first-order and higher-order logic, model theory, set theory, proof theory, computability theory, formal languages, definability, interplay of syntax and semantics, constructive logic, intuitionism, philosophical logic, modal logic, completeness, Gödel incompleteness, decidability, undecidability, ...

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6
votes
3answers
703 views

Decidability of matrix algebra

Take multi-sorted first-order logic with equality, complex scalars, 1xn vectors, nx1 vectors, nxn matrices, addition and multiplication for each pair of sorts they make sense for, and hermitian ...
6
votes
0answers
117 views

On an unpublished result of Magidor

In 1970th, Magidor proved the following important results: (1) Assuming the existence of a supercompact cardinal, it is consistent that $\aleph_\omega$ is strong limit and ...
77
votes
9answers
13k views

Solutions to the Continuum Hypothesis

Related MO questions: What is the general opinion on the Generalized Continuum Hypothesis? ; Completion of ZFC ; Complete resolutions of GCH Background The Continuum Hypothesis (CH) posed by Cantor ...
3
votes
1answer
213 views

On generic forcing conditions

Let $P$ be a forcing poset, and $Q \in V^P$ a forcing poset in $V^P$. Let $M \prec H(\lambda)$ ($\lambda$ sufficiently large) countable with $P,Q \in M$. What I want to know is if then the following ...
6
votes
1answer
259 views

Tree property and singular strong limit cardinals

I heard that the following theorem is proved recently by Foreman-Magidor, which answers a famous old open question: Theorem. It is consistent, relative to the existence of large cardinals, that ...
1
vote
1answer
145 views

Notation: $Sigma$ and $Pi$ of intersections

In Jech - Set Theory, the proof of Theorem 31.7, I came along some notations I wish to understand correctly. For a countable elementary substructure $M \prec H_\lambda$ and $A \in M$ and a generic ...
3
votes
1answer
350 views

Preservation of $\diamondsuit$ by ccc forcings of size $\leq \omega_1$

This is essentially exercise H8 (p.248) of Kunen's Set Theory: An Introduction to Independence Proofs (old edition), or exercise IV.7.58 (p.307) of Kunen's Set Theory (new edition). Suppose $P$ is ...
2
votes
1answer
108 views

Precise interpretability strength of $\mathcal P_{DF}(\omega)$ and $L_{\omega_1^{CK}}$

I am curious about the relationship between the definable power set of $\omega$ and the $\omega_1^{CK}$th level of the constructible sets $L$. In short, $\omega_1^{CK}$ is the least nonrecursive ...
4
votes
1answer
279 views

What are key $\Sigma^0_2$ or $\Pi^0_3$ theorems?

I am researching a logical system that is limited to $\Pi^0_2$ sentences and I am busy to prove that FOL + PA is a conservative extension of that system. Meaning that with $\Sigma^0_n$ sentences (that ...
2
votes
0answers
116 views

RCS iteration such that the RCS limit is semi-proper

For a countable ordinal $\eta$ and an ordinal $\gamma$ let $\langle P_\alpha, \dot{Q}_\alpha \colon \alpha < \gamma \rangle$ be an RCS iteration with RCS limit $P_\gamma$, such that ...
3
votes
0answers
90 views

Cardinality based results in Topological Vector Spaces?

Given a topological vector space $V$, let its density be the smallest cardinal $A$ such that a set of cardinality $A$ is dense in $V$. Naively, it seems one of two things happen: TVS's $V$ of ...
3
votes
1answer
171 views

May open sentences be eliminated?

Saul Kripke famously invoked a free logic to avoid validating the Barcan Formula and its converse. In that context he adduced a generality interpretation of free variables. The converse of the Barcan ...
5
votes
1answer
188 views

A question regarding strong cardinals and measure sequence

Let $E$ be a $(\kappa, \lambda)$-extender such that $j: V\to M\simeq Ult(V,E)$ is the corresponding elementary embedding with critical point $\kappa$, $M\supset V_{\kappa+2}$, $M^\kappa\subset M$. Let ...
3
votes
1answer
82 views

Understanding Corollary 3, Sec. 5.6, of Papadimitriou's Computational Complexity

I am struggling to understand Corollary 3 from Section 5.6 of Papadimitriou's Complexity Theory book (Addison-Wesley, 1993). It got me completely confused... If anyone out there has read it and ...
5
votes
2answers
225 views

Is every non-empty $\Delta_0$ set provably the range of some primitive recursive function?

Suppose $A(x)$ is a $\Delta_0$ formula defining a non-empty set of natural numbers. It's an easy theorem that there is a primitive recursive function $f:\mathbb{N} \rightarrow \mathbb{N}$ such that ...
136
votes
27answers
16k views

What are some reasonable-sounding statements that are independent of ZFC?

Every now and then, somebody will tell me about a question. When I start thinking about it, they say, "actually, it's undecidable in ZFC." For example, suppose A is an abelian group such that every ...
4
votes
1answer
279 views

“set of all irreducible representations of a group”, set-theoretic issues [closed]

I am working on a problem related to representations of the Weil group of a local field $\mathcal{W}_F$. In many articles one introduces the set $\hat{\mathcal{W}}_F$ of all equivalence classes of ...
6
votes
0answers
208 views

A Banach-Tarski game

This is partially inspired by the question http://math.stackexchange.com/questions/1383397/cutting-a-banach-tarski-cake, which I find intriguing if unclearly written. A paradoxical family of subsets ...
11
votes
0answers
255 views

Adding a saturated ideal

Is it consistent that there is no $\omega_2$-saturated ideal on $\omega_1$, but one is introduced by an $\omega_2$-closed forcing? Some motivation: If $\delta$ is a Woodin cardinal, then it remains ...
3
votes
3answers
664 views

In which sense “closure” is a closure?

In predicate and first-order logic, if $\phi$ is a sentence, then $\forall X . \phi$ is said to be the (universal) closure of $\phi$. Is the use of the word "closure" incidental, or is there a ...
2
votes
0answers
114 views

How distributive are the bad Laver tables?

Suppose that $n\in\omega\setminus\{0\}$. Then define $(S_{n},*)$ to be the algebra where $S_{n}=\{1,...,n\}$ and $*$ is the unique operation on $S_{n}$ where $n*x=x$ $x*1=x+1\,\text{mod}\, n$ and if ...
10
votes
2answers
375 views

Constructing an $\omega_1$-sequence of functions that almost extend all previous functions

I want to construct a sequence of functions $$f_\alpha: \alpha \rightarrow \omega,\ \alpha < \omega_1$$ such that for all $\alpha < \omega_1$ the following holds: $f_\alpha$ is injective. ...
6
votes
1answer
186 views

Reverse of a termspace forcing fact

Suppose $\kappa$ is an inaccessible cardinal. Consider the termspace forcing for adding a Cohen subset of $\kappa$ after $\mathbb P= Col(\omega,<\kappa)$. Members are Levy names for countable ...
15
votes
1answer
513 views

Is the class of additive groups of rings axiomatizable?

I know that it is impossible to axiomatize the multiplicative structures of rings, called $R$-semigroups. Is anything known about the first-order axiomatizability of the class of abelian groups which ...
2
votes
1answer
151 views

A question on the name of a property

What is the name of the property that a system $T$ has if $\vdash_{T}\exists x F(x)$ only if there is a term $a$ so that $\vdash_{T} F(a)$? If I recall correctly Heyting Arithmetics has the ...
1
vote
0answers
98 views

Oracle queries asked in parallel

Definition: Assume that $\phi(q)$ is of the form $\exists y \leq 2^{p(n)} \varphi(q,y)$, where $p$ is a polynomial and $n = |q|$ (i.e. $n$ is the length of the binary representation of $q$). Then a ...
14
votes
1answer
755 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 ...
12
votes
1answer
583 views

Does “cardinal arithmetic is well-defined” imply axiom of choice?

Let me quickly explain what I mean with my question. Let $(\kappa_i)_{i\in I}$ be a collection of cardinal numbers, indexed by elements of some set $I$. We can try to define $\sum\limits_{i\in ...
3
votes
1answer
209 views

History of unstable formulas [closed]

There are many equivalent definitions for stability, one of them being that being unstable is equivalent to the existence of a formula having the order property. While intuitively it makes sense that ...
9
votes
4answers
2k views

intuitionistic interpretation of classical logic

Basically intuitionistic logic is classical logic minus the law of the excluded middle, i.e. $\neg A\vee A$ is not necessarily valid for all formulas. So I would take this to mean that classical logic ...
25
votes
4answers
1k views

Does every set of reals contain a measure-zero set of the same cardinality? Does it contain a meager set of the same cardinality?

This question arises from an issue in my post on Ashutosh's excellent question on Restrictions of the null/meager ideal. Question 1. Does every set of reals contain a measure-zero subset of the same ...
7
votes
1answer
177 views

Consistency strength of being strong cardinal and indestructible under collapses

What is the consistency strength of the following statement: $\kappa$ is a strong cardinals and it is indestructible under $Col(\kappa, <\theta),$ where $\theta> \kappa$ is some fixed ...
10
votes
1answer
248 views

Does stationary reflection imply Mahloness?

Suppose $\kappa$ is strongly inaccessible and every stationary subset of $\kappa$ reflects. Must $\kappa$ be Mahlo? Remarks: It is possible for every stationary subset of $\kappa$ to reflect, but ...
4
votes
1answer
161 views

Can we always add sets without collapsing cardinals or adding [very] bounded sets?

Given a model of $\sf ZFC$, and an infinite ordinal $\alpha$. Can we prove that there is always a cardinal $\kappa$, and a forcing $\Bbb P$, such that: $\Bbb P$ does not add sets of rank ...
6
votes
2answers
405 views

Collapsing the cardinals between two singular cardinals

Question 1: Is it consistent that there is a forcing notion collapsing $\aleph_{\omega\cdot 2}$ to $\aleph_\omega$ without collapsing $\aleph_\omega$ or $\aleph_{\omega\cdot 2 + 1}$? If the ...
-1
votes
3answers
243 views

A question about sentences undecidable in Peano's Arithmetic

Many examples are now known of sentences undecidable in Peano's Arithmetic (PA) assuming that PA is consistent. Are some or all of these sentences also undecidable in Second Order Arithmetic (SOA) if ...
6
votes
0answers
174 views

$\alpha$-minimal degrees for singular $\alpha$

An important question in $\alpha$-recursion theory is whether there is a minimal $\alpha$-degree at $\alpha=\aleph_\omega.$ Question 1. Who first introduced the above question, and where can I find ...
13
votes
4answers
1k views

Notation in Frege's Grundgesetze der Arithmetik: The U with a flourish

In the Grundgesetze der Arithmetik, Frege used a number of strange characters for notation. I would be most interested to know anything about the typography (origin, usage and so on) of the strange U ...
3
votes
1answer
192 views

Löwenheim-Skolem for many-sorted theories

Let $L$ be a many-sorted first order language, and let $\kappa$ be an infinite cardinal which is greater than or equal to the number of function and relation symbols in $L$. Let $T$ be a complete ...
9
votes
1answer
553 views

Are there discontinuities in the large cardinal hierarchy?

Suppose that $\Phi(x,y)$ is a formula in the language of set theory so that for each natural number $n$, the axiom $\exists x\Phi(n,x)$ is a large cardinal axiom (for example consider $n$-huge ...
2
votes
1answer
97 views

Classification of commutative ring ideal closure operators?

First, some setup: So: given a commutative ring $R$, let $Ideals(R)$ be set of ideals of $R$ and let $IdealClosure(R)$ be the set of closure operations $cl: \mathcal{P}(R) \rightarrow Ideals(R)$. In ...
0
votes
0answers
118 views

Minimum regular open set containing a given set in a T0 Alexandrov topological space

What is known about the minimum regular open set containing a given set in a T$_0$ Alexandrov topological space? I'm particularly interested in the condition for the minimum set happening to be ...
15
votes
0answers
374 views

What sort of cardinal number is the Löwenheim-Skolem number for second-order logic?

In their paper "On Löwenheim-Skolem-Tarski numbers for extensions of first order logic", Magidor and Väänänen make the following statement: "For second order logic, $LS(L^{2})$ ...
7
votes
1answer
281 views

A question related to Woodin's $HOD$ conjecture

Recall that an $\kappa$ is $\omega$-strongly measurable in $\text{HOD}$ if there exists $\lambda < \kappa$ such that $(2^{\lambda})^{\text{HOD}} < \kappa$ and such that there is no partition of ...
5
votes
1answer
264 views

Diagonalizing against a non stationary set of functions

Suppose $\kappa$ is regular uncountable (assume $2^{< \kappa} = \kappa$ and $\kappa$ is weakly Mahlo if needed). Is the following true? For every sequence $\langle f_i: i \to 2 \mid i \in A ...
7
votes
2answers
335 views

Vopenka's Principle for non-first-order logics

(For simplicity, the background theory for this post is NBG, a set theory directly treating proper classes which is a conservative extension of ZFC.) Vopenka's Principle ($VP$) states that, given any ...
0
votes
0answers
80 views

A question on two modal formulas

I want to find out the correspondences for the following two formulas or whether they are already derivable in the modal logic $KD4.2$, i.e. whether the formulas are valid in serial, transitive and ...
3
votes
5answers
857 views

Set theories that do require the existence of urelements?

I am looking for an axiomatic set theory that not only admits the existence of urelements/atoms (via two-sortedness or an additional unary predicate) but requires it, e.g. by an axiom like "for each ...
5
votes
0answers
230 views

Proving that a subgroup is normal

This question is partly inspired by the recent question on measurability and the axiom of choice. Suppose I come up with a way to define a subgroup of a group, in a way that involves "no arbitrary ...
0
votes
2answers
205 views

Propositional logic without negation

As part of a bigger project I am researching a propositional logic, without a negation. And I would like to know, whether this already exists, to avoid double work and have proper references. In this ...