**4**

votes

**0**answers

96 views

### Cohen's model yet again

It has been discussed already whether a countable OD set necessarily contains an OD element. See e.g.
A question about ordinal definable real numbers
.
A negative answer was obtained in Archive for ...

**8**

votes

**2**answers

278 views

### Normal subgroups of Aut(M)

Let $S$ be the set of all finite permutations of $\mathbb{N}$, i.e. they fix all but a finite set, and $A\subset S$ the set of all even permutations.
Theorem The normal subgroups of $S_\infty$ are ...

**-3**

votes

**0**answers

49 views

### Is ∃x.px=>q equal to ∃x.(px=>q)? [on hold]

I wondered if the sentences
∃x.px=>q
equal to
∃x.(px=>q).
I think they are equal because of the following example:
The instance of the first sentence:
If there exist a cow, then it ...

**3**

votes

**1**answer

129 views

### Number of distinct variables used in axiomatizating (Classical) Propositional Logic

The first part of the present question is concerned with Classical Propositional Logic (CPL). The second part involves its fragments or alternative logical systems.
There are in the literature many ...

**6**

votes

**0**answers

189 views

### Tree property using side conditions

The following problems were asked during the high and low forcing workshop:
Question 1. Can one force tree property at $\kappa^{++}$ for $\kappa$ singular using side conditions?
Question 2. ...

**0**

votes

**0**answers

67 views

### (∀x.(p(x)⇒∀x.p(x)) )= ((∃x.p(x))⇒(∀y.p(y)))? [closed]

In dealing with my homeowrk, someone has told me
∀x.(p(x)⇒∀x.p(x))
could be transformed to
(∃x.p(x))⇒(∀y.p(y))
However, intuitively, this doesn't make sense to me, could anyone give me a ...

**4**

votes

**0**answers

77 views

### Semi-algebraicness of cells involved in integrals of semi-algebraic functions

Background: In "Stability under integration of sums of products of real globally subanalytic functions and their logarithms", by R. Cluckers and D.J. Miller, it is shown that the integral of a ...

**7**

votes

**1**answer

208 views

### Uncountable models of Kelley-Morse set theory with only a countable number of sets

The Kelley-Morse set theory can be thought as the "full-secondorderification of $\sf ZFC$", where we switch from sets to classes and allow the comprehension schema to include quantifiers on class ...

**4**

votes

**0**answers

222 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 ...

**10**

votes

**1**answer

320 views

### Forcing PFA with ccc forcing

Is it consistent (from suitable large cardinals) that there is a ccc poset which forces PFA?
This seems quite implausible to me. If we could force PFA via ccc forcing, the ground model would have to ...

**5**

votes

**2**answers

278 views

### Why do we need a transitive model in forcing arguments?

One major approach to the theory of forcing is to assume that ZFC has a countable transitive model $M \in V$ (where $V$ is the "real" universe). In this approach, one takes a poset $\mathbb{P} \in M$, ...

**7**

votes

**0**answers

98 views

### Choice and the Baire property in non-separable complete metric spaces

It's known to be consistent with ZF+DC that every subset of $\mathbb{R}$ has the Baire property (BP). (E.g. Shelah's model). If so, then every subset of every complete separable metric space has ...

**5**

votes

**2**answers

356 views

### Consistent sentences with no arithmetically definable models

I've seen a construction of a sentence of first order logic that is consistent, but has no models with underlying set $\mathbb{N}$ and recursive functions and relations. Do there also exist consistent ...

**5**

votes

**1**answer

275 views

### Comparing the sizes of uncountable sets of reals under AD

Working in ZF+AD, let $$\theta_0(X)=\min\{\alpha\in ON: \not\exists f: X\rightarrow \alpha\mbox{ surjective and OD}\}$$ be the least ordinal onto which $X$ does not surject in an OD way, for $X\...

**19**

votes

**4**answers

1k views

### Nuances Regarding Naturality

It's frequently said, informally, that a natural isomorphism is one that doesn't depend on arbitrary choices.
But the phrase "arbitrary choices" lends itself to different interpretations. Consider ...

**9**

votes

**0**answers

278 views

### What ccc forcings add a Suslin tree?

In a comment to Miha's question in Forcing PFA with ccc forcing, I suggested that if such situation is even possible, it might be achieved by screwing with PFA by some ccc forcing (e.g. adding a Cohen ...

**5**

votes

**1**answer

150 views

### Spreading sets - especially without choice

For what follows, I work in ZF+AD+DC. However, the questions below are not obviously trivial in ZFC, so I'm also interested in results in that system.
Suppose I have a set $X\subseteq \mathbb{R}$. ...

**5**

votes

**1**answer

118 views

### Dropping “generic” from the definition of forcing

Back when I was first learning about forcing and trying to understand the need to consider generic filters, I came up with the following question. Suppose we have a countable transitive model $M$. ...

**3**

votes

**0**answers

123 views

### Compactness beyond extendibility

By a result of Magidor, $\kappa$ is extendible if and only if the infinitary $n$th-order logic over the language $L_{\kappa,\kappa}$ is compact for every $n < \omega$, where by compact, we mean ...

**2**

votes

**1**answer

138 views

### How can two theories $T$ and $T+\phi$ be mutually interpretable?

Following Koellner in http://plato.stanford.edu/entries/independence-large-cardinals/, "a theory $T_1$ is interpretable in $T_2$ ($T_1 \leq T_2$) when, roughly speaking, there is a translation $\tau$ ...

**7**

votes

**1**answer

267 views

### Proof-theoretic ordinals after liberalizing induction to $RCA_0$

This is kind of a follow-up to this question.
For a class $\Gamma$ of second-order formulas (here either $\Sigma_n^0$ or $\Sigma_n^1$), let $X\Gamma$ be a formal theory consisting of $RCA_0$ together ...

**1**

vote

**1**answer

266 views

### Further research on $\mathrm L_{\infty}$

In the mathoverflow question , "Godel's Constructible Universe in Infinitary Logics (A Possible Solution to $HOD$ Problem), Prof Hamkins answered user46667's question 2
What is $\mathrm L_{\infty}$...

**5**

votes

**1**answer

905 views

### Surreals and NSA: some foundational issues

Surreals and NSA: some foundational issues.
A.
Leaving aside the whole internal machinery of surreals (with funny questions like is $\omega$ an entire number and if yes is it odd or even, simple, a ...

**5**

votes

**0**answers

144 views

### Radin forcing and large cardinals

Assume $\kappa$ is a $(\kappa+2)$-strong cardinal and let $j: V \to M \simeq Ult(V, E) \supseteq V_{\kappa+2}$ witness this where $E$ is a $(\kappa, \kappa^{++})$-extender. Also let $u$ be the measure ...

**14**

votes

**1**answer

936 views

### Can Vopenka's principle be violated definably?

One form of Vopenka's principle (a large cardinal axiom) states that no locally presentable category contains a full subcategory which is large (= a proper class) and discrete (= contains no ...

**6**

votes

**2**answers

534 views

### An Extender is a Generalization of an Ultrafilter?

I'm not sure whether this belongs here or on math stackexchange, but I'll give a try here.
I've heard that an extender is a generalization of an ultrafilter. This is not to say that an ultrafilter ...

**4**

votes

**1**answer

175 views

### What countable ordinals are called $\kappa_\alpha$?

Jervell has a notation for countable ordinals up to the small Veblen ordinal using trees:
• Herman Ruge Jervell, How to wellorder finite trees
and get good ordinal notations, Berkeley Logic ...

**9**

votes

**1**answer

238 views

### Cofinal monotone maps from $\omega^\omega$ to $\kappa^\kappa$

Given a cardinal $\kappa$ consider the set $\kappa^\kappa$ of all functions from $\kappa$ to $\kappa$, endowed with the partial order $f\le g$ iff $f(\alpha)\le g(\alpha)$ for all $\alpha\in\kappa$.
...

**6**

votes

**0**answers

100 views

### On the use of Weisfeiler-Leman refinement in Babai's GI proof

This question is for those familiar with the methods behind Babai's recent proof that graph isomorphism can be decided in quasipolynomial time. I am a newcomer to the GI problem, so I apologize if my ...

**7**

votes

**1**answer

172 views

### On the cardinal arithmetic of accessible categories

If $\lambda, \mu$ are regular cardinals, say that $\lambda \trianglelefteq \mu$ if $\lambda \leq \mu$ and
$$\forall X, \, |X| < \mu \implies \mathrm{cf} (P_\lambda(X)) < \mu$$
Here $P_\lambda(X)...

**6**

votes

**1**answer

154 views

### $\mathfrak b_a$: a new cardinal characteristic of the continuum?

By a partial function from $\omega$ to $\omega$ we understand a function $f:dom(f)\to\omega$ defined on an infinite subset of $\omega$.
A family $\mathfrak F$ of partial functions from $\omega$ to $\...

**5**

votes

**1**answer

185 views

### Ordinal-definable witnesses to the perfect set property?

This possibly a very basic descriptive set-theory question; if it is too basic for MO, feel free to migrate.
Throughout we work in ZF+AD. My question is:
If $A$ is an uncountable OD set of reals,...

**5**

votes

**0**answers

94 views

### Preservation of Baumgartner's I-ultrafilters under various forcings

For $I\subset \mathcal{P}(2^\omega)$, an ultrafilter $U$ on $\omega$ is said to be an I-ultrafilter if for all $f:\omega \to 2^\omega$, there exists $A\in U$ such that $f''[A]\in I$ [Baumgartner]. In ...

**57**

votes

**2**answers

5k views

### How would you solve this tantalizing Halmos problem?

$1-ab$ invertible $\implies$ $1-ba$ invertible has a slick power series "proof" as below, where Halmos asks for an explanation of why this tantalizing derivation succeeds. Do you know one?
...

**17**

votes

**2**answers

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 ...

**5**

votes

**1**answer

320 views

### History of the notation for substitution

One of the very common notations for syntactic substitution is $[\ /\ ]$.
However, there seems to be an inconsistency in the literature about its usage.
Many write $[t/x]$ for "substitute $t$ for $...

**3**

votes

**1**answer

218 views

### A kind of saturation property related to forcing notions

Forcing is typically done over well-founded models. There are lots of good reasons for this, but it can feel confining at times. Fortunately, we can equally well force over non-well-founded models! It ...

**8**

votes

**2**answers

597 views

### Difference between Laver's and Mathias's forcing

Mathias forcing consists of conditions of the form $(s,A)$ where $s$ is a finite subset of $\omega$, $A$ is an infinite subset of $\omega$ such that $\max(s) < \min(A)$ and $(s,A)\leq (t,B)$ iff $s\...

**13**

votes

**1**answer

367 views

### Turing degree of finding independent formulas

In this Paper of D. Isaacson, it is proved that the true arithmetic($Th(\mathbb{N}$)) is the only $\omega$-consistent and complete extension of $Q$ (Robinson's arithmetic). This, together with the ...

**5**

votes

**1**answer

250 views

### Why relative consistency results by forcing arguments are provable in finitistic metatheory

It is claimed in many textbooks that relative consistency results, such as $\text{Con}(\text{ZFC})\rightarrow\text{Con}(\text{ZFC}+2^{\aleph_0}\geq\aleph_2)$, are provable in the finitistic metatheory....

**6**

votes

**1**answer

299 views

### What do we call this quantifier (“binder”)?

There's a quantifier ("binder", whatever), call it $\alpha$, defined as follows: $\alpha x.\tau$ is the (usually infinite) expression obtained by applying the substitution $\{x \mapsto \tau\}$ to the ...

**42**

votes

**5**answers

4k views

### Is the Riemann Hypothesis equivalent to a $\Pi_1$ sentence?

1) Can the Riemann Hypothesis (RH) be expressed as a $\Pi_1$ sentence?
More formally,
2) Is there a $\Pi_1$ sentence which is provably equivalent to RH in PA?
Update (July 2010):
So we have ...

**3**

votes

**0**answers

164 views

### Construction of model of arithmetic from an arbitrary model

Let $M$ be a non-standard model of $PA$, $a\in |M|$ be an arbitrary non-standard number and $T$ be a theory of arithmetic. We want to choose a subset $M'\subsetneq M$ such that:
$M'\models PA^-$ (or ...

**6**

votes

**1**answer

178 views

### O-minimal spectrum is a spectral space

I'm trying to understand a proof on "Sheaves of Continuous Definable Functions" (Pillay, Anand. "Sheaves of continuous definable functions." The Journal of symbolic logic 53.04 (1988): 1165-1169.)
...

**2**

votes

**1**answer

130 views

### Model existence and consistency conditions for $\Pi_1^0$ oracles

Let a $\Pi_1^0$ sentence be a sentence asserting that some given Turing machine never halts at the empty input tape. Let Q1 be a (potentially consistently lying) oracle for deciding $\Pi_1^0$ ...

**4**

votes

**1**answer

205 views

### Spectrum Problem for Higher-Order Logic

Definitions. Given a sentence $\varphi$ of $n$th-order logic, we define the spectrum of $\varphi$ to be the set of cardinalities of finite structures that satisfy $\varphi$. A set $X\subseteq\mathbb ...

**15**

votes

**12**answers

5k views

### The best text to study both incompleteness theorems

Hi!
What text on both incompleteness theorems you would recommend for beginner?
Specifically, I'm looking for the text with the following properties:
1) The proofs should be finitistic, in Godel's ...

**5**

votes

**1**answer

97 views

### Complexity of integer programming with added predicates

A classical theorem in Integer Programming by Lenstra says that any integer system
$$A x \le b$$
can be solved in polynomial time, where $A \in \mathbb{Z}^{m \times n}, x \in \mathbb{Z}^n, b \in \...

**7**

votes

**1**answer

194 views

### Definability using rudimentary function

Denote by RUD the set of all rudimentary functions, together with the function that takes any set to its transitive closure.
Assume that I know that a binary relation $R$ is definable by some ...

**20**

votes

**4**answers

943 views

### Is there a non-Hopfian lacunary hyperbolic group?

The question's in the title and is easily stated, but let me try to give some details and explain why I'm interested. First, a disclaimer: if the answer's not already somewhere in the literature then ...