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.

20
votes
1answer
970 views

Is Dependent Choice all we really need?

http://en.wikipedia.org/wiki/Axiom_of_dependent_choice Is DC sufficient for the understanding of objects that are countable in some suitable sense? For example, is DC sufficient for the full ...
21
votes
6answers
2k views

How can category theory help my research in set theory?

How can category theory help my research in set theory? I rarely use category theory as such in my current work, and one almost never sees any category theory in set-theoretic research papers or at ...
0
votes
1answer
330 views

Determining set membership from ordering relationships among disjoint sets

Suppose we have a set $P$ (an infinite set), and we have a partition of $P$ into (finitely many) disjoint subsets $P_i$, so that $P = \cup_i P_i$, and $P_i \cap P_j = \emptyset$ for $i \neq j$. ...
11
votes
1answer
1k views

V=L and a Well-Ordering of the Reals

A fairly simple question: I've read in multiple sources that Godel proved that if we accept the axiom of constructibility in ZFC, then we can create an explicit formula that well-orders the real ...
2
votes
4answers
1k views

nonstandard set theories

Does anyone know of good references for nonstandard set theories and their applications to various branches of mathematics like category theory, algebra, geometry, etc.? Edit: What I mean by "...
4
votes
2answers
1k views

Hausdorff dimension vs. cardinality

What is the relationship between the Hausdorff dimension and cardinality of a set? Specifically, assuming the Continuum Hypothesis, if a set has Hausdorff dimension greater than zero does, that imply ...
7
votes
1answer
899 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 ...
18
votes
10answers
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 ...
20
votes
7answers
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-...
17
votes
8answers
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 ...
14
votes
2answers
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?
5
votes
6answers
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 ...
25
votes
4answers
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 ...
3
votes
5answers
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 ...
21
votes
8answers
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 ...
17
votes
4answers
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 ...
3
votes
2answers
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? ...
4
votes
2answers
661 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 ...
19
votes
1answer
868 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. ...
6
votes
3answers
933 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 ...
12
votes
1answer
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 ...
8
votes
1answer
586 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. ...
10
votes
3answers
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 ...
7
votes
1answer
492 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 $M$,...
6
votes
2answers
421 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 &...
219
votes
31answers
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 ...
4
votes
7answers
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 ...
2
votes
5answers
803 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 ...
9
votes
4answers
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 ...
7
votes
10answers
2k views

What do models where the CH is false look like?

Additionally, is there any intuitive way to visualize the cardinalities that result?
27
votes
6answers
4k views

Does finite mathematics need the axiom of infinity?

A statement referring to an infinite set can sometimes be logically rephrased using only finite sets/objects. For example, "The set of primes is infinite" <-> "There is no largest prime". ...
10
votes
5answers
1k views

Questions about ordering of reals and irrationals

Three problems from G.Rosenstein "Linear orderings" (from the end of Chapter 2 and beginning of Chapter 4): 1) Is there a nondecreasing function from irrationals onto reals? 2) Is there a ...