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.
40 questions from the last 30 days
2
votes
0
answers
30
views
Cardinal characteristics and $\mathfrak{c} < \aleph_\omega$
Let $\mathsf{R}$ denote some finitely many relations about finitely many cardinal characteristics (e.g. $\mathfrak{a} \leq \mathfrak{s}$, $\mathfrak{a} < \mathfrak{d} = \mathfrak{r}$, $\mathfrak{b} ...
- 1,392
2
votes
0
answers
52
views
Diamonds on supercompact $\kappa$ after a $\kappa$-c.c. forcing
Let $\kappa$ be supercompact. Then the (supercompact) Laver diamond holds at $\kappa$: There is $f:\kappa\to V_\kappa$ such that for all $\lambda\geq \kappa$ and $x\in H(\lambda^+)$ there is $j:V\to M$...
- 51
3
votes
1
answer
82
views
Diamonds at $\omega_2$ under PFA
Let $\lambda$ be a cardinal, and $S\subset \lambda^+$ be stationary. A $\diamondsuit^+(S)$-sequence is a sequence $\langle \mathcal{A}_\alpha\mid \alpha\in S\rangle$ such that each $\mathcal{A}_\alpha\...
- 51
4
votes
3
answers
227
views
An uncountable measurable subset of $\Bbb R$ containing no nonempty perfect set
$\newcommand\R{\Bbb R}$Assuming the axiom of choice, is there an uncountable Lebesgue-measurable subset $S$ of $\R$ that contains no nonempty perfect set?
Of course, such a set $S$, if it exists, ...
- 128k
0
votes
0
answers
92
views
How near are a groupoid and its 'preorderification'?
As remarks, a groupoid is a category with only (categorical) isomorphisms as its morphisms and a preorder is a category only having one morphism between each object. If we choose one isomorphism by ...
5
votes
0
answers
89
views
Query about extender embeddings
This seems as though it should be a result which is possible to prove but I was just wondering if I have it right and also if there is a source for it.
Suppose that $j:V_{\alpha} \rightarrow V_{\beta}$...
- 2,125
-2
votes
0
answers
72
views
There is a typo in Stall's textbook on Set Theory: unable to prove the trichotomy of sets (m ∈ n or m = n, or n ∈ m) [migrated]
Here is the textbook, chapter 7, page 300. This lemma seems very of important, and I've spend about 8 hours trying to figure it out, but I'm unable to prove even the weaker version of the lemma (only ...
- 105
2
votes
0
answers
62
views
Consistency of Sigma-V-2 uniformization with AD
Is ZF + AD consistent with: For every real $r$, every true $Σ^V_2(r)$ statement has a $Δ^V_2(r)$ example?
DC is provable in ZF + every true $Σ^V_2$ statement has a $Δ^V_2$ example (i.e. witness). ...
- 7,545
3
votes
1
answer
136
views
Forcing equivalence and equal generic extensions
Two forcing notions $\Bbb P$ and $\Bbb Q$ could be defined to be forcing equivalent if the associated complete Boolean algebras are isomorphic (so, the CBA's formed by considering the regular opens of ...
- 393
2
votes
2
answers
123
views
Existence of k-complete uniform ultrafilter over a regular cardinal, k is strongly compact
This is a question about set theory. Let $\kappa\leq \lambda$ be infinite cardinals such that $\kappa$ is strongly compact and $\lambda$ is regular. My question is: how to construct a $\kappa$-...
- 61
5
votes
1
answer
148
views
Does there exist a section of $\mathcal{P}(\kappa)\to\mathcal{P}(\kappa)/(\text{fin})$ that is "nearly Boolean"?
The following might be a somewhat esoteric question:
Does there exist an infinite cardinal $\kappa$ and a section $f$ of the quotient map $\pi:\mathcal{P}(\kappa)\to\mathcal{P}(\kappa)/(\text{fin})$ (...
- 2,830
8
votes
0
answers
194
views
Can $\mathbb{C}$ have a "doppelganger" in $L(\mathbb{R})$ with countable automorphism group?
Working in $\mathsf{ZFC}$ + large cardinals (a proper class of Woodins, to be precise), is there a field $F\in L(\mathbb{R})$ such that $V\models F\cong\mathbb{C}$ and $L(\mathbb{R})\models\vert\...
- 21.2k
6
votes
1
answer
550
views
Destroying scales
Suppose $\lambda$ is a cardinal with uncountable cofinality and $C\subseteq\lambda$ is a club of ordertype cf$(\lambda)$. Let $f=\langle f_\xi\mid\xi<\lambda^+\rangle$ be a scale consisting of ...
- 533
-2
votes
0
answers
81
views
Every well-ordered set is isomorphic to an unique ordinal? [closed]
Every well-ordered set $W$ is isomorphic to a unique ordinal
Proof: We start with some well-ordered set $W$. We attempt to construct a bijective map from $W$. Let us consider this class
$$\{(x, \...
- 1
1
vote
0
answers
79
views
How strong is separation + reflection without transitivity?
Consider a theory $T$ with a binary relation $\in$ and the following axiom schemas:
$\exists u \forall x (x \in u \leftrightarrow x \in a \land \phi)$ where $u$ is not free in $\phi$. This is the ...
- 2,203
6
votes
1
answer
464
views
NBG, ZFC+I, and Global Choice
In Shulman's 2008 paper 'Set Theory for Category Theory', he includes amongst the axioms of $\sf NBG$ the axiom of limitation of size. Being well known to imply the axiom of global choice, it seems to ...
- 63
1
vote
0
answers
98
views
Is this theory synonymous with ZF + Global Choice?
$\textbf{Logic:}$ Mono-sorted first order logic with equality.
$\textbf{Extralogical Primitives: } <, \in$
Define: $x > y \iff y < x \\ x \leq y \iff x < y \lor x=y \\ x \not > y \iff \...
- 11.3k
10
votes
1
answer
368
views
Reference request: The non-productivity of Lindenbaum numbers
For a set $X$, the Lindenbaum number of $X$, $\aleph^*(X)$, is the least non-zero ordinal $\alpha$ such that there is no surjection $X\to\alpha$. It seems to be well-known that for infinite sets $X$ ...
- 2,196
9
votes
2
answers
384
views
Iteration of $\aleph_2$-properness
Let us say a forcing $P$ is proper for a class of models $\mathcal C$, if for large enough regular $\theta$ and $M \prec H_\theta$ in $\mathcal C$ with $P \in M$, every $p \in P \cap M$ can be ...
- 18.6k
10
votes
0
answers
204
views
4-quantifier formula not decided by ZF
This interesting question asks the minimum number of quantifiers required to state the Axiom of Choice, and recalls that any sentence having three or fewer quantifiers is already decided by ZF. This ...
- 1,124
3
votes
1
answer
132
views
Is the interval topology on ${\cal P}(\omega)/(\text{fin})$ connected?
If $(P,\leq)$ is a poset and $x\in X$, we let $\downarrow x = \{p\in P: p \leq x\}$, and $\uparrow x$ is defined dually. The collection $$\Big\{P\setminus (\downarrow x): x\in P\Big\} \cup \Big\{P\...
- 48.9k
3
votes
2
answers
157
views
Restriction of a locally finitely supported function on an ordinal is finitely supported?
This is a question about set theory. Let $f:\delta\rightarrow H$ be a function from an ordinal $\delta< \omega_1$ to an arbitrary abelian group $H$. Endow $\delta$ with the order topology. Let $f$ ...
- 61
1
vote
0
answers
89
views
About synonymy relationships around these two theories?
The following question is about patterns of synonymy relationships around two theories, $T^+$ and $\sf PA$.
For purposes of self inclusiveness I'll re-iterate $T$ and its extensions.
$\textbf{Logic:}$ ...
- 11.3k
3
votes
0
answers
93
views
What axioms are needed to show that the range of a finitely additive diffuse measure on $\mathbb N$ is not closed?
The other day I learned of a small error in the book Theory of Charges: A Study of Finitely Additive Measures. Example 11.4.1 goes as follows.
Let $\mu_0$ be a finitely additive probability measure ...
- 869
17
votes
2
answers
2k
views
Do the surreal numbers enjoy the transfer principle in ZFC?
The surreal field $\newcommand\No{№}\No$ is definable in ZFC, and it is easy to see that the surreal order is $\kappa$-saturated for every cardinal $\kappa$, precisely because we fill any specified ...
- 236k
4
votes
0
answers
107
views
Partial uniformization under AD
Under ZF + AD, and especially $\text{AD}^+$, even if uniformization fails for reals, in some ways it must almost hold.
For a notion of small, we say that uniformization holds on a co-small set of ...
- 7,545
6
votes
1
answer
986
views
How many colors do we need?
How many different colors do we need so that the set of all possible colorings of $\mathbb{R}^3$ is greater than the powerset of $\mathbb{R}$.
Countably many doesn't seem to be enough and even $|\...
- 131
14
votes
0
answers
389
views
Can the axiom of choice be expressed in 4 quantifiers?
This 2007 paper presents a 5-quantifier $(\in, =)$-expression that is ZF-equivalent to the axiom of choice, but leaves open the 4-quantifier case:
Thus the gap is reduced to the undecided case of a 4 ...
- 2,203
8
votes
1
answer
245
views
Is $\operatorname{non}(\mathcal{M}) < \mathfrak{a}$ consistent?
Let $\operatorname{non}(\mathcal{M})$ be the least cardinality of a non-meagre subset of the reals. Let $\mathfrak{a}$ be the least cardinality of an infinite maximal almost disjoint family (i.e. $\...
- 1,392
-4
votes
0
answers
133
views
Which arithmetic\set theory is synonymous with this theory?
$\textbf{Logic:}$ Mono-sorted first order logic with equality.
$\textbf{Extralogical Primitives: } <, \in$
Define: $x > y \iff y < x$
Define: $x \leq y \iff x < y \lor x=y$
$ \textbf{...
- 11.3k
-4
votes
1
answer
173
views
To which arithmetic\set theory this theory is bi-interpretable?
$\textbf{Logic:}$ Mono-sorted first order logic with equality.
$\textbf{Extralogical Primitives: } <, \in$
$ \textbf{Axioms:}$
$ \textbf{Order:} \ x < y < z \to x < z $
$ \textbf{...
- 11.3k
12
votes
1
answer
227
views
Is there a $\Pi_2$ sentence $A$ such that $\text{ZFC}^- + A$ proves powerset?
This is a follow-up to this question.
Let $\text{ZFC}^-$ be ZFC without powerset and with collection rather than replacement, as described here.
Is there a $\Pi_2$ (or perhaps $\Sigma_2$) sentence $A$ ...
- 2,203
5
votes
0
answers
158
views
If $\omega_1$ is not inaccessible in $L$, how hard can it be to find a non-measurable $\Sigma^1_3$ set of reals?
In his wonderfully titled paper Can you take Solovay's inaccessible away? Shelah showed that if every $\mathbf{\Sigma}^1_3$ set of reals is Lebesgue measurable, then $\omega_1$ is an inaccessible ...
- 12.5k
3
votes
2
answers
158
views
Is there a sparse almost disjoint family over $\omega$ of cardinality $2^{\aleph_0}$?
Is there an almost disjoint family $\mathcal{F}$ of subsets of $\omega$ of cardinality $2^{\aleph_0}$ satisfying the following property?
For all $A,B\in\mathcal{F}$ with $A\neq B$ and every $k\in\...
- 1,782
-4
votes
0
answers
189
views
Can ZFC be interpreted in this infinitary logic theory?
Working in language $\mathcal L_{\Omega^+,\Omega^+}$ where $\Omega$ is the first strongly inaccessible cardinal. If we add a primitive partial $\Omega$-ary function $F$ and a primitive constant $\...
- 11.3k
2
votes
1
answer
161
views
Are Cohen Generics Minimal Covers?
Are Cohen generics (in $2^\omega$) minimal covers?
I'm ultimately interested in this question for some more effective notion of forcing but I realized I wasn't sure how to show this even assuming full ...
- 3,029
16
votes
1
answer
969
views
Pedagogically intuitive reformulation of Zorn's Lemma for functional analysis
While teaching an applied functional analysis class, I’ve noticed that students often struggle to develop an intuitive understanding of Zorn’s lemma. It’s relatively straightforward to explain why ...
- 5,824
-5
votes
0
answers
250
views
Can Cardinality Theory capture ZFC?
Cardinality Theory "CT" is a theory of sets of cardinals and links between them, only sets of cardinals can be assigned cardinalities. The links are unordered edges linking cardinals, they ...
- 11.3k
4
votes
1
answer
144
views
Stably embedded clone
Let $M$ be a first-order structure, considered as an element of the ambient set-theoretic universe $V$.
Clearly, for any $L_{\infty,\infty}$-formula $\varphi(\bar x)$ (with $\bar x$ finite, say) in ...
- 1,338
6
votes
0
answers
188
views
Is there a characterization of measurables in terms of indiscernibles?
There is a characterization of $\alpha$-Erdős cardinals in terms of sets of indiscernibles of order type $\alpha$. There is also a characterization of Ramsey cardinals in terms of sets of good ...
- 2,031