set-theory

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....

In Salvador I talk to Samuel (or perhaps more precisely he talks to me) about lots of things: set theory, the axiom of choice, topological spaces, topological systems and weakenings, nearly countable cardinals and how to show inequalities between them, dialectica categories of different shapes... This time I gave a Dept Seminar on the Curry-Howard Correspondence and why I think more than simply c…

basic constructions: strong axioms further The axiom of choice is the following statement: This means: for every surjection of sets, there is a function (a section), such that Note that a surjection of sets can be regarded as a -indexed family of inhabited sets, while the existence of a section is equivalent to a choice of one element in each set of this family. This reproduces the more classical…

This is a short note listing some basic facts on set theory and set theory notations, mostly about cardinality of sets. The discussion in this note is useful for proving theorems in topology and in many other areas. For more … Continue reading →

This is a basic discussion on the first uncountable ordinal and its immediate successor . At heart, the ordinal is the smallest possible uncountable well ordered set. We use it to list uncountably many things (points, sets, spaces, functions, etc). Because is … Continue reading →