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.
1,111 questions
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The existence of definable subsets of finite sets in NBG
This question is motivated by my preceding MO-question on (in)consistency of NBG theory of classes.
Let $\varphi(x,Y,C)$ be a formula of NBG with free parameters $x,Y,C$ and all quantifiers running ...
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Is this compactness property for "satisfiability on $\mathbb{R}$" consistent?
This was originally part of this older question of mine, but in retrospect that question should have been broken into two parts - this is the still-unanswered part.
Let $\Sigma$ be the language of ...
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What is the height (or depth) of $[\mathbb{N}]^\infty$?
(This question assumes familiarity with combinatorial cardinal characteristics of the continnum.)
Let $[\mathbb{N}]^\infty$ be the family of infinite subsets of $\mathbb{N}$,
partially ordered by $\...
- 3,104
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Absoluteness, reflection to ctms, and choice in outer models
Last night I was thinking about some related statements which follow from ZF+DC, but it actually seems they only need DC to hold in some outer model of the universe. In particular, let $M \models ZF.$ ...
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Limitations of determinacy hypotheses in ZFC
When considering (set-theoretic) games, we have three parameters we can adjust:
Definability of the payoff set
The set of legal moves
The length of the game
When working in $\textsf{ZFC}$, what are ...
- 1,170
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268
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Meeting a set of lines in $\mathbb{R}^n$
Fix an integer $n\ge 2$ and suppose that ${\cal L}$ is a set of lines in $\mathbb{R}^n$. Is there a set $M\subseteq \mathbb{R}^n$ with the following properties?
$M$ intersects all the elements of ${\...
- 48.9k
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Ramsey-theoretic properties of Erdős cardinals
The $\beta$-Erdős cardinal is defined as the least ordinal $\eta$ such that $\eta \to (\beta)^{\lt \omega}_2$, that is for every function $f: [\eta]^{\lt \omega} \to 2$, there is an $f$-homogeneous ...
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406
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Existence of a "diagonal" set in certain set systems
Let $\kappa\geq\aleph_0$ be an infinite cardinal, and suppose that ${\cal A}$ is a collection of subsets of $\kappa$ such that for all $A\in {\cal A}$ we have $|A| = \kappa$ and for $A,B\in {\cal A}$ ...
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How short can we state the Axiom of Choice?
How short can we state a principle which is equivalent with the Axiom of Choice under $ZF$? The principle should be a sentence in the language of set theory with only $\in$ and$=$ as extralogical ...
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Restricted notions of set-theoretic geology
We say that $W$ is a ground of the universe $V$ if $W$ is a model of ZFC and there is a poset $P\in W$ such that $W[G]=V$ for some $G$ which is $P$-generic over $W$. The Ground Axiom ($\text{GA}$) ...
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By replacing the Laver tables with nearly distributive fake Laver tables, can one produce algebras where the period of 1 grows fast?
I am now generalizing the notion of the classical Laver table and the fake Laver table to a larger class of algebras. A sequence of algebras $(\{1,...,2^{n}\},*_{n})_{n\in\omega}$ is said to be a ...
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Elementary submodels of V
Consider the claim:
(C) There is a transitive set $S \in V$ such that the structure $(S, \in)$ is an elementary submodel of $(V,
\in)$.
Obviously, this claim cannot be a theoreom of ZFC, by Godel's ...
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A question on simple $P_{\aleph_2}$-points
This question is motivated by discussion surrounding this MO question.
An ultrafilter $U$ on $\omega$ is a simple $P_{\aleph_2}$-point if it is generated by a sequence $\langle X_\alpha:\alpha<\...
- 7,125
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570
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Ultraproducts in the category of structures and elementary embeddings
A previous question on the categorical nature of ultraproducts had great answers, mostly categorically characterizing ultraproducts in the category of $L$-structures and homomorphisms for a fixed ...
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3
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When does the generalized Cantor space embed in a $\kappa$-compact space
The generalized Cantor space is the space $2^\kappa$, with basic open sets
$$
[\sigma] := \{f\in 2^\kappa : \sigma\subseteq f\},
$$
for $\sigma\in 2^{<\kappa}$.
A space is $\kappa$-compact if ...
- 3,104
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Minimal Hausdorff topologies compatible with a bunch of functions
Let $X$ be an infinite set, let ${\cal F}$ be a set of functions $f: X\to X$. We say that a topology $\tau$ is compatible with ${\cal F}$ if every $f\in {\cal F}$ is a continuous function $f:(X, \tau)\...
- 48.9k
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886
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Why are some axioms preserved in generic extensions?
It is a known theorem that for a model of $ZF$, $M$, if $M\models AC$ and $G$ is a $P$-generic filter over $M$, for some $P\in M$, then $M[G]\models AC$.
On the other hand, it is long known that ...
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Consistency results using nonstandard models
Are there any consistency results in set theory (or in mathematics) that can be proved using nonstandard models of ZFC but not using transitive models of ZFC?
- 32.2k
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An axiom for collecting proper classes
I'm currently working on some universal algebra using proper classes (in MK class theory), and I repeatedly run into situations where I want to collect together some proper classes as the members of a ...
- 10.1k
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622
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A weak (?) form of Shelah cardinals
The following definition of a large cardinal property combines parts of the definitions of "Shelah cardinal" and "Woodin cardinal":
A cardinal $\kappa$ is weakly Shelah if for all $f : \kappa \to \...
- 5,471
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Does a nonlinear additive function on R imply a Hamel basis of R?
A function is additive if $f(x+y) = f(x) + f(y)$. Intuitively, it might seem that an additive function from R to R must be linear, specifically of the form $f(x) = kx$. But assuming the axiom of ...
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Definable map from all the ordinals to the surreal numbers with a dense image?
I'm trying to understand analogies and disanalogies between ${\Bbb R}$, the reals numbers, and ${\bf No}$, the surreal numbers.
${\Bbb R}$ admits countable dense sets such as the rationals. This ...
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419
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When is there an unbounded tower in $[\mathbb{N}]^\infty$?
(Edit: I'm splitting the question, leaving here only what is answered by Ashutosh, and moving the rest to another question.)
This question assumes familiarity with combinatorial cardinal ...
- 3,104
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What is the smallest number of hyperplanes covering $\ell_2$?
For a Banach space $X\ne \{0\}$, let $\mathrm{cov}_H(X)$ be the smallest number of hyperplanes covering $X$.
By a hyperplane in a Banach space I understand any closed affine subspace of codimension ...
- 42k
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value of Theta in ZF+AD
Since I found out about it, I've always been interested in the Axiom of Determinacy rather than the Axiom of Choice. Along these lines, I've kept flipping back to http://en.wikipedia.org/wiki/%CE%98_%...
user5810
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Is each compactification of $\mathbb N$ soft?
Definition. A compactification $c\mathbb N$ of the countable discrete space $\mathbb N$ is defined to be soft if for any disjoint sets $A,B\subset\mathbb N\subset c\mathbb N$ with $\bar A\cap\bar B\ne\...
- 42k
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What is the proof of consistency of anterior reflection?
Let Anterior Reflection be the following principle: $$\forall \vec{v}~ \exists X: \operatorname {transitive} (X) \land \, (\varphi \to \varphi^{X"}) $$
where $\varphi$ is a formula in $\sf FOL(=,\in)$ ...
- 11.3k
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When is the generalized Cantor space $\kappa$-compact?
My M.Sc. student has the following question, that I assume has an answer in the literature, and we are looking for references.
The generalized Cantor space is the space $2^\kappa$, with basic open ...
- 3,104
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471
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Comparing the sizes of uncountable sets of reals under AD
Working in ZF+AD, let $$\theta_0(X)=\min\{\alpha\in ON: \not\exists f: X\rightarrow \alpha\mbox{ surjective and OD}\}$$ be the least ordinal onto which $X$ does not surject in an OD way, for $X\...
- 20.5k
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431
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Cardinal characteristics without choice
(I'm taking my definition of a cardinal characteristic from Blass' excellent article http://www.math.lsa.umich.edu/~ablass/need.pdf, which cites Vojtas/Fremlin/Miller; theirs is more general, but I'm ...
- 20.5k
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special extremally disconnected spaces with only finite isolated points
We Know that a cardinal $\kappa$ is measurable if there is a set $X$ with cardinal $\kappa$ and a {0,1}-measure $\mu: P(X) \rightarrow ${$0,1$} so that for all $x \in X$, $\mu(x)=0$ and $\mu(X)=1$. ...
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Cardinal arithmetic in $L(\mathbb{R})$
I asked this on math.stackexchange but did not receive an answer, so I'm asking here.
Assume large cardinals. Can we have $\omega_2^{L(\mathbb{R})}=\omega_2$?
Note that $\omega_1=\omega_1^{L(\mathbb{...
- 20.5k
4
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1
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Is ZFC interpretable in a kind of an extended form of second order arithmetic?
Informally the following theory is a kind of extension of second order arithmetic, where numbers are not limited to naturals, instead here we have formation of further numbers by setting limits on ...
- 11.3k
4
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1
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Chromatic number of regular linear hypergraphs on $\omega$
For any cardinal $\alpha \in \omega\cup \{\omega\}$, let $[\omega]^\alpha$ denote the collection of subsets of $\omega$ having cardinality $\alpha$.
A linear hypergraph $H=(V,E)$ is a hypergraph such ...
- 48.9k
4
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The example of the idempotent filter or subsets family with finite intersections property
From the answer of Andreas Blass and comments of Ali Enayat on my question Selective ultrafilter and bijective mapping it became clear that for any free ultrafilter $\mathcal{U}$ we have $\mathcal{U}\...
- 1,133
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3
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Approximation of infinite set in generic extension
Suppose $M$ is a c.t.m and suppose $P$ is $Fn(I,2)$ where $I$ is infinite. Now suppose $G$ is $P$-generic, and $A \in M[G]$ is infinite set.
Is it guaranteed that the exist $B \in M$ such that $B \...
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Is the ordering principle preserved in generic extensions?
The ordering principle says that every set can be linearly ordered.
In a previous question Why are some axioms preserved in generic extensions? Asaf Karagila asked which axioms are preserved in ...
- 16.2k
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Strongly minimal covers
Let $H=(V,E)$ be a hypergraph, that is $V$ is a set and $E\subseteq \mathcal{P}(V)$. We say that $C\subseteq E$ is a cover of $H$ if $\bigcup C = V$.
A cover $M\subseteq E$ is said to be strongly ...
- 48.9k
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595
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Upward reflection of rank-into-rank cardinals
Rank-into-rank cardinals have the rather intriguing property that they reflect upwards. I would be interested to know how far the upward reflection goes:
1) Does "There exists a rank-into-rank ...
- 675
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Is ZFGC, minimally modified to allow two Quine atoms instead of the empty set, synonymous\bi-interpretable with ZFGC?
Working in $\sf ZFGC$, remove Foundation, stipulate the existence of exactly two Quine atoms. Restrict Separation to fulfillable formulas, i.e. $\{x \in A \mid \phi \}$ exists as long as $\phi$ holds ...
- 11.3k
3
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1
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260
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Choice sets from above and below
Let $X\neq \emptyset$ be a set and let ${\cal S} \subseteq {\cal P}(X)\setminus\{\emptyset\}$ be a collection of non-empty subsets of $X$. We say $C\subseteq X$ is a choice set for ${\cal S}$ if $|C\...
- 48.9k
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2
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622
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Can uncountable sets be proved to exist in this variant of ZFC with definability restrictions?
If we restrict all parameters in the set building axioms of $\sf ZFC$ to definable sets, would the celebrated Cantor's theorem still apply? Can existence of uncountable sets be proven at all?
The set ...
- 11.3k
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543
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Is the Ordering Principle equivalent to a selection principle?
Working in the context of set theory $\sf ZF$, selection may be defined as a function from nonempty sets to their elements. Formally:
$\operatorname {selective}(c) \iff \operatorname {function}(c) \...
- 11.3k
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3
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952
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Which notions of forcing add a cofinal branch to an $\omega_1$-tree?
I'd like to know more about forcing to add a cofinal branch to an $\omega_1$-tree.
Question 1:
What kinds of forcings add cofinal branches to $\omega_1$-trees? What kinds of forcings cannot?
...
user10290
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Centralizer of a single element in the monoid of self-maps of a set
This is a follow-up to this question: For what sets $X$ do there exist a pair of functions from $X$ to $X$ with the identity being the only function that commutes with both?
Let $X$ be a set, and $X^...
- 63.9k
3
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268
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Minimal covers in hypergraphs with finite edges
Let $H=(V,E)$ be a hypergraph. We say that $C\subseteq E$ is a cover if $\bigcup C = V$. Let $H$ be a hypergraph with the following properties:
$\bigcup E = V$,
all members of $E$ are finite, and
$d,...
- 48.9k
2
votes
1
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452
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Can we choose an element from a class?
Let $H$ be a complex Hilbert space and $H_1,...,H_n$ be closed subspaces of $H$.
Set $H_0:=H_1\cap H_2\cap...\cap H_n$ and let
$P_i$ be the orthogonal projection onto $H_i$, $i=0,1,2,...,n$.
I study ...
- 935
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Is there a model of ZF+ACC where transfer fails for the definable hyperreals?
In 2003 Kanovei and Shelah constructed a definable hyperreal field. The ultrapower used exploits a fairly large index set so that it is clear that the usual proof of Los and transfer does not go ...
- 16.6k
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1
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702
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Is ZFC plus a truth predicate capable of variable substitution consistent?
Let us define a truth predicate that allows one to substitute in variables. For example, for a 3-ary predicate $p$ and sets $A$, $B$, and $C$, then $\text{true}(\ulcorner p \urcorner, A, B, C)$ (where ...
- 6,353
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1
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Does adding definability axiom expressed in infinitary language to ZF, let all models be pointwise definable?
This posting is generally related to a prior posting titled "Are all constructible from below sets parameter free definable?"
If we work in infinitary language $\mathcal L_{\omega_1, \omega}$...
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