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-1
votes
1answer
173 views

Is there a proof (maybe formulated by Feferman) which says that a proof about the (in)consistency of ZFC is unachievable? [on hold]

Is there a proof (maybe formulated by Feferman) which says that a proof about the (in)consistency of ZFC is unachievable? A professor said it to me a long time ago, but I don't have any references. ...
10
votes
2answers
675 views

Witness to a failure of Fubini/Tonelli

Is it provable in ZFC that there is a subset of the plane all of whose vertical cross sections have Lebesgue measure zero and all of whose horizontal cross sections are complements of sets of Lebesgue ...
0
votes
1answer
127 views

Is there any infinite set which is Dedekind finite and weakly Dedekind finite?

Is there any infinite set which is Dedekind finite and weakly Dedekind finite? If $X$ is a weakly Dedekind finite amorphous set, can we show that $\mathcal P(X)$, the power set of $X$, is also weakly ...
5
votes
1answer
205 views

Is this weak form of $V=L$ (in)consistent with large cardinals?

I have been considering a (definability-free) weak form of the constructibility axiom, which is intended to capture the coarse structure of the constructible hierarchy. This means that this weak form ...
3
votes
1answer
151 views

Reference request: The consistency of a tall tower in $\mathbb{N}^\mathbb{N}$

A $\kappa$-tower in $\mathbb{N}^\mathbb{N}$ is a sequence $\langle a_\alpha : \alpha<\kappa\rangle$ in $\mathbb{N}^\mathbb{N}$ that is $\le^*$-increasing with $\alpha$ and has no $\le^*$-upper ...
6
votes
0answers
118 views

Examples of analytic $\mathcal{I}$-mad families

If $\mathcal{I}$ is an ideal (proper and containing the finite sets) on $\omega$, call a family of subsets $\mathcal{A}\subseteq[\omega]^\omega$ $\mathcal{I}$-almost disjoint if for all distinct ...
8
votes
0answers
182 views

Inner models and strongly compact cardinals

The following question is motivated by a result of Magidor that it is consistent that the least strongly compact cardinal is the least measurable cardinal. Question. Assume $\kappa$ is a strongly ...
5
votes
1answer
139 views

Martin-Solovay Tree of Weakly Homogeneous Tree under $\mathsf{AD}_\mathbb{R}$

A tree $T$ on $\omega \times \lambda$ is weakly homogeneous if there is a countable set $\sigma$ of countably complete measures on ${}^{<\omega}\lambda$ so that $x \in p[T]$ if and only if there is ...
10
votes
1answer
259 views

Precipitous ideals and GCH

It is well known that ZFC + "There is a measurable cardinal" is equiconsistent with ZFC + "There is a precipitous ideal on $\omega_1$." Is ZFC + "There is a measurable $\kappa$ such that $2^\kappa ...
4
votes
0answers
175 views

Unbounded towers and combinatorial cardinal characteristics of the continuum

Update: Perhaps the question is too difficult. I would appreciate, thus, even just comments or related observations. This question assumes familiarity with combinatorial cardinal characteristics of ...
5
votes
1answer
288 views

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 ...
4
votes
1answer
214 views

Is it consistent that the gaps between cardinals $\kappa$ and $2^\kappa$ “get larger and larger”?

Is the following statement consistent in $\mathsf{ZFC}$? For every ordinal $\beta$ there is an ordinal $\lambda_0$ such that for all ordinals $\lambda\geq\lambda_0$ we have ...
7
votes
1answer
203 views

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 ...
-8
votes
1answer
349 views

Missing Axiom: There are no other axioms. Leads to a proof of CH within ZFC [closed]

I have come to the conclusion that there is often an implied axiom: There are no other axioms. Failure to explicitly state this axiom and to consider its consequences can result in the misleading ...
0
votes
1answer
154 views

Can epsilon-induction be derived from the transitive closure operator?

I was wondering (and could not seem to prove or disprove) if epsilon-induction could be derived from the transitive closure operator for binary relations, if we do not have the Foundation Axiom. The ...
10
votes
1answer
370 views

A question on subsets of $\omega_1$

Is there a family $\{A_\alpha:\alpha<2^{\omega_1}\}\subset [\omega_1]^{\omega_1}$ in ZFC such that for each countable set $I\subset 2^{\omega_1}$ and $\alpha\in 2^{\omega_1}\setminus I$ we have ...
1
vote
1answer
156 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 ...
14
votes
0answers
344 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 ...
4
votes
2answers
321 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
104 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
258 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
188 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
161 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, ...
14
votes
1answer
464 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 ...
4
votes
1answer
263 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
234 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, ...
4
votes
1answer
209 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 ...
5
votes
1answer
272 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 ...
6
votes
1answer
352 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 ...
7
votes
1answer
200 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 ...
3
votes
1answer
233 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 ...
6
votes
0answers
126 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 $ ...
6
votes
1answer
199 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 ...
3
votes
1answer
193 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 ...
5
votes
1answer
119 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 ...
8
votes
1answer
222 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 ...
8
votes
1answer
272 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
240 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 ...
38
votes
1answer
989 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, $f(S)\subset Y$ ...
22
votes
0answers
454 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 ...
9
votes
1answer
293 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 ...
6
votes
1answer
199 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 ...
5
votes
1answer
176 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 $[0,1]$ ...
9
votes
2answers
224 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 ...
2
votes
0answers
120 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
2answers
628 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 ...
12
votes
0answers
243 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 ...
5
votes
1answer
232 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 ...
3
votes
1answer
89 views

Levels of L resembling each other, take 2

(Everything below is assuming $V=L$.) Fix an uncountable regular cardinal $\kappa$, and let $$E_\kappa=\{\mu<\kappa: \mbox{there is an elementary substructure of $L_\kappa$ isomorphic to ...
3
votes
1answer
90 views

Generic sections of non-null sets are non-null

Consider the Cantor space $\mathcal C={}^\omega2$ with the usual product measure, and let $r$ be a random real (over a transitive model $V$ of ZFC). Let $B\subset \mathcal C^V\times\mathcal C^V$ a ...