**3**

votes

**1**answer

210 views

### When can you canonically extend an ultrafilter after forcing?

Suppose that $V$ is a model of $\sf ZFC$, and fix some regular $\kappa$, say $\omega_1$ for practical purposes.
Let $\cal U$ be an ultrafilter on $\omega_1$ in $V$ which is non-principal and even ...

**1**

vote

**2**answers

360 views

### Was “arithmetical translation” (coding in the Goedel sense) ever a part of Hilbert's Program?

Was "arithmetical translation" (that is, coding in the Goedel sense) ever a part of Hilbert's Program? I ask this question for several reasons:
i) it gives the numerals |, ||, |||,.... an ersatz ...

**0**

votes

**0**answers

147 views

### Is the positive existential theory undecidable?

Could you tell if the positive existential theory of $\mathbb{C}[e^{\mu x} \mid \mu \in \mathbb{C}]$ is undecidable in the language $\{+, \cdot , \frac{d}{dx} , 0, 1, e^x\}$ ?
How can we prove the ...

**2**

votes

**2**answers

263 views

### Time Hierarchy Theorem and P vs NP

One obvious strategy for proving P not equal to NP would be to show that there is some problem in NP which is hard for a time class strictly containing P (the origin of this question is the recent ...

**4**

votes

**1**answer

74 views

### Proving moduli of uniform continuity in RCA_0

Simpson's Subsystems of Second Order Arithmetic (pp. 134ff.) uses RCA$_0$ to prove various theorems of analysis for all continuous functions with a suitable modulus of uniform continuity. And he ...

**1**

vote

**0**answers

53 views

### Fully residually free groups and completion

Let $G$ be a fully residually free group with a finitely generated profinite completion. Is $G$ necessarily finitely generated?

**8**

votes

**2**answers

566 views

### Did Bishop, Heyting or Brouwer take partial functions seriously?

The partial μ-recursive functions which may or may not be provably total seem to have some direct relation to the initial motivations for intuitionistic mathematics. (Following Kronecker, one ...

**4**

votes

**2**answers

294 views

### A property of uncountable almost disjoint families

Let $\mathcal{A}$ be an uncountable almost disjoint family (not necessarily maximal) of infinite subsets of $\mathbb{N}$. Denote by $\mathcal{A}_{\subseteq}=\{ B\subseteq\mathbb{N}:|B|=\omega \wedge ...

**4**

votes

**0**answers

174 views

### Theories introduced by a class of forcing notions

The following notion is introduced by Mohammad Golshani. Let $V$ be a model of set theory and let $\mathcal{P}$ be a class consisting of non-trivial forcing notions in $V$. Let
$$Th(V, ...

**-5**

votes

**1**answer

132 views

### New hilbert system and theorem [closed]

Let we have following axioms and modus ponens :
$$(A1):(B ⇒ (C ⇒B ))$$
$$(A2):((B ⇒ (C ⇒D )) ⇒ ((B ⇒C ) ⇒ (B ⇒D )))$$
$$(A3):( ( B ⇒C) ⇒(¬C ) ⇒ (¬B ))$$
now can we prove following theorem ?
$\vdash ...

**4**

votes

**1**answer

295 views

### Show that the positive existential theory is undecidable

To show that the positive existential theory of $\mathbb{C}[t, e^{\lambda t} \mid \lambda \in \mathbb{C}]$ in the language $\{+, \cdot , ' , 0 , 1, t\}$ is undecidable we have to prove the following: ...

**6**

votes

**0**answers

95 views

### Finitely presented algebras with isomorphic semilattices of congruences

Let $\mathbb{T}$ be a finitary algebraic theory. For each $\mathbb{T}$-algebra $A$, let $Q (A)$ be the join semilattice of finitely generated congruences on $A$. There is an evident pushforward ...

**3**

votes

**1**answer

237 views

### How can the critical point of an elementary embedding be omega_1?

I've seen an example of an elementary embedding such that $\omega_1$ is the critical point.
I was wondering what's wrong with the following proof that this cannot be:
Let $\phi(x_1,x_2)$ be the ...

**2**

votes

**0**answers

110 views

### Is a finitely generated residually free group “almost LERF”?

Let $G$ be a finitely generated residually free group.
(i.e. for each $1 \neq g \in G$ there exists a homomorphism $\tau \colon G \to F$ such that $F$ is a free group, and $\tau(g) \neq 1$.)
...

**19**

votes

**3**answers

641 views

### Is being rational decidable?

Given a real number uniquely defined by a finite system of equations and inequalities with rational coefficients involving the standard elementary functions only. Is it decidable whether this number ...

**14**

votes

**1**answer

338 views

### Joyal's construction of the spectrum of a commutative ring

I am trying to understand bits and pieces of Lawvere's article Continuously Variable Sets; Algebraic Geometry = Geometric Logic. I'm not doing very well.
I know this is a lot to ask, but basically, I ...

**2**

votes

**0**answers

69 views

### A Question on Provability Logic and Co-Necessitation

The provability logic $GL$ has the characteristic axioms:
$K\hspace{15pt}\Box(\alpha\rightarrow \beta)\rightarrow(\Box\alpha\rightarrow\Box\beta)$
$L\hspace{15pt}\Box(\Box \alpha\rightarrow ...

**3**

votes

**3**answers

233 views

### Semantic reflection

Let $\ulcorner \cdot \urcorner$ be a fixed encoding of formulas by numbers, e.g.
let $\ulcorner \varphi \urcorner$ denote the Godel number of $\varphi$.
Let $T$ be a first-order arithmetic theory, ...

**7**

votes

**0**answers

221 views

### Proving Richardson's theorem for constants

(I asked this a little over 3 months ago on math.SE, and when I initially re-asked here, no one had responded there. $\:$ After I re-asked here, Eric Towers responded there, since I had forgotten to ...

**4**

votes

**1**answer

197 views

### What is the right notion of generalized element of a category?

I've been working out how the internal language of a category C extends to taking the category itself as a type.
The most obvious way to interpret $X : \mathbf{C}$ is, of course, that $X$ is an ...

**18**

votes

**2**answers

540 views

### Removing large cardinals from an uncountable transitive model

The usual way of removing large cardinals from a given model of set theory is to cut off the model below the least large cardinal of interest. But this method may have dramatic effects on the external ...

**6**

votes

**2**answers

249 views

### The role of the index set in the product of uncountably many topological spaces

Let $\langle X_i,\mathcal{T}_i \rangle_{i\in I}$ be a family of topological spaces. Consider $X=\prod_{i\in I} X_i$ with product topology.
Question. Is there a topological property that holds in ...

**3**

votes

**2**answers

233 views

### Do limit groups satisfy Howson's theorem?

Let $G$ be a limit group, and let $A,B \leq G$ be finitely generated
subgroups generating $G$ (i.e. $\langle A \cup B \rangle = G$). Must
$A \cap B$ be finitely generated?
Recall that a limit ...

**2**

votes

**1**answer

218 views

### Effectively non-recursiveness of some sets

A set $A$ is completely productive if there exists a computable function $f$ such that for every $e$, $f(e)\in (A-W_e)\cup (W_e-A)$. A set is effectively non-recursive if it is r.e. and its ...

**10**

votes

**1**answer

198 views

### Is there a (first-order) sentence which admits $(\aleph_2,\aleph_0)$ iff a Kurepa tree exists?

In Chang and Keisler's Model Theory I came across the following theorem (Theorem 7.2.13):
Theorem There exists a (first-order) sentence $\sigma$ such that for all infinite cardinals $\alpha$, ...

**7**

votes

**1**answer

132 views

### Consistency Strength of “HC is elementary in V[G]”

Let $P$ be the Levy-collapse of the ordinals, so $P$ is a class forcing notion that makes every ordinal countable.
Note that since $P$ is weakly homogeneous, for any formula $\phi(\overline{a})$ ...

**5**

votes

**0**answers

130 views

### Sacks minimality without choice

The usual argument for the minimality result for Sacks forcing uses choice.
Theorem (Sacks): Let $s \subseteq \mathbb S_\kappa$ be generic for the forcing to add a Sacks subset to $\kappa$, where ...

**11**

votes

**1**answer

406 views

### Is it consistent with ZFC that no nontrivial forcing notion has automatic mutual genericity?

A nontrivial forcing notion $\newcommand\Q{\mathbb{Q}}\Q$ exhibits
automatic mutual genericity, if whenever $G,H\subseteq\Q$ are
distinct $V$-generic filters (existing, say, in some forcing
extension ...

**7**

votes

**2**answers

204 views

### How to extend Morley's omitting type theorem to uncountable languages?

In his 1965 paper Omitting Classes of Elements (found in The Theory of Models: Proceedings of the 1963 International Symposium at Berkeley, published by North-Holland Publ. Co., Amsterdam (1965)), ...

**7**

votes

**0**answers

150 views

### $V$ as a $HOD$ of its class generic extension

By an old result of Roguski, The theory of the class $HOD$, any model $V$ of $ZFC$ has a class generic extension $V[G]$ such that $HOD$ of $V[G]$ equals $V$.
This result is also stated and generalized ...

**9**

votes

**0**answers

223 views

### (A little bit) Beyond the E-recursive

The E-recursive functions are a particular generalization of classical recursion theory to the entire set-theoretic universe, $V$. They are defined via a schemes: see ...

**10**

votes

**1**answer

368 views

### Extending an infinite simple group

Maybe the question does not fit here.
Yesterday in my logic course, I presented a nice example about an application of model theory to group theory. The example is due to Hodges and as following: For ...

**7**

votes

**0**answers

204 views

### A strengthening of Chang's conjectures

In the Handbook of Set Theory, Foreman has (essentially) the following proposition (3.9):
Suppose $\kappa_n>...>\kappa_0$ and $\lambda_n>...>\lambda_0$ are regular cardinals and ...

**2**

votes

**1**answer

194 views

### Is Extensionality needed for the incompleteness of very weak set theories?

$ST$ is the weak set theory built upon identity theory and containing
the axiom for empty set,
the axiom for adjunction and
the axiom for extensionality.
It is known that $ST$ interprets ...

**6**

votes

**0**answers

166 views

### PCF conjecture and fixed points of the $\aleph$-function

Recently Moti Gitik refuted Shelah's PCF conjecture, by producing a countable set $a$ of regular cardinals with $|pcf(a)| \geq \aleph_1.$ See his papers
Short extenders forcings I and Short ...

**-2**

votes

**1**answer

105 views

### A question on a presumably admissible/derivable rule for (usual) formal systems [closed]

Suppose formal system $X$ has the rule $\vdash\beta\Rightarrow\hspace{2pt}\vdash\gamma$. Is the rule $\vdash\alpha\wedge\beta\Rightarrow\hspace{2pt}\vdash\alpha\wedge\gamma$ (usually) just admissible ...

**12**

votes

**0**answers

171 views

### Is the game Hanabi NEXPTIME-complete?

The game Hanabi is a cooperative, hidden-information game. You can read the rules elsewhere, but broadly speaking the players are attempting to cooperatively build a fireworks display by playing cards ...

**2**

votes

**0**answers

108 views

### A question regarding an analogue of the Kleene $T$-predicate for Koepke's ordinal computability

Does Koepke's notion of ordinal computability admit an analogue of the Kleene $T$-predicate? If so, is the existence of such a $T$-predicate independent of $ZFC$? Also, if one assumes the existence ...

**13**

votes

**0**answers

139 views

### Complexity classes for BSS machines

Given a first-order structure $\mathcal{S}$, a Blum-Shub-Smale machine on $\mathcal{S}$ is essentially a Turing machine where
Cells on the tape can hold arbitrary elements of $\mathcal{S}$.
The ...

**1**

vote

**0**answers

184 views

### How may we define a bijection from $\wp(\mathbb{Q})$ to $\mathbb{R}$ in $ZFC$? [closed]

Some expressed difficulty understanding that there are more members in $\mathbb{R}$ than in $\mathbb{Q}$ according to classical set theories because there between all real numbers is a rational ...

**10**

votes

**0**answers

154 views

### Can we find minimal-diameter metrics without computability?

A beautiful argument by Nabutovsky and Weinberger (see http://math.uchicago.edu/~shmuel/fractal.ps) shows that, if $M$ is any smooth compact manifold of dimension $\ge 5$, then the diameter functional ...

**13**

votes

**2**answers

420 views

### Logical complexity of algebraically closed fields

One can define fields using a finite list of axioms that quantify over the field itself. However, the obvious way to define algebraically closed fields involves either an infinite list of axioms, or ...

**7**

votes

**3**answers

154 views

### Ordinal-indexed transitive antichain of sets with urelements

Operate in ZFC. Can we find a function-class $\phi$ whose domain is the class of ordinals such that the following properties hold?
If $x \in \phi(\alpha)$, then either $x \in \mathbb{N}$ or there ...

**2**

votes

**0**answers

66 views

### Kripke semantics for fuzzy logics

I am interested in Fuzzy logic.
I have a problem about Gödel Logic, I'm studying Kripke semantics for fuzzy logics and have found the necessary and sufficient conditions on Kripke frames for ...

**35**

votes

**4**answers

2k views

### Hilbert's (cancelled) 24th problem

Hilbert's 23 problems, ten of which were presented at the 1900 ICM in Paris, are too famous for any mathematician to not know. If one reads the descriptions of the problems in Hilbert's paper, one ...

**25**

votes

**0**answers

547 views

### Where do uncountable models collapse to?

Suppose $T$ is a complete first-order theory (in an finite, or at worst countable, language). Given any model $\mathcal{M}\models T$ of cardinality $\kappa$, we can ask whether $\mathcal{M}$ can be ...

**12**

votes

**2**answers

393 views

### Woodin on Posner-Robinson for the hyperjump and sharp

The Posner-Robinson theorem states that, if $X$ is noncomputable, there is some $G$ such that $X\oplus G=G'$; that is, even though genuine jump inversion only works above $0'$, every (nontrivial) $X$ ...

**11**

votes

**1**answer

465 views

### Can you have many independent reals?

Working in $\sf ZFC$, is it provable, or at least consistent (say, over $L$), that you have $\aleph_1$ forcings, $\Bbb P_\alpha$ such that:
$\Bbb P_\alpha$ is c.c.c.
$\Bbb P_\alpha$ adds a real ...

**3**

votes

**0**answers

383 views

### “Nicely” strong measure zero sets

This question is essentially an expanded version of the unanswered half of Two strengthenings of "strong measure zero".
A set $X$ of reals is strong measure zero if, for any $f: ...

**7**

votes

**1**answer

337 views

### Is every ordinal potentially definable?

It is easy to see that, if $V\models\alpha>\omega_1^{CK}$, then $\alpha$ is not recursive in any forcing extension of $V$. The argument goes as follows:
The relation "$\Phi_e=r$" is $\Pi^0_2$.
...