The set-theoretic platinum standard

It is commonly asserted that ZFC is the set-theoretic gold standard.
You can always assume these axioms, without specifying that this is
what you are doing.

However, in some cases, mathematicians have been quietly taking
Tarksi-Grothendieck set theory as the standard instead. This is
ZFC+inaccessible cardinals. In algebraic geometry, and therefore in
proofs that rely on algebraic geometry, use is sometimes made of such
proceedures as using the axiom of replacement within a universe within
a universe. The Mizar project, which was discussed on a JST thread,
bases its rock-solid, computer-verified proofs on Tarski-Grothendieck
set theory.

Does anyone want to vote in favor of the proposition that ZFC is OK,
but Tarski-Grothendieck isn't? Is Tarksi-Grothendieck the unofficial
platinum standard of mathematics?

.

Relevant Pages

  • Re: infinitesimal calculation ?
    ... >non standard objects based on the ZFC axioms. ... That sounds like you're talking about axioms for a nonstandard set ... equivalents of all sets in ZFC, not just sets of reals. ...
    (sci.math)
  • Re: infinitesimal calculation ?
    ... > can define everything that you need within ZFC. ... non standard objects based on the ZFC axioms. ... collection of all infinite countable real sequence is a ZFC set (I ...
    (sci.math)
  • Re: Godel proved maths inconsistent not incompleteness theorem
    ... abstract algebra at least uses the axioms of ZFC (not ... claim that abstract algebra might also use axioms that are ... for the claim that standard mathematics uses the axioms of ZFC set ...
    (sci.logic)
  • Re: Godel proved maths inconsistent not incompleteness theorem
    ... abstract algebra at least uses the axioms of ZFC (not ... claim that abstract algebra might also use axioms that are ... for the claim that standard mathematics uses the axioms of ZFC set ...
    (sci.logic)
  • Re: Godel proved maths inconsistent not incompleteness theorem
    ... claim that abstract algebra might also use axioms that are not ... used by Abstract Algebra is the ZFC Axiom of Extensionality? ... The system ZFCA uses FOL, the standard logic axioms and rules, ... for the claim that standard mathematics uses the axioms of ZFC set ...
    (sci.logic)