**1**

vote

**1**answer

30 views

### References for the Keisler Order

Are there any good/modern references on the Keisler order. I have been reading Keisler's original paper, "Ultraproducts which are not Saturated", which introduces the order. However it is somewhat ...

**4**

votes

**0**answers

46 views

### The (global) theory of Borel equivalence relations

What do we know about the complexity of the theory of Borel equivalence relations, with the Borel reducibility order $\leq_B$?
That is, let $\mathcal{B}$ be the set of all Borel equivalence ...

**4**

votes

**1**answer

208 views

### Set-theoretic tautologies

Let us consider unquantifed formulas of a set theory (for example, NBG), more precisely,
the formulas, constructed from variables and the constants $\emptyset, V$ (the empty set
and the class of all ...

**0**

votes

**1**answer

132 views

### Refutation of $A \land \lnot\lnot\lnot A$ by resolution [on hold]

$ A \land \lnot\lnot\lnot A $ this is a very simple example. Resolution is refutation complete. So it should be able to refute this formula. However, I don't see how would it do that without using ...

**0**

votes

**0**answers

30 views

### Is the pseudomenon a statement? [migrated]

I'm asking this because I'm teaching a class on paradoxes for kids, and I realized I have no idea what the answer to this question is. It is a research question in the pedagogical sense, I suppose.
...

**2**

votes

**1**answer

68 views

### Is quasivariety generated by all perfect graphs finitely axiomatizable?

Fix logic $L$ with equality and a binary relation symbol $E$.
The class of graphs can be identified with the class of models of the universal first-order Horn $L$-sentences $\forall x,y\; E(x,y) ...

**2**

votes

**0**answers

63 views

### Countable model theory for $\omega$-stable theories?

This is a bit of a fishing expedition, because I'm not sure what I'm looking for. Very vaguely stated, here's the driving question:
What conditions on an $\omega$-stable theory make the class of ...

**4**

votes

**1**answer

219 views

### How do we know if Vaught's Conjecture is Absolute?

Please note that this might be some confusion on my part about the work surrounding Vaught's conjecture.
First of all, Vaught Conjecture states that if a first-order complete theory $T$ in a ...

**4**

votes

**1**answer

242 views

### Question about “Coding the universe”

The following is a result which I know as a weak form of Jensen's coding lemma$^*$ (first published in the book "Coding the universe"; also see http://www.jstor.org/stable/2273986):
For any class ...

**7**

votes

**1**answer

107 views

### Strongly compact cardinal with bad covering properties

This is a continuation of the question covering properties of strongly compact embedding.
Recall that a cardinal $\kappa$ is $\nu$-strongly compact cardinal if there is an elementary embedding ...

**8**

votes

**0**answers

179 views

### Main Gap Phenomenon

Shelah's Main Gap Theorem states that for all first-order, complete theories, T, in a countable language, we have that either $$I(T,\aleph_\alpha)=2^{\aleph_\alpha}$$ or ...

**15**

votes

**0**answers

347 views

### If all reals are generic, is the set of reals generic?

Let $W\subseteq V$ be two models of $\sf ZFC$ with the same ordinals. Is the following situation consistent:
For every $x\in\Bbb R^V$ there is some $P_x\in W$ such that for some $G\subseteq P_x$ ...

**10**

votes

**1**answer

373 views

### Harrington's unpublished note “The constructible reals can be anything”

Around 1974, Leo Harringto wrote an unpublished note entitled "The constructible reals can be anything", in which he proved that it is consistent that being $\Delta^1_n$ is the same as being ...

**7**

votes

**2**answers

479 views

### Is there one binary operation foundational for set theory?

The membership relationship "$\epsilon$" is foundational for set theory, in the sense that the axioms of any set theory are formulated in the language of "$\epsilon$". Naturally, the question arises ...

**4**

votes

**1**answer

572 views

### solvable word problem without algorithm

Let $G$ be a finitely generated group. I wonder if there are examples where:
1) The word problem is known to be solvable in $G$ but there is no algorithm known.
2) The word problem is known to be ...

**7**

votes

**2**answers

181 views

### Does forcing with recursively pointed perfect trees add a Turing degree that is minimal over $V$?

A tree $T$ on $\omega$ is recursively pointed if it is recursive in each of its branches. We can consider a variant of Sacks forcing where the conditions are recursively pointed perfect trees ordered ...

**6**

votes

**1**answer

208 views

### Does V=L imply transitive containment over, say, Z?

In Zermelo set theory, the axiom of constructibility V=L seems to imply every set has a transitive closure. But does that argument have to assume transitive closure in the first place, to get an ...

**1**

vote

**1**answer

174 views

### Is anyone talking about partial interpretations of theories? (Edited)

Consider the concept of a module. This can be understood a multisorted algebraic theory $\mathsf{Mod}$ on two sorts, a scalarsort $S$ and a vectorsort $V$. An interpretation of $\mathsf{Mod}$ in ...

**7**

votes

**1**answer

179 views

### Forcing, cuts, and Dedekind-finite cardinalities

Tl;dr version: there are two natural classes of cuts in the nonstandard model of arithmetic consisting of the Dedekind-finite sets (if, in fact, they constitute such a model); both these classes are ...

**13**

votes

**1**answer

835 views

### How much of GCH do we need to guarantee well-ordering of continuum?

It's well known that, if GCH holds, then every cardinal can be well-ordered. However, I'm sure we don't need full power of GCH to prove it for specific cardinal, e.g. continuum. I have been wondering, ...

**0**

votes

**0**answers

58 views

### What is semantics of “type”? Do “types” of “type theory” semantically differ from “set” of set theory? [migrated]

"To be of a (certain type)" is a fundamental relationship for ontology and the computer science "ontologies" are in the core of Semantic Web (which is my interest). But I did not encounter a ...

**2**

votes

**2**answers

552 views

### Meta$^{n{-}th}$ mathematics [duplicate]

Metamathematics has a reasonably clear connotation,
enough to have a Wikipedia page,
with Gödel, Tarski, and Turing playing leading roles;
Kleene's book (Introduction to Metamathematics (Amazon ...

**10**

votes

**1**answer

262 views

### What is the large cardinal strength of the assertion that every $\kappa$-complete filter on $\kappa$ extends to a $\kappa$-complete ultrafilter?

It is well-known that an uncountable regular cardinal $\kappa$ is strongly compact if and only if every $\kappa$-complete filter on any set extends to a $\kappa$-complete ultrafilter on that set. The ...

**3**

votes

**1**answer

96 views

### If questions are formalized as ideals of a boolean algebra, what kind of algebra of questions appears from Stone representation theorem?

Affirmative propositions make up a Boolean algebra, and Boolean algebras became part of classical algebra for over one century ago - in this sense they are "simple". But I did not encounter in ...

**3**

votes

**1**answer

150 views

### Elementary chains of $\aleph_1$-saturated models

If $X$ and $Y$ are two sets linearily ordered by $<$, $X$ is called cofinal in $Y$ if $X \subseteq Y$ and and for every $y \in Y$, there is a $x \in X$ with $y < x$.
If $M$ is some model and ...

**-2**

votes

**0**answers

19 views

### Can variables of quantification be repeated? [migrated]

For example, is the following valid?
$\forall x \forall x \alpha$
Is so, then how is it evaluated user the basic semantic definition?

**0**

votes

**0**answers

40 views

### How to study first order logic, without the use of sets? [migrated]

I am interested in learning first order logic. In particular, I would like to be comfortable enough with first order logic in order to fully tackle, say, the Axioms of Zermelo-Fraenkel, as in Jech's ...

**0**

votes

**1**answer

63 views

### A question on Carnap's modal semantics on the basis of Cochiarelli's primary semantics

I believe I learned that Carnap's state description semantics for propositional modal logic suffered from validating $\lozenge p$ for all atomic variables p. Re-reading Nino Cochiarelli's primary ...

**6**

votes

**0**answers

186 views

### Does Sageev's result need an inaccessible?

In 1981, building on work by Ellentuck in 1974, Sageev showed ("A model of ZF + there exists an inaccessible, in which the Dedekind cardinals constitute a natural non-standard model of arithmetic," ...

**3**

votes

**1**answer

213 views

### What axioms (other than choice) have a taming effect on the ordering of cardinalities?

Axiom of choice arranges all cardinalities into a well-ordered chain but without it their ordering can be wild in general ZF models, e.g. two cardinalities may not even have inf or sup. However, ...

**0**

votes

**0**answers

86 views

### A Question on an extension of the Friedman-Sheard set theory and truth theory with complements of unions

The Friedman-Sheard logic of Truth has been studied much; see e.g. Andre Cantinis $\textit{Logical Frameworks fro Truth and Abstraction}$, Elsevier 1996 §66 and a number of articles by Volker Halbach. ...

**0**

votes

**2**answers

214 views

### Valid statement in submodel? (Consistency lemma in Cohen's CH book)

In Paul Cohen's
Set Theory and the Continuum Hypothesis
(Page 71) there is a lemma with the assumption that
$\exists x\, P(x)$.
The ''proof'' there, uses the following argument:
Intuitively the ...

**0**

votes

**0**answers

94 views

### How to prove a Lefschetzesque principle for relative homology and absolute homology

Here is my conjecture:
If H is some homology theory from some abelian category A to some abelian category B, E is the class of all epimorphism in A, E' is some closed class of epimorphisms in A and ...

**10**

votes

**1**answer

413 views

### Does Turing determinacy imply full determinacy?

The axiom of Turing determinacy is a weakening of the full axiom of determinacy, $AD$, in which only games with payoff sets which are $\equiv_T$-invariant are demanded to be determined.
In "Turing ...

**2**

votes

**1**answer

135 views

### The number of countable models [duplicate]

Let $\mathcal{L}$ be a countable first order language. For a natural number n, can we find a complete $\mathcal{L}$-Theory $T$ which has exactly n non-isomorphic countable models ?

**3**

votes

**0**answers

84 views

### Is every union-closed family of set the set of solutions of some co-HORNSAT formula?

Related to the Union-closed sets conjecture.
Let $\phi$ be a co-HORNSAT
on variables $x_1 \ldots x_n$ in CNF format.
This means in every close at most one literal is negative.
The solutions of ...

**8**

votes

**2**answers

298 views

### When is $A$ “$L$-ish” whenever $B$ is “$L$-ish”?

My question is about a kind of relative constructibility in set theory.
Fix a countable transitive model $W\models ZFC$ which is much bigger than $L^W$. There is a natural way within $W$ to compare ...

**12**

votes

**1**answer

352 views

### Transfinitely extending $\sf PA$ — can we get stronger than $\sf ZFC$?

Let $\sf PA$ denote the theory of natural numbers with constants $(0, 1)$ and binary operators $(+,\times)$ based on the first-order predicate calculus with equality, having the following axioms, ...

**10**

votes

**1**answer

292 views

### Elements of the method of forcing in some papers of N. N. Luzin

In the paper
Eléments de la méthode de forcing dans quelques travaux de N. N. Lousin. (French) [Elements of the method of forcing in some papers of N. N. Luzin] Amphora, 469–479, Birkhäuser, ...

**10**

votes

**1**answer

201 views

### Higher recursion theory and reverse mathematics: What is to $\Pi^1_1-CA_0$ as $RCA_0$ is to $ACA_0$?

There is an extremely rich and well-understood analogy between "recursively enumerable" and "$\Pi^1_1$" - indeed, this is the starting point of metarecursion theory, and $\alpha$-recursion theory in ...

**6**

votes

**1**answer

186 views

### Stationary sets in HOD

My questions concern the following quote from “The HOD Dichotomy”, page 8.
"… notice that $\ cof(\omega)\cap\lambda$ belongs to $HOD$ even though it might mean
something else there. Also, ...

**5**

votes

**1**answer

317 views

### On Consistency of an Existence

Let $\omega \leq \kappa <2^{\omega}$ , $\omega \leq\lambda \leq \kappa$ and $D(\kappa, \lambda)$ be the statement:
For all $ \mathfrak{B} \subseteq \mathbf{P}(\omega)$ with ...

**3**

votes

**1**answer

303 views

### “the” random permutation

I recently looked at Permutations on the random permutation which seems to talk about the notion of random permutuation as a notion from logic rather than probability.
The random permutation is ...

**18**

votes

**2**answers

941 views

### Is the Invariant Subspace Problem arithmetic?

Invariant Subspace Conjecture: A bounded operator on a separable Hilbert space has a non-trivial closed invariant subspace.
Can this conjecture be reformulated as an arithmetic statement, that is, ...

**1**

vote

**0**answers

68 views

### Interesting fragments of first-order logic induced by sorting?

In first approximation, modal logic (I'm using the term loosely)
can be understood as an interesting fragment of first-order logic
(for simplicity I ignore e.g. how modal logic relates to
...

**4**

votes

**2**answers

140 views

### Models of PRA/EFA with induction on $X$ but not $\omega^X$

As I currently understand it, induction on formulas containing $N+1$ first-order quantifiers is required to prove the well-ordering of the ordinal $(\omega \uparrow\uparrow N) < \epsilon_0$, that ...

**6**

votes

**0**answers

151 views

### Co-Heyting Valued Models of Paraconsistent Set Theory

I've been trying to do some forcing arguments in intuitionistic ZF using Heyting valued models where the Heyting algebra I'm using is actually a bi-Heyting algebra (both a Heyting algebra and a ...

**12**

votes

**2**answers

301 views

### Producing no non-constructible reals

The following is stated without proof in Shelah's book "Cardinal arithmetic" (page 276), and is attributed to Uri Abraham:
Suppose that $L[A], L[B]$ have no non-constructible reals and that ...

**0**

votes

**0**answers

95 views

### What do we call functions that behave like predicate symbols?

Assume a metatheory that supports lambda-abstraction, and an object language that is merely first-order. Now let $\varphi$ denote a formula in the object language with one free variable $x$. Then we ...

**2**

votes

**1**answer

205 views

### Partial interpretation of an iteration

Suppose that $\langle\mathbb{P_\alpha,\dot Q_\beta}\mid \beta<\delta,\alpha\leq\delta\rangle$ is a system of iterated forcing.
Let $\dot a$ be a name in $\mathbb P_\delta$, and let $G_\alpha$ be a ...