# Questions tagged [set-theory]

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.

3,476 questions
889 views

### Sets, Universes, and the small Yoneda Lemma

Suppose we fix a universe $U$ and a $U$-small category $C$. The regular Yoneda lemma gives us some locally small (not necessarily locally U-small?) functor category $C'=[C^{op},Sets]$ with a fully ...
9k views

### Set theory for category theory beginners

I am wondering how much set theory is needed to read the basics of category theory, and what (preferrably short) book would be recommended. Usually I would just use naive set theory without worrying ...
2k views

### Does every set admit a rigid binary relation? (and how is this related to the Axiom of Choice?)

Let us say that a set B admits a rigid binary relation, if there is a binary relation R such that the structure (B,R) has no nontrivial automorphisms. Under the Axiom of Choice, every set is well-...
2k views

### The Importance of ZF

It seems as though many consider ZF to be the foundational set of axioms for all of mathematics (or at least, a crucial part of the foundations); when a theorem is found to be independent of ZF, it's ...
3k views

### Basis of l^infinity

Is it possible to exhibit a (Hamel) basis for the vector space l^infinity, given by the bounded sequences of real numbers?
2k views

### Independence of the continuum hypothesis on ZFC

Can anyone point out some good reference to understand how Paul Cohen proved that the continuum hypothesis is independent of ZFC? I know he used the so called forcing technique to construct two ...
2k views

### What is induction up to epsilon_0?

This is a question asked out of curiosity, and because I can't understand the Wikipedia page. I have often been told that PA cannot prove the validity of induction up to $\epsilon_0$, which has been ...
1k views

### Independence from Set Theory Axioms

I have often heard of various statements being independent from the axioms of set theory (typically ZFC). Some examples include The continuum hypothesis is probably the most famous The independence ...
2k views

### Can we disallow finite choice?

When people work with infinite sets, there are some who (with good reason) don't like to use the Axiom of Choice. This is defensible, since the axiom is independent of the other axioms of ZF set ...
2k views

### How many of the true sentences are provable?

Is there a natural measure on the set of statements which are true in the usual model (i.e. $\mathbb{N}$) of Peano arithmetic which enables one to enquire if 'most' true sentences are provable or ...
329 views

### Definition modifications without choice

What definitions or equivalencies between definitions for standard set theory objects (such as large cardinals) do not hold or do not carry through in the expected manner to the world without choice? ...
654 views

### Godel's 1st incompleteness theorem - clarification.

This should be a trivial question for people who know Gödel's 1st incompleteness theorem. I quote the statement the theorem from wikipedia: "Any effectively generated theory capable of expressing ...
862 views

### Logical endofunctors of Set?

What set-theoretic assumptions are necessary and sufficient to ensure the existence of a nontrivial (i.e. not isomorphic to the identity) endofunctor of the category Set which is logical (i.e. ...
927 views

### Characterizations of non-wellfounded models?

My question is whether there are any characterizations of non-wellfounded models of set theory. A wellfounded model is one that does not have any \epsilon-descending infinite sequences. I'm not asking ...
2k views

### How should we define “locally small”?

Let U be a Grothendieck universe, and U+ its successor universe (assume Grothendieck's universe axiom). Everybody agrees that a U-small category is a category whose sets of objects and morphisms are ...
584 views

### Controlling Ultrapowers

Say I start with some a transitive model of a large fragment of ZFC (say enough to run Łoś' Theorem externally) and a specific set x∈M. Now let's say I'm going to pick some M-ultrafilter U on x. ...
1k views

### Intuitionistic Lowenheim-Skolem?

Is there a version of the Löwenheim-Skolem theorem in intuitionistic logic? I'm particularly interested in the "downward" form. The standard proof I know uses the Tarski-Vaught test for ...
491 views

### Actions of finite permutation groups on hereditarily finite sets.

Model theorists have a lot to say about so-called definable imaginary elements of a structure. One way to formulate imaginaries is the following: Suppose $\mathcal{M}$ is a structure with universe $M$,...
419 views

### Cofinality of Theta if sharps exist

If ℝ# exists then why is cof(θL(ℝ)) = ω? Also I have the same question for the L(Vλ+1) generalization (if it's actually a different proof; I presume it isn't), i.e. if &...
30k views

### What are some reasonable-sounding statements that are independent of ZFC?

Every now and then, somebody will tell me about a question. When I start thinking about it, they say, "actually, it's undecidable in ZFC." For example, suppose A is an abelian group such that every ...
1k views

### What can't be described by categories?

I've been reading some "introduction to categories" type materials and have been impressed with the all-encompassing nature, but the skeptic in me wonders: is there any mathematical object that ...
801 views

### Models of the reals which have no unmeasurable sets

I recall being told -- at tea, once upon a time -- that there exist models of the real numbers which have no unmeasurable sets. This seems a bit bizarre; since any two models of the reals are ...
1k views

### Does Cantor-Bernstein hold for classes?

In Bonn, we've been have a discussion on the topic in the title: Suppose that A and B is are classes and that there are injections from A to B and fom B to A. Does it follow that there is a ...
2k views

### What do models where the CH is false look like?

Additionally, is there any intuitive way to visualize the cardinalities that result?