**4**

votes

**2**answers

339 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 M\...

**6**

votes

**1**answer

262 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

**0**answers

193 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 ...

**9**

votes

**5**answers

312 views

### Probability theory without deductive closure

Human knowledge is not deductively closed. Uncertainty can arise from that just as much as from lack of brute facts. (When a Harvard graduate was reported to have thought that the earth is farther ...

**14**

votes

**1**answer

475 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

**1**answer

269 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 ...

**15**

votes

**4**answers

2k views

### Languages beyond enumerable

A language is a set of finite-length strings from some finite alphabet $\Sigma$.
It is no loss of generality (for my purposes) to take $\Sigma=\{0,1\}$; so a language is a set of bit-strings.
...

**4**

votes

**2**answers

236 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, $\mathsf{...

**4**

votes

**1**answer

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 ...

**7**

votes

**2**answers

384 views

### Difference between constructive Dedekind and Cauchy reals in computation

If the Axiom of Countable Choice (ACC)
$$ \forall n\in \mathbb{N} . \exists x \in X . \varphi [n, x] \implies \exists f: \mathbb{N} \longrightarrow X . \forall n \in \mathbb{N} . \varphi [n, f(n)] $$
...

**6**

votes

**1**answer

359 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 ...

**3**

votes

**1**answer

242 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

**1**answer

200 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 $G \in V$...

**9**

votes

**0**answers

463 views

### Algebraic proofs of algebraic theorems about algebraically closed fields

It is well-known that the first order theory of algebraically closed fields admits quantifier elimination, whence the theory $ACF_p$ of algebraically closed fields of given characteristic $p$ is ...

**3**

votes

**1**answer

197 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 ...

**2**

votes

**0**answers

41 views

### Quantifier elimination of pp-subgroups of modules

This is a model-theoretic questions. Let $M$ be a $R$-module. Our language will be the standard language of modules, i.e. the language of abelian groups together with an unary function symbol for ...

**8**

votes

**1**answer

229 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 ...

**5**

votes

**1**answer

194 views

### Easier Girard's paradox in a circular pure type system (PTS)

System U is an inconsistent PTS in that one has a term of type $\bot = \forall p\colon \ast \ldotp p$, and such a term is explicitly constructed in Hurkens' A Simplification of Girard's Paradox.
One-...

**22**

votes

**0**answers

460 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

**1**answer

299 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 are ...

**6**

votes

**1**answer

203 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

**1**answer

156 views

### Posets (partially ordered sets) in equational logic

I know about equational logic, cf. https://en.wikipedia.org/wiki/Lattice_(order)#Lattices_as_algebraic_structures, and understood that lattices are expressed equationally, i.e., in terms of equational ...

**2**

votes

**0**answers

121 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

**2**answers

641 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 ...

**14**

votes

**0**answers

368 views

### What is the Cantor-Bendixson rank of the space of first order theories?

Let $L$ be the language $\{R\}$ containing a single binary relation symbol. Consider the space $S_0(L)$ of complete, first-order $L$-theories. This is a seperable, compact Hausdorff space; what is its ...

**12**

votes

**0**answers

249 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

**1**answer

236 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

**1**answer

91 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 $L_\mu$}\}...

**3**

votes

**1**answer

92 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 ...

**6**

votes

**1**answer

207 views

### Fine structure question: when do levels of $L$ look “a lot” like each other?

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

**9**

votes

**1**answer

318 views

### Coherence and rewriting

In category theory there are numerous coherence theorems (https://ncatlab.org/nlab/show/coherence+theorem). One example is the Mac Lane's coherence theorem for monoidal categories. This and probably ...

**3**

votes

**1**answer

100 views

### How to get $\omega$-regular expression from buchi automaton

Is there an algorithm or a trick on how to get $\omega$-regular expressions from Buchi automatons? If yes, is there also some way to do create minimal such regular expressions?
It is extremely ...

**4**

votes

**0**answers

217 views

### Is the two variable fragment of arithmetic, i.e., theory of ($\mathbb{N}, + ,\times$), decidable?

Any references would be appreciated. Most places only address different vocabularies (e.g. a survey of arithmetical definability by Bes).

**5**

votes

**1**answer

223 views

### A button for individual reals

Hamkins introduced the notion of a "button" in forcing. This is a set-theoretic statement that can be forced, and can never be made false by further forcing. An example is $V \not= L$. Another ...

**8**

votes

**1**answer

240 views

### Is there a modern account of Veblen functions of *several* variables?

Veblen $\phi$ functions extend the $\xi \mapsto \phi(\xi) := \omega^\xi$ and the $\xi \mapsto \phi(1,\xi) := \varepsilon_\xi$ functions on the ordinals by repeatedly taking fixed points (I won't ...

**3**

votes

**0**answers

173 views

### Borel equivalence relations in models of determinacy

The following appears as fact 3.1 in the slides from Hjorth's 2010 Tarski Lectures. Assume ${L(\mathbb R)} \models \mathrm{AD}$.
Fact 3.1 For E and F Borel equivalence relations one has
$$E \...

**14**

votes

**1**answer

483 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 ...

**10**

votes

**0**answers

291 views

### A question concerning model theory of groups

Several days ago, Professor Martin Bridson gave a very nice talk in my department. Some questions concerning his talk came into my brain Since I am neither a model theorist nor a algebraist, I am not ...

**2**

votes

**1**answer

155 views

### Non-completeness of the Borel-Lebesgue measure and countable choice

Is it possible to prove the non-completeness of the Borel-Lebesgue measure on $\mathbb{R}$ (restricted to the Borel $\sigma$-algebra) without the full axiom of choice, but still with Countable Choice ?...

**10**

votes

**0**answers

229 views

### A variant of strong ideals, is it consistent?

Is it consistent relative to large cardinals that there is a precipitous ideal on $\omega_1$ forcing a generic elementary embedding $j : V \to M \subseteq V[G]$, such that $j(\omega_1) = \omega_n^V$ ...

**4**

votes

**1**answer

218 views

### Reference request, proof about well-ordering of ordinals

I am looking for a second order proof that ordinals are well-ordered up to $\epsilon_0$.
So, starting with encoding those ordinals in numbers, defining the < relation on those encoded ordinals and ...

**3**

votes

**0**answers

333 views

### A question about the Chaitin constant of a theory

Chaitin's famous incompleteness theorem states that for every r.e. theory $T\supseteq Q$ in the language of arithmetic, there is a constant $d_T$ such that for any $m\geq d_T$ and any $x$, $T$ can ...

**12**

votes

**1**answer

404 views

### Changing cofinalities above supercompact cardinals

Question. Suppose $\kappa$ is a supercompact cardinal and $\lambda > \kappa$
is measurable (or even larger large cardinal if necessary).
Is there a set generic extension of the universe in ...

**1**

vote

**1**answer

154 views

### A question on the provability predicate of Q

I am not familiar with Robinson's construction as I do not have access to his text or to precise accounts of this, but I have come to understand that the proof predicate of Robinson arithmetic is non-...

**4**

votes

**2**answers

179 views

### Which Heyting algbras arise out of some elementary topos which satisfies the ultrafilter principle?

It is known that AC implies LEM constructively, and also that AC implies the ultrafilter principle. Is there a similar relationship between the ultrafilter principle and classical logic? In other ...

**11**

votes

**1**answer

338 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 ...

**1**

vote

**0**answers

166 views

### “Kolmogorov complexity” of models of computation

This question was partly inspired by my learning about John Tromp's binary lambda calculus and similar minimal languages such as Jot. A more detailed discussion of some of these ideas is in Michael ...

**2**

votes

**0**answers

338 views

### A question on the consistency of a (seemingly) very weak set theory

I have preoccupied myself some with very weak set theories that suffice to interpret Robinson Arithmetic, as in this question Is Extensionality needed for the incompleteness of very weak set theories?....

**6**

votes

**1**answer

440 views

### Are there 'finitistic' nonrecursive functions (assuming Church's Thesis is false)?

[Note: In what follows, I will be using the same type of argument Laszlo Kalmar did in his paper "An Argument Against the Plausibility of Church's Thesis" found in Constructivity in Mathematics, (...

**9**

votes

**2**answers

272 views

### Bounded Arithmetic vs Complexity Theory

In this post, when I talk about bounded arithmetic theories,
I mean the theories of arithmetic according to "Logical Foundations of Proof Complexity", which capture the complexity classes between $AC^...