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 […]

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 […]

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 […]

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 […]

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 […]

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?

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?

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?

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 […]

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 […]

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