Articles of set theory

Is this a set in New Foundations theory of sets: A={x∈X: x∉f(x)} when X=universal set and f(x)=x?

Does the formula A={x∈X: x∉f(x)} define a set in New Foundations theory of sets (NF) when X=universal set and f(x)=x? If not: WHY? It can be seen that many elements of universal set (X) satisfy the condition of the formula: If x=∅(it is possible since ∅ is an element of universal set(∅∈X)): Then: f(∅)=∅ and […]

Best known upper and lower bounds for $\omega_1$

What are the best known lower and upper bounds for $\omega_1$? Are there sharper lower bounds than $\varepsilon_0$? And are there any known upper bounds which can be explicitly constructed like $\varepsilon_0$? Since $\varepsilon_0$ is countable, $\varepsilon_0$ must be less than $\omega_1$. But in all of the 45 books on logic and set theory on […]

Which ZF axioms are satisfied in $V_{\omega+\omega}$?

Promoted to a separate question from $\mathbb R$ vs. $\omega+\omega$. Show that all axioms of ZF except the scheme of Replacement hold in $V_{\omega+\omega}$. Extension: Holds for all subsets of the universe. Separation: $x\in V_{\omega+\omega}$ means $x\in V_{\omega+\alpha}$ for some $\alpha$. But then also $y\in V_{\omega+\alpha}$ for any $y\subseteq x$, ie $V_{\omega+\omega}$ is closed under […]

Do the proofes in set theory rely on the semantics of the formulas used in the axioms?

Motivation: The Axiom of separation $$\forall w_1,\ldots,w_n \, \forall A \, \exists B \, \forall x \, ( x \in B \Leftrightarrow [ x \in A \wedge \phi(x, w_1, \ldots, w_n, A) ] )$$ is used to guaranty the existence of subsets by constructing formulas $\phi$ which, I assume, say true or false for every […]

In NBG set theory how could you state the axiom of limitation of size in first-order logic?

Limitation of size: “For any class $C$, a set $x$ such that $x=C$ exists if and only if there is no bijection between $C$ and the class $V$ of all sets.” In Von Neumann–Bernays–Gödel set theory how could you state the axiom of limitation of size in first-order logic similarly to the axioms stated in […]

The class of all functions between classes (NBG)

Is it possible in NBG (von Neumann-Bernays-Gödel set theory) to construct the class of all functions $X \to Y$ between two (proper) classes $X,Y$? I guess that this does not work. In the special case $Y=\{0,1\}$ we would get the class of all subclasses of $X$, which does not exist. Can someone confirm this?

Is there a constructible flat pairing function?

In ZFC set theory, is there a Skolem function f such that ZFC can prove f is a flat pairing function? And if so, can someone explicitly give me a formula?

Uncountable linearly independet family in $K^\mathbb{N}$

Let $K$ be a field. Consider the vector space $K^\Bbb{N}$ of $K$-sequences. Is there an uncountable linearly independent set of vectors in this vector space? If Yes, can you name it explicitely? Does this work for modules as well?

Relative merits, in ZF(C), of definitions of “topological basis”.

Typical Terminology: A basis $\mathcal{B}$ for a topology on a set $X$ is a set of subsets of $X$ such that (i) for all $x\in X$ there is some $U\in\mathcal{B}$ such that $x\in U$, and (ii) if $x\in U\cap V$ for some $U,V\in\mathcal{B}$, then there is some $W\in\mathcal{B}$ such that $x\in W\subseteq U\cap V$. The […]

Are $\sigma$-algebras that aren't countably generated always sub-algebras of countably generated $\sigma$-algebras?

It is well known that there are countably generated sigma-algebras containing sub-sigma-algebras that cannot be countably generated. (Some tail sigma-algebras serve as examples.) Question. Suppose $\mathcal{A}$ is a sigma-algebra of subsets of $X$ that cannot be countably generated. Does there exist a countably generated sigma-algebra $\mathcal{B}$ on $X$ such that $\mathcal{A}$ is a sub-$\sigma$-algebra of […]