set theory - Understanding ZFC - Mathematics Stack Exchange
Nov 23, 2023 · In terms of ZFC, this theory describes the properties of sets, and that include what sets can be constructed or otherwise proven to exist under the axioms of ZFC. It is, of course, …
logic - Why did we settle for ZFC? - Mathematics Stack Exchange
Jan 10, 2018 · $\begingroup$ 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. …
ZFC + "There exists an inaccessible cardinal" proves Con (ZFC)
This relies exactly on looking at the way we're defining ZFC in the object language. (Note that I said "axiom of ZFC," not "theorem of ZFC.") We can indeed whip up a code for a theory ZFC' …
About ZFC, peano's axioms, first order logic and completeness?
Mar 11, 2018 · ZFC is an effective theory in first-order logic which is sufficiently strong for the incompleteness theorems to apply (this is much weaker than being able to interpret Peano …
How many axioms does ZFC have? - Mathematics Stack Exchange
Dec 10, 2020 · ZFC doesn't have finitely many axioms because it has some axiom schemas. Let's look at Axiom Schema of Specification: $$ \forall X \exists Y \forall a \quad (a \in Y \iff a \in X …
What formula of ZFC defines the set of natural numbers?
Oct 19, 2021 · However, A is potentially wrong in that A assumes that the set $\mathbb{N}$, defined in ZFC, has anything to do with the "actual natural numbers". If ZFC were inconsistent, …
How strong is this second-order version of ZFC?
Mar 25, 2020 · If you prefer, we can embed this whole question inside first-order ZFC ("Does ZFC prove that ...?") since the relevant definitions (specifically, the definition of second-order logic …
elementary set theory - How to prove from ZFC that the …
Oct 28, 2020 · Stack Exchange Network. Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for …
Are category-theory and set-theory on the equal foundational …
That said, choice of foundations actually matters here. The category of (ZFC) sets does not exist in ZFC. Typically, to formulate category theory as used into something like ZFC requires …
In relatively simple words: What could be a model of $\\sf {ZFC}$?
Mar 29, 2016 · There are many options for such a metatheory. One option would be ZFC itself, except that (by the second incompleteness theorem), ZFC can't prove that there is a model of …