set theory - Understanding ZFC - Mathematics Stack Exchange
Nov 23, 2023 · ZFC does generate a “theory”. Formally speaking, a theory is simply a set of sentences in the given formal language. Any set of axioms generates a theory, namely the …
Why did we settle for ZFC? - Mathematics Stack Exchange
Jan 10, 2018 · ZFC is named after Zermelo, but it is not like he went up to the mountain alone and came back with stone tablets that have been used unchanged ever since. He was one link in a …
Why can't ZFC prove an internal version of ZFC to be consistent?
Dec 8, 2025 · ZFC proving this statement can be shown in a metatheory to be equivalent to ZFC's consistency. Not quite. That is equivalent to ZFC being inconsistent.
logic - Consistency-proof of ZFC - Mathematics Stack Exchange
Sep 9, 2024 · There's no tension: "some theories stronger than ZFC prove the consistency of ZFC" doesn't in any way imply "every theory which proves the consistency of ZFC is strictly …
logic - Can the independence of a statement in ZFC itself be ...
Dec 23, 2024 · Under this interpretation, a standard example of a statement that is (assumed to be) independent of ZFC but this independence cannot be proved in ZFC is the existence of a …
Why is establishing absolute consistency of ZFC impossible?
Apr 17, 2016 · Why is establishing the absolute consistency of ZFC impossible? What are the fundamental limitations that prohibit us with coming up with a proof? EDIT: This post seems to …
Continuum Hypothesis in specific models of $ZFC$
Mar 25, 2025 · When we talk about $\mathbb {N}$ and $\mathbb {R}$ that is (supposedly) because we produced syntactically valid constructions of those sets from the ZFC axioms; as …
Why is ZFC incapable of interpreting second-order logic?
May 8, 2022 · The point is that using ZFC we can interpret (in the non-technical sense) second-order logic. ZFC gives us a way to understand the notion of sets and relations which are …
How many axioms does ZFC have? - Mathematics Stack Exchange
Dec 10, 2020 · The suggestion that "definite" should mean "first-order expressible" was made by Skolem about a decade later. Zermelo didn't like it at all; if I remember correctly, he …
Why are some Busy Beaver values independent from ZFC if ZFC …
Jan 1, 2024 · But then why can't ZFC come up with the value of BB (745)? After all, the 745-state BB is a halting Turing Machine (by definition of Busy Beaver), and we have established that …