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8
votes
1answer
214 views

“Clubiness” of projective sets of ordinals

I'm sure this is just my google-fu failing me, but: what are sufficient, non-overkill large cardinal axioms which guarantee "Every (boldface) $\Pi^1_n$ set of (real codes for) countable ordinals ...
11
votes
1answer
320 views

Singular in $V$ regular in $HOD$

Prikry forcing can be used to produce a model $V$ of $ZFC$ such that fo rsome cardinal $\kappa$ we have: (1) $\kappa$ is singular in $V$ of cofinality $\omega,$ (2) $\kappa$ is regular (and in fact ...
10
votes
0answers
306 views

c.c.c forcing notions and adding minimal generic reals

Is the following statement consistent: ``There is no non-trivial c.c.c forcing notion adding a minimal generic real''? The question is related to Prikry's question: Is it consistent that any ...
9
votes
1answer
283 views

Two questions about higher Souslin trees

Assume $V=L$ and let $\kappa$ be a Mahlo cardinal. Let $L[G]$ be the generic extension obatined by Mitchell forcing to make $2^{\aleph_0}=\aleph_2=\kappa.$ It is known that in the extension there ...
1
vote
1answer
148 views

Completing class-sized Fields

Let's say that an ordered Field is a class (proper or not) which satisfies the axioms of ordered fields. We work in NBG set theory with global choice. Let's say that an ordered Field is real closed ...
17
votes
0answers
702 views
+100

Souslin trees and weakly compact cardinals

In Souslin trees on the first inaccessible cardinal it is asked if it is consistent that there are no $\kappa-$Souslin trees at the least inaccessible cardinal $\kappa$. In this question I would like ...
4
votes
1answer
259 views

Is tree property inconsistent with Berkeley cardinals in the absence of Axiom of Choice?

On one hand due to Kunen's inconsistency theorem it is known that within $\sf ZF$, large cardinal axioms beyond Reinhardt cardinal are inconsistent with $\sf AC$. Also some recent results of Bagaria, ...
12
votes
0answers
251 views

The axiom $I_0$ in the absence of $AC$

It is well-known that if $AC$ holds and if $j: L(V_{\lambda+1}) \to L(V_{\lambda+1})$ is a non-trivial elementary embedding with $crit(j) < \lambda,$ then $\lambda$ has countable cofinality (and in ...
17
votes
2answers
1k views

Analogues of Primitive Recursive Functions

Let $\mathbf{A}$ be an admissible set (possibly with urelements). I am wondering if there is some good notion of "primitive recursive arithmetic" relative to $\mathbf{A}$. More precisely, I would like ...
9
votes
0answers
231 views

Does Con(ZF + Reinhardt) really imply Con(ZFC + I0)?

The question is: if I assert in ZF that there exists a Reinhardt cardinal, do I really get a theory of higher consistency strength than when I assert in ZFC that there exists an I0 cardinal (the ...
15
votes
1answer
333 views

Namba forcing and semiproperness

This question is the result of leaving "Proper and Improper Forcing" on my nightstand by accident. Is the statement "Namba forcing is semiproper" known to be equiconsistent with some more standard ...
8
votes
1answer
283 views

How long does the slow inefficient algorithm for computing the product in classical Laver tables take?

Let $(A_{n},*)$ denote the $n$-th classical Laver table. Let $X_{n}$ be the set of all finite sequences of elements from $A_{n}$. Define a function $E_{n}:X_{n}\rightarrow X_{n}$ by letting ...
5
votes
2answers
657 views

A specific Model of ZFC

In his paper "Some Second Order Set Theory", Joel Hamkins asked whether there is a model of set theory $V$ that is elementary equivalent to $V[G]$, Whenever $G$ is $V$-generic for the collapse of a ...
4
votes
2answers
305 views

Embedding property of weakly compact cardinals

One of the characterizations of $\kappa$ being a weakly compact cardinal is being inaccessible, and for every $\kappa$-model $M$, there is a [$\kappa$-model] $N$ and an elementary embedding $j\colon ...
-1
votes
1answer
97 views

Subgraphs of $\mathbb{R}^2$ in the Hadwiger-Nelson problem

In the setting of the Hadwiger-Nelson problem, two points of $\mathbb{R}^2$ form an edge if and only if their distance is $1$. The resulting graph $G$ has chromatic number $\chi(G)\in \{4,5,6,7\}$ and ...
6
votes
1answer
239 views

moving up a consequence of PFA

The Proper Forcing Axiom (PFA) implies that every forcing which adds a subset of $\omega_1$ either adds a real or collapses $\omega_2$. Is it consistent that every forcing which adds a subset of ...
6
votes
0answers
168 views

Nonexistence of generic objects over $L(\mathbb{R})$

A well known result (stated and credited to Todorcevic in "Semiselective Coideals", by Farah, Mathematika, 1997, but with antecedents going back to Mathias) says that, under the appropriate large ...
4
votes
2answers
153 views

Can one satisfaction class code another?

Let $M$ be a model of ${\sf ZFC}$. A satisfaction class $S$ for $M$ is subset of $M$'s ordered pairs which satisfies in $M$ the standard Tarskian compositional axioms. E.g.: $M\vDash \forall \phi, ...
34
votes
1answer
718 views

Does there exist a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?

Let $(X,\tau), (Y,\sigma)$ be two topological spaces. We say that a map $f: \mathcal{P}(X)\to \mathcal{P}(Y)$ between their power sets is connected if for every $S\subset X$ connected, $f(S)\subset Y$ ...
3
votes
1answer
175 views

Representation of meager sets in Cohen extensions

Let $M$ be a transitive model of ZFC and $c\in {}^\omega2$ a Cohen real over $M$. Let $A$ be a meager Borel subset of $^\omega2$ in $M[c]$. I would like to prove that there exists a meager Borel set ...
4
votes
1answer
245 views

Large cardinals without choice?

For any given extension $T$ of ZFC (or perhaps NBGC or something), we can ask whether there is an extension $T'$ of ZF which does not prove AC such that $Con(T) \leftrightarrow Con(T')$ $Con(T) \to ...
4
votes
2answers
218 views

Is injectivity of $2^{(\ldots)}$ weaker than $\mathsf{GCH}$? [duplicate]

The following statement cannot be proven in $\mathsf{ZFC}$: (S) : If $A, B$ are sets with $|A| < |B|$, then $2^{|A|} = |{\cal P}(A)| < |{\cal P}(B)| = 2^{|B|}$. Obviously, ...
10
votes
2answers
284 views

Slim Kurepa tree at a singular strong limit cardinal of uncountable cofinality

For a strong limit cardinal $\kappa$ the notion of $\kappa$-Kurepa tree is trivial: the full binary tree is a $\kappa$-Kurepa tree. Accordingly, we consider the following strengthening: A slim ...
4
votes
1answer
200 views

Does “Every infinite set is splittable” imply $\mathsf{AC}$? [duplicate]

We say an infinite set $X$ is splittable if there are $X_1, X_2\subseteq X$ with $X_1\cap X_2 = \emptyset$, $X_1\cup X_2 = X$ and there are bijections $\varphi:X_1\to X_2$ and $\psi:X_1\to X$. Does ...
156
votes
27answers
18k 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 ...
6
votes
1answer
340 views

Does the consistency strength hierarchy coincide with the “arithmetic consequence” hierarchy at ZF + Reinhardt?

In these slides (see especially slide 26), Steel emphasizes the phenomenon that for all known "natural" extensions of ZFC, the ordering by consistency strength agrees with the ordering by containment ...
6
votes
3answers
371 views

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 ...
5
votes
1answer
266 views

Cardinality of connected Hausdorff topologies

Let $X$ be an infinite set and let $C(X)$ denote the collection of connected Hausdorff topologies on $X$. Suppose $N\subseteq C(X)$ has the property that whenever $\tau\neq\sigma \in N$ then ...
7
votes
1answer
461 views

When are unions of isomorphic groups isomorphic?

I was thinking about how to prove $\operatorname{Br}(K)\cong H^2(\operatorname{Gal}(\bar{K}/K),\bar{K}^*)$ without having to introduce inductive limits and all the profinite stuff. So, I started ...
3
votes
2answers
567 views

Group & modules of arbitrary cardinality [closed]

How do I see that there is a group of an arbitrary cardinality? Is this also true for abelian groups? Also, given a commutative ring $R\neq 0$ how do I see that there is an $R$-module of arbitrary ...
7
votes
1answer
192 views

$\mathsf{AD}_\mathbb{R}$ and Elementary Embeddings

Suppose $\mathsf{AD}_\mathbb{R} + V = L(\mathscr{P}(\mathbb{R})) + \mathsf{DC}$ holds. (We can use more if it is helpful.) I believe under $\mathsf{AD}_\mathbb{R}$, every $A \subseteq \mathbb{R}$ is ...
23
votes
0answers
443 views

Can one divide by the cardinal of an amorphous set?

This question arose in a discussion with Peter Doyle. It is provable in ZF that one can divide by any positive finite cardinal $k$: if $X \times \{1,\ldots,k\} \simeq Y \times \{1,\ldots,k\}$ then $X ...
8
votes
1answer
208 views

Must $L_\alpha$ be correct about well-foundedness?

If $R \in L_\alpha$ is a binary relation so that $L_\alpha$ thinks $R$ is well-founded, must $R$ truly be well-founded? (Edit) That is, if $L_\alpha$ thinks that every nonempty subset of the domain of ...
6
votes
0answers
118 views

Forcing in GBC, the ctm approach

There is a nice, detailed survey about forcing in GBC in the appendix of the dissertation of Jonas Reitz. At page 115 the author wrote: " If $ \Gamma $ is a finite collection of sentences forced by $ ...
3
votes
1answer
188 views

“Lexicographic” ordering on ${\cal P}(\omega)$

For $A\neq B\in {\cal P}(\omega)$ we set $$\mu(A,B) = \min\big((A\setminus B)\cup (B\setminus A)\big).$$ We define $A < B$ if and only if $A \neq B$, and $A = B\cap \mu(A,B)$ (that is $A$ is an ...
6
votes
1answer
191 views

Theorem of Bukovsky characterizing ground models

It was mentioned in a talk that Bukovsky proved the following are equivalent for inner models $M \subseteq V$: (1) There is a partial order $\mathbb P \in M$ and a $\mathbb P$-generic filter $G \in ...
5
votes
1answer
107 views

Can There be Rudin-Keisler Immediate Sucessors?

There are several well-studied orderings on the set $\omega^*$ of ultrafilters on the natural numbers. Three popular ones are $\le_i$ for $i = 1,2,3$. We define $\mathcal U \le_i \mathcal V$ to mean ...
7
votes
1answer
300 views

Is the dual of the product of infinite cyclic groups a free abelian group ?

By a theorem of Specker, the group $\mathrm{Hom}(\prod_{\aleph_0} \mathbb{Z},\mathbb{Z})$ is isomorphic to $\bigoplus_{\aleph_0}\mathbb{Z}$ and is in particular a free abelian group. I wonder, if this ...
8
votes
1answer
263 views

Interpreting Robinson arithmetic in a very weak set theory

It is known that adjunctive set theory interprets Robinson arithmetic, and that extensionality is not needed for that. (Montagna and Mancini, "A minimal predicative set theory", Notre Dame Journal of ...
4
votes
2answers
220 views

Least inner model of ZF without power set axiom

I'm interested in the existence and properties of an analogue version of $L$ for models of ZF$^-$ (ZF without the power set axiom), which for simplicity I'll call $L^-$. By "analogue" I mean the least ...
15
votes
2answers
609 views

$\mathfrak{ufo}$: An unidentified combinatorial cardinal characteristic of the continuum?

An ultrafilter ornament is a chain of free filters on $\mathbb{N}$ that are not ultrafilters, whose union is an ultrafilter. Let $\mathfrak{ufo}$ be the minimal cardinality of an ultrafilter ...
9
votes
2answers
219 views

Limits of rearranged sequences along ultrafilters

Suppose that a bounded sequence of real numbers $s_i$ ($i\in\omega$) has a limit $\alpha$ along some ultrafilter $\mu_1\in \beta{\Bbb N}\setminus{\Bbb N}$. Then given another ultrafilter $\mu_2\in ...
5
votes
1answer
171 views

Cofinality of countable ordinals in ZF, and in toposes

A countable limit ordinal $\kappa$ has cofinality $\omega$. One proves this in ZF, say, using the usual trick for representing $\kappa$ as a countable set of reals having closed convex span $[0,1]$ ...
6
votes
1answer
192 views

Fat stationary sets

Recall a stationary subset $S$ of a regular cardinal $\kappa$ is fat when for every $\alpha < \kappa$, and every club $C$, there is a closed set of order type $\alpha$ contained in $S \cap C$. It ...
2
votes
4answers
814 views

The paradox with the first uncountable ordinal

Suppose we have a set $M = (0,1) \subset R$ of reals well-ordered as the first uncountable ordinal. Let $M(a) = \lbrace x \in M : x < a \rbrace$. For every $a \in M$ set $M(a)$ is countable. ...
5
votes
1answer
222 views

Iteration of random reals

Consider two random reals $x, y$ over a transitive model $V$ of ZFC. More specifically, if $\mathcal C^V={}^\omega2$ is the Cantor space, composing the canonical homeomorphism with the projections ...
14
votes
3answers
815 views

Set-theoretic forcing over sites?

All texts I have read on set-theoretic independence proofs consider several different sorts of constructions separately, such as Boolean-valued models (equivalently, forcing over posets), permutation ...
2
votes
0answers
115 views

When do wide initial segments ruin admissibility?

Suppose $\alpha$ is admissible and $\beta<\alpha$. We know that $L_\alpha$ is an admissible set (by definition), but adding subsets of $\beta$ to $L_\alpha$ might break admissibility: while set ...
15
votes
1answer
469 views

Is there a class of mathematical structures with non-isomorphic natural representations as a standard Borel space?

Background. The field of Borel equivalence relation theory provides a robust, unifying theory that organizes most of the classification problems of classical mathematics into a hierarchy, allowing us ...
12
votes
0answers
239 views

If $U,D$ are $\kappa$-complete nonprincipal ultrafilters on $\kappa$ and $j_U(U) = j_D(D)$, is $U=D$?

Here, $\, j_U, \, j_D$ are the canonical elementary embeddings induced by $U,D$ respectively. I note that it is consistent with the existence of a measurable that the answer be yes: it is true in the ...