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.
5,268
questions
2
votes
1
answer
79
views
A problem with a $\Pi_1$ formula of the Lévy hierarchy
Let $M\equiv N$ means that $(M,\in_M)$ and $(N,\in_N)$ satisfy the same sentences of the language of set theory, with $\in_M$ and $\in_N$ being the standard membership relation restricted to $M\times ...
- 91
11
votes
2
answers
787
views
Why is inner model theory evidence for consistency of large cardinals?
I want to understand the viewpoint that existence of canonical inner model for a large cardinal notion is strong evidence for its consistency. For example, below is Trevor Wilson's answer to What &...
- 525
1
vote
1
answer
171
views
Can there be a minimal remote cardinal?
Working in $\sf Z- Reg.$ we can have sets bigger than every well-founded set, let's label such sets as "remote". Now, suppose we'll add the axiom of existence of transitive closures, and the ...
- 9,256
3
votes
1
answer
85
views
Chromatic numbers realised by almost disjoint subsets of $\omega$
If $H=(V,E)$ is a hypergraph then the chromatic number $\chi(H)$ is defined to be the least cardinal $\kappa \leq |V|$ such that there is a map $c:V\to \kappa$ such that for all $e\in E$ with $|e| \...
- 45.4k
3
votes
1
answer
387
views
Can we have an axiom that refers to itself and the prior axioms of the theory it is an axiom of?
I know that this question is little bit imprecise, I'll try to present it in the best I can.
Can one have an axiom which is self referential with respect to itself and the theory in which it belongs?
...
- 9,256
0
votes
0
answers
41
views
Is stratified Z - Infinity + there is a set as big as its powerset, consistent if NF is consistent?
The question of consistency of $\sf NF$ can be seen to be equivalent to the question of whether the theory "Stratified $\sf Z$ - Regularity - Infinity + There exists a set as big as its powerset&...
- 9,256
2
votes
1
answer
153
views
Cardinality of maximal diverse families
Let $\kappa\geq \aleph_0$ be a cardinal. We say a collection ${\cal E} \subseteq {\cal P}(\kappa)$ is diverse if $|(A \setminus B) \cup (B \setminus A)| = \kappa$ whenever $A\neq B\in {\cal E}$. A ...
- 45.4k
-1
votes
0
answers
85
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Can existence of uncountable sets be proved in a ZFC variant with mild definability restriction?
Starting with ZF[C], if we restrict Separation and Replacement to parameter free versions from Parameter free definable sets, re-write Infinity asserting existence of $\omega$. Add to it axioms of ...
- 9,256
4
votes
0
answers
118
views
Consistency of definability beyond P(Ord) in ZF
Is it consistent with ZF that the satisfaction relation of $L(P(Ord))$ is $Δ^V_2$ definable? More generally, is it consistent with ZF that there is a $Δ^V_2$ formula (taking $α$ as a parameter) that ...
- 6,102
5
votes
0
answers
74
views
Is the derived category of sheaves localised at pointwise homotopy equivalences locally small?
In order to define the cup and cross products in sheaf cohomology, Iversen makes computations in an intermediate derived category. If $K(X;k)$ is the triangulated category of cochain complexes of ...
- 681
6
votes
1
answer
166
views
Strengthening of a classical set mapping theorem of Lázár
We are writing a paper in which we need to use the following theorem to prove that a certain poset satisfies c.c.c. As far as I remember I learned this theorem from András Hajnal.
Theorem 1: If $\...
- 1,477
5
votes
1
answer
263
views
Would strengthening Foundation and Choice in NBG, make it equi-consistent with MK?
This is a follow up to an earlier question about strengthening of foundation in relation to proving the consistency of ZFC. It was shown that it would achieve that, but it may fail short of MK. Here, ...
- 9,256
5
votes
1
answer
200
views
Long chains of Dedekind finite sets
This is a variation on this question with amorphous cardinals replaced with dedekind finite sets.
Dedekind finite sets are sets that have no countable subset, and it is well known that this is a ...
- 285
2
votes
1
answer
305
views
Transversal of $\mathbb{N}\times\mathbb{N}$
Motivation. I am trying to make an interesting infinite version out of this fascinating problem from the Russian mathematical olympiad:
There are $c$ flavours of cookies, we are given $n$ cookies of ...
- 45.4k
6
votes
2
answers
468
views
Is strengthening Foundation in NBG sufficient to make it prove Con(ZFC)?
Can $\sf NBG$ class theory prove the foundation scheme:
Foundation schema: if $\Phi(X)$ is a formula in which "$X$" occurs free and only free, and in which "$Y$" doesn't occur, ...
- 9,256
-2
votes
0
answers
114
views
What happens if we restrict inputs in Separation and Replacement axioms to definable sets?
If we replace the axiom of Foundation by
Foundation schema: if $\varphi(x)$ is a formula in which "$x$" occurs free and only free, and in which "$y$" doesn't occur, whose free ...
- 9,256
13
votes
1
answer
530
views
Long chains of amorphous cardinalities
An amorphous set is an infinite set that cannot be partitioned into 2 infinite subsets. An amorphous cardinality is the cardinality of an amorphous set. Working in $\sf ZF$, it is consistent that ...
- 285
9
votes
1
answer
277
views
Do precipitous ideals "always" come from collapsing?
It's well-known that if $\kappa$ is a measurable cardinal, then there is a poset $\mathbb{P}$ that forces $\kappa$ to carry a precipitous ideal.
Suppose that $\omega_1$ carries a preciptous ideal $I$.
...
- 1,143
2
votes
0
answers
86
views
Uniformization and functions on Turing degrees
Assuming Martin's Conjecture on functions between Turing degrees, is AD + DC consistent with existence of an $f:\mathcal{D}_t → \mathcal{D}_t$ of rank $Θ$ ?
$\mathcal{D}_t$ is the set of Turing ...
- 6,102
3
votes
2
answers
600
views
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 ...
- 9,256
11
votes
1
answer
386
views
Building the real from Dedekind finite sets
It is well known that the real numbers can be countable union of countable sets by starting with GCH and taking a finite support permutations while collapsing all of $\aleph_n$ for natural $n$.
The ...
- 1,643
10
votes
1
answer
466
views
Does proper forcing preserve properness under PFA?
I'm interested in forcing classes $\Gamma$ which preserve membership in themselves, i.e. for all posets $\mathbb{P}, \mathbb{Q}\in \Gamma$, we have $\Vdash_{\mathbb{P}}\check{\mathbb{Q}}\in\Gamma$. ...
- 435
6
votes
1
answer
375
views
Second-order ordinal definability
As is familiar, a set $S$ is ordinal definable ($S \in \mathsf{OD}$) just in case there exists a formula $\varphi(x, \vec{z})$ of the first-order language of set theory with parameters $\vec{z} = z_1, ...
- 1,135
9
votes
1
answer
332
views
Consistency strength of strongly compact cardinal
Where can I find a proof that strongly compact cardinal has higher consistency strength than Woodin cardinal, or even just strong? Recall that a strongly compact cardinal itself may not be strong, ...
- 323
1
vote
0
answers
146
views
What is the strength of adding this de-schematizing inference rule to Ackermann's set theory?
Language: first order logic with equality, membership, and a constant symbol $W$.
Axioms:
Extensionality: $\forall z \, (z \in x \leftrightarrow z \in y) \to x=y$
Comprehension: $\exists x \forall y \,...
- 9,256
-5
votes
0
answers
86
views
Can ur-elements be used as/to construct infinitesimals?
Background material:
Truss[95], "The structure of amorphous sets."
Harrison-Trainor and Kulshreshtha[22], "The Logic of Cardinality Comparison Without the Axiom of Choice."
...
2
votes
1
answer
318
views
Convergence of distance
Consider these sets
$$
A\equiv \bigcap_{\delta>0} \liminf_{n\rightarrow \infty} \{x \in X: d(p_n, [\ell(x), u(x)])\leq \delta\}
$$
$$
C_n(L_n)\equiv \{x \in X: d(p_n, [\ell(x), u(x)])=0\}
$$
where:
...
- 76
1
vote
0
answers
108
views
Is this modified H. Friedman theory bi-interpretable with ZFC + ORD is Mahlo?
The following theory is a modification of Harvey Friedman $\sf K(W)$ theory.
Language: first order logic with equality, membership, and a constant symbol $W$.
Axioms:
Extensionality: $\forall z \, (z ...
- 9,256
6
votes
0
answers
113
views
From HODs to corresponding models of AD
If $M$ is HOD of a model $N$ of $\text{AD}^+ + V=L(P(ℝ))$, what kind of forcing construction in $M$ gives back such an $N$?
HODs for $\text{AD}^+ + V=L(P(ℝ))$ are conjectured (and under anti-large ...
- 6,102
-4
votes
0
answers
181
views
CH vs Not CH, What is the Consequence?
EDIT: Altered (3) to say that $\beth_1$ can't be $\aleph_{\omega}$ or $\aleph_0$ but removed statement that it must be less than $\aleph_{\omega}$.
Let us assume ZFC. We now consider 2 transfinite ...
- 99
6
votes
1
answer
160
views
An iteration of proper forcing without proper iterands
Famously, a countable support iteration with proper iterands is again proper. This is mostly stated as follows: Suppose $(\mathbb{P}_{\alpha},\dot{\mathbb{Q}}_{\alpha})_{\alpha<\lambda}$ is a ...
- 1,612
5
votes
0
answers
189
views
Friedman's proof of covering lemma for $L$
There is a two-page proof of the covering lemma for $L$ using $\Sigma_n$ substructures (Theorem 3.10) in Sy Friedman's Fine Structure and Class Forcing, compared to the proof that spans about twenty ...
- 323
6
votes
1
answer
138
views
Is there a Bernstein subset $X$ of $\mathbb{R}$ such that no continuous map $f : X → [0,1]$ is surjective?
Is there a Bernstein subset $X$ of $\mathbb{R}$ such that no continuous map $f : X → [0,1]$ is surjective ?
- 1,071
2
votes
2
answers
196
views
Name for a certain type of cardinal
I'm not a set-theorist, but I hope this question is appropriate. This is just a question about names:
Fix a cardinal $\lambda$. I'd like to know if there is a name for regular cardinals $\kappa$ such ...
- 13.6k
12
votes
1
answer
704
views
Can proper classes have different sizes?
I'm presently working in a non-ZF set theory, where there are proper classes. (Think MK or VNBG.) And I'm interested in how to think about the possibility (or impossibility) of proper classes with ...
1
vote
0
answers
163
views
How does the cardinality of a set and its powerset compare in the hereditarily rank-concordant constructible world?
Working in the constructible universe "$L$", if we define two kinds of ranks for any constructible set $x$, one being the ordinal index of the first $L_\alpha$ where $x$ appears as a subset ...
- 9,256
5
votes
1
answer
209
views
Is the partition tiling relation transitive?
The following is motivated by an (as of yet) unanswered question on optimal colorings of graphs. I am convinced that the question below has a positive answer in $\newcommand{\ZF}{{\sf (ZF)}}\ZF$, but ...
- 45.4k
4
votes
1
answer
196
views
Simplified method of building an Aronszajn tree
There is a very interesting method to build an Aronszajn tree in Judith Roitman's "Introduction to Modern Set Theory", on pages 100-102. In short, we build a tree $T$ in which the nodes are ...
- 4,525
8
votes
1
answer
187
views
A reference for forcing projections
The idea of a projection $\pi\colon\mathbb{Q}\to\mathbb{P}$ of forcing notions is something like a combinatorial stand-in for the fact that forcing with $\mathbb{Q}$ produces a generic for $\mathbb{P}$...
- 2,289
3
votes
1
answer
171
views
Is there a metric separable space with the following properties...?
Let $\omega_1<\mathfrak{q}_0$ where $\mathfrak{q}_0:=\min\{|Y|:Y\subseteq \mathbb{R}$, $Y$ is not a $Q$-space$\}$.
Is there a metric separable space $X$ with the following properties:
$|X|\geq\...
- 1,071
6
votes
0
answers
217
views
Do maximal compact logics exist?
By "logic" I mean regular logic in the sense of abstract model theory (see e.g. the last section of Ebbinghaus/Flum/Thomas' book). My question is simple:
Is there a logic $\mathcal{L}$ ...
- 19.1k
5
votes
1
answer
243
views
How far does the intuition "non-stationary = null sets (of certain measure)" go?; what position do non-stationary ideals take in measure theory?
When I was taught undergraduate set theory I was told that the idea for club sets is that they are "of full measure" and the non-stationary sets are "of null measure". (There were ...
2
votes
0
answers
72
views
(MK) universal class = von Neumann universe?
I am studying MK set theory. By the axiom schema of class comprehension, which roughly states that
Given a monadic predicate $\phi$ of MK, then there is a class $C = \{x: \phi(x)\}$,
the universal ...
- 115
11
votes
1
answer
207
views
Is ground model $\Sigma_n$ truth $\Sigma_n$-definable in every forcing extension?
It is well-known (and quite useful) that for any formula $\phi$, forcing poset $\mathbb{P}$, $p\in\mathbb{P}$, and $\mathbb{P}$-name $\dot{a}$, $p\Vdash_{\mathbb{P}}\phi(\dot{a})$ is definable as a ...
- 435
7
votes
2
answers
619
views
Ideals generated by Turing independent sets
Recall that $X \subseteq 2^{\omega}$ is Turing independent if no $y \in X$ is computable from the Turing join of any finite subset of $X \setminus \{y\}$.
Question 1. Can we construct a Turing ...
- 71
5
votes
1
answer
190
views
Chromatic number of the infinite Erdős–Hajnal shift-graph
For any set $X$, let $[X]^2= \big\{\{x,y\}: x\neq y \in X\big\}$. Let $\kappa$ be an infinite cardinal. Let $G_\kappa = ([\kappa]^2, E_\kappa)$ where $E_\kappa = \big\{\{a,b\}\in \big[[\kappa]^2\big]^...
- 45.4k
9
votes
0
answers
218
views
Naive way to violate $\mathsf{SCH}$ at $\aleph_\omega$
I asked this question on MSE and got a partial answer. Shamefully I still haven't figured out myself a full answer, so I would like to ask it here. The usual way to get the failure of $\mathsf{SCH}$ ...
- 727
6
votes
0
answers
148
views
Complexity of transfinite 5-in-a-row and other games
Suppose that 5-in-a-row is played on an infinite board, and after an infinite number of moves, if no one won yet and there is an empty square, the game just continues. At limit steps, it is the first ...
- 6,102
12
votes
1
answer
370
views
Partition into antichains
I've read that the following statement is a result of Balcar, but I am unable to find a reference or a proof:
Theorem: If $\kappa\ge \lambda$ are infinite cardinals, then $[\kappa]^{<\lambda}$ can ...
- 1,477
2
votes
1
answer
115
views
Closed unbounded sets and partitions
Let $\kappa$ be a regular, uncountable cardinal. Let $S\subseteq \kappa$ be a closed and unbounded set. Suppose that we partition $S$ into $<\kappa$ pieces. Does one of those pieces contain a ...
- 18.1k