth is defined relative to ZFC.”