set-theory

In the Holmes' book, elementary set theory with universal set, is mentioned how NFU avoids the Burali-Forti paradox and is said that: let $\Omega$ the ordinal of all ordinals with $\leq$, then $\leq$ ...

I am going through the book Set Theory for the Working Mathematician by Ciesielski, and on p. 8 the author defines the intersection of a family $\mathcal{F}$ of sets as follows: $$ \bigcap \mathcal{F} ...

Horsten and Brickhill (2024) propose two methods for constructing non-Archimedean probability functions on the set-theoretic universe V: the finite snapshot approach and the bootstrapping approach. The mathematical execution is competent and the results within their framework are correct. This note raises three foundational concerns that the paper does not adequately address. First, the proposed …

A prime ideal theorem is a theorem stating that every proper ideal is contained in some prime ideal. A prime ideal theorem is typically equivalent to the ultrafilter principle (UF), a weak form of the axiom of choice (AC). We say ‘a’ prime ideal theorem (PIT) instead of ‘the’ prime ideal theorem, since we have not said what the ideals are in. We list some representative examples of prime ideal th…

A cardinal $\kappa$ is real-valued measurable (rvm) if there is a $\kappa$-additive probabilistic measure on $\kappa$ which vanishes on singletons and is moreover atomless, i.e., any positive set can ...

nLab weak limited principle of omniscience Context Foundations foundations The basis of it all mathematical logic deduction system , natural deduction , sequent calculus , lambda-calculus , judgment type theory , simple type theory , dependent type theory collection , object , type , term , set , element equality , judgmental equality , typal equality universe , size issues higher-order logic Set…

nLab lesser limited principle of omniscience Context Foundations foundations The basis of it all mathematical logic deduction system , natural deduction , sequent calculus , lambda-calculus , judgment type theory , simple type theory , dependent type theory collection , object , type , term , set , element equality , judgmental equality , typal equality universe , size issues higher-order logic S…

I am studying set theory right now, and my intuition is that the Axiom of Replacement is used to generate all sorts of sets starting from the Inductive set (from the Axiom of Infinity). We use the ...

Gödel's Constructible Universe L is an interesting model of ZF, since it showed the consistency of GCH and AC with ZF's axioms. However, that isn't important right now. Seeing this Wikipedia page, ...

I am reading some articles about the measure problem for my bachelor's degree thesis. In particular, I am interested in the fact that the axiom of choice implies the impossibility of solving the ...

I noticed that certain proofs for Cantor's theorem and $\not\exists S, P(S)\subset S$ are incredibly similar, and I made a general outline for proofs of this form, that I've been giving many names ...

Let $f : X \to Y$ and let $B \subseteq Y$. Then according to Wikipedia, the preimage of $B$ under $f$ is defined like \begin{align*} f^{-1}\bigl[B\bigr] = \bigl\{ x \in X \;:\, f(x) \in B \bigr\} ...

A strong subtree $T \subseteq 2^{<\omega}$ is a perfect tree of infinite height such that for all $s,t \in T$ for which $|s| = |t|$, $s$ splits in $T$ iff $t$ splits in $T$. The Ramsey theory of strong subtrees was studied by Milliken in A Partition Theorem for the Infinite Subtrees of a Tree. Let $\mathcal{M}$ denote the set of all strong subtrees. What is known about the forcing $(\mathcal{M},\…

The Cichon’s Diagram is a diagram that shows the relationships among ten small cardinals – four cardinals associated with the -ideal of sets of Lebesgue measure zero, four cardinals associated with the -ideal of sets of meager sets, the bounding … Continue reading →

This is the third in a series of four posts leading to a diagram called The Cichon’s Diagram. This post focuses on the -ideal of meager subsets of the real line. The links to the previous posts: the first post … Continue reading →

Andrei Chekmasov explores order and infinity The post Curiosities of linearly ordered sets appeared first on Chalkdust .

This post puts a spot light on a little corner in the world of set-theoretic topology. There lies in this corner a simple topological statement that opens a door to the esoteric world of independence results. In this post, we … Continue reading →

We explore the concept of emptiness in set theory, and explain how zero went from "nothing" to "something" The post Can you make something out of nothing? appeared first on Chalkdust .

Now there&#8217;s a simple theorem in set theory whose proof has always appeared a bit cloudy to me, since I&#8217;ve never been able to find it written in a straightforward manner. This theorem is the Schroeder-Bernstein Theorem, whose statement is utterly intuitive: Schroeder-Bernstein Theorem: Let and be sets. If there exists an injection , and an [&#8230;]

Set-Theoretic Foundations of Mathematics It is important to realize that in standard mathematics we attempt to characterize everything in terms of sets. This means notions such as natural numbers, integers, and real and rational numbers are defined in mathematics to be certain sets. Also, the very notion of a function is defined as a set....