# zfc

1. ### Axiomatic set theory ZFC is inconsistent thus mathematics ends in contradiction

Axiomatic set theory ZFC is inconsistent, thus mathematics ends in contradiction: http://gamahucherpress.yellowgum.com/wp-content/uploads/MATHEMATICS.pdf Axiomatic set theory ZFC was in part developed to rid mathematics of its paradoxes, such as Russell's paradox. The axiom in ZFC developed...
2. ### ZFC Axiom of regularity

"In mathematics, the axiom of regularity (also known as the axiom of foundation) is an axiom of Zermeloâ€“Fraenkel set theory that states that every non-empty set A contains an element that is disjoint from A. In first-order logic, the axiom reads: {\displaystyle \forall x\,(x\neq \varnothing...
3. ### What effect does including the axiom of choice have on ZF?

I imagine the axiom of choice does something for Zermelo-Freankel set theory, as in it modifies it in some non-negligible way, otherwise there would be no point in having a ZFC in addition to ZF. What is its effect? What theorems can be proved with the axiom of choice that can't be proved...
4. ### Godel incompleteness in set theory ZFC

I am trying to understand what Godel incompleteness means in terms of ZFC. Let A be a set whose members satisfy predicate p. I suppose we can understand the predicate p as an axiom or a set of axioms that entail A - is that correct? If A is subject to Godel first incompleteness theorem...
5. ### Are all standard models of ZFC transitive?

Are all standard models of ZFC transitive?
6. ### Must all models of ZFC be at least countable?

Must all models of ZFC (in a standard formulation) be at least countable? Why I think this: there are countably many instances of Replacement, and so, if a model is to satisfy Replacement, it must have at least countably many satisfactions of it. Does my question only apply to first-order...
7. ### An example of a model that satisfies the axioms of ZFC?

From what I know, a model is a set of things that "give" an "interpretation" of the axioms such that the axioms are true of the "things" in the set. Is this right? I know there are many, many examples of such "sets of things," models, interpretations, structures, etc. of ZFC, but I don't have...
8. ### regular cardinal. ZFC

Suppose that \kappa is a regular cardinal such that 2^{\lambda} < \kappa for all \lambda < \kappa. Prove that H(\kappa)=\mathcal{V}_{\kappa}. Which axioms of ZFC does \mathcal{V}_{\kappa} satisfy? Here, for a given cardinal \kappa, H(\kappa) denotes the collection of sets whose transitive...
9. ### cardinality, ZFC

Prove that H(\beth_{\omega}) has cardinality \beth_{\omega}. Which axioms of ZFC are satisfied by H(\beth_{\omega})? Here, for a given cardinal \kappa, H(\kappa) denotes the collection of sets whose transitive closure has cardinality less than \kappa. Also, \beth_{\alpha} is defined by...