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Working, or at least claiming to work, in ZFC is fairly standard, but that doesn’t make it the definition of mathematical truth. As a sibling comment mentioned

Working, or at least claiming to work, in ZFC is fairly standard, but that doesn’t make it the definition of mathematical truth. As a sibling comment mentioned, most mathematicians have a sense of truth that is not bound to any axiom system. I don’t think it’s contradictory to work in ZFC whilst simultaneously having a non-axiomatic notion of mathematical truth. I would hazard a guess that most (all?) working research mathematicians would accept the truth of the Gödel sentence for their preferred axiom system (and deductive calculus), be it ZF/ZFC or TG or something else entirely, so I cannot accept the claim that they see the “standard” notion of truth as being relative to all models of some axiom system. You might think this is just nit-picking, but if it’s fine (in the sense that this is still “standard”) to add an arbitrarily large set of Pi_1 formulae to ZFC from repetitions of Gödel 1, then I don’t think we can say that ZFC is the standard basis of mathematical truth because this cannot be justified in ZFC; there must be some other (standard) notion of mathematical truth used to justify this.

constantcrying 3y ago
>Working, or at least claiming to work, in ZFC is fairly standard Which is why I called it "standard". > I don’t think we can say that ZFC is the standard basis of mathematical truth You literally just said that working in ZFC is "standard". A standard is a social agreement, standards can be completely false and absurds, while being standards. >As a sibling comment mentioned, most mathematicians have a sense of truth that is not bound to any axiom system. Which I agree with.
- denotational 3y ago
  You said: > 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.”
  - constantcrying 3y ago
    >I think this is at odds with: 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.

constantcrying 3y ago

>Working, or at least claiming to work, in ZFC is fairly standard Which is why I called it "standard". > I don’t think we can say that ZFC is the standard basis of mathematical truth You literally just said that working in ZFC is "standard". A standard is a social agreement, standards can be completely false and absurds, while being standards. >As a sibling comment mentioned, most mathematicians have a sense of truth that is not bound to any axiom system. Which I agree with.

denotational 3y ago
You said: > 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.”
- constantcrying 3y ago
  >I think this is at odds with: 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.