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Results tagged with set-theory
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user 290
forcing, large cardinals, descriptive set theory, infinite combinatorics, cardinal characteristics, forcing axioms, ultrapowers, measures, reflection, pcf theory, models of set theory, axioms of set theory, independence, axiom of choice, continuum hypothesis, determinacy, Borel equivalence relations, Boolean-valued models, embeddings, orders, relations, transfinite recursion, set theory as a foundation of mathematics, the philosophy of set theory.
13
votes
Does anyone still seriously doubt the consistency of $ZFC$?
This is basically a long comment. I like Nik Weaver's careful distinction between the question of whether ZFC is consistent and the question of whether it is (e.g. arithmetically) sound, and would lik …
4
votes
Conditions equivalent to finiteness
Here are two fun ones off the top of my head. I don't know if you want these to be in separate answers:
A set $X$ is finite iff the free vector space $k[X]$ is finite-dimensional, for any field $k$. …
2
votes
Product decompositions and maps from product of initial object with itself
To my mind it's cleaner to take opposite categories and talk about coproduct decompositions of affine schemes, where $\text{Spec } \mathbb{Z} \times \mathbb{Z}$ is just "two points" (the coproduct $2 …
- 118k
19
votes
Are there non-diagonal proofs for Cantor's continuum and Godel's incompletness theorems?
This isn't an answer but a proposal for a precise form of the question. First, here is an abstract form of Cantor's theorem (which morally gives Godel's theorem as well) in which the role of the diago …
- 118k
16
votes
Accepted
Can this informal argument (for the fact that almost all reals in the unit interval are irra...
You can make sense of the uniform probability distribution on lots of infinite sets, notably any compact topological group $G$, where "uniform probability distribution" should mean "normalized Haar me …
- 118k
7
votes
Accepted
Another adjoint pair: Definable sets and set-builder formulas
Given any pair of sets $X, Y$ (let me not be too specific about what "sets" means because what follows is robust with respect to changes in the definition), any relation $R : X \times Y \to 2$ whatsoe …
- 118k
7
votes
Spectra of elements of a Banach algebra and the role played by the Hahn-Banach Theorem.
I do not believe that the Hahn-Banach theorem is necessary. At some point I had planned on writing up a blog post verifying this but I lost the motivation...
The idea is that you can prove Liouville …
- 118k
5
votes
Choice Function on the Powerset of the Reals
A choice function on $\mathcal{P}(\mathbb{R}) \backslash \{ \emptyset \}$ lets you construct a well-ordering of $\mathbb{R}$, and a well-ordering of $\mathbb{R}$ lets you construct a non-measurable su …
- 118k
3
votes
What are interesting families of subsets of a given set?
Some people are interested in coarse structures. I am told they allow one to study the "large-scale" rather than "small-scale" structure of spaces. The Wikipedia article has references.
- 118k
4
votes
Easy proof of the uncountability of bijections on natural numbers
Bijections of the natural numbers are equinumerous with functions $\mathbb{N} \to \mathbb{N}$, which are equinumerous with continued fraction expansions of positive irrational numbers.
Edit: Here's …
- 118k
10
votes
Confusion over a point in basic category theory
It would be best to talk about the category of sets first, I think. Any isomorphism class of sets shows up so many times that a given isomorphism class doesn't itself form a set - for example, $\{ 1, …
- 118k
2
votes
How do we construct (in a vector space) a chain of countable dimensional subspaces that ca...
Some subset of a basis of $V$ can be put into bijection with the first uncountable ordinal $\omega_1$. Consider the subspaces spanned by each initial segment of $\omega_1$, all of which have countabl …
- 118k
61
votes
Accepted
What practical applications does set theory have?
The purpose of set theory is not practical application in the same way that, for example, Fourier analysis has practical applications. To most mathematicians (i.e. those who are not themselves set th …
- 118k
7
votes
Independence from Set Theory Axioms
I'm not qualified to answer your question, but the clearest introduction I've ever read to these ideas is Timothy Chow's A beginner's guide to forcing, which tries to explain the mathematical techniqu …
- 118k