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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5
votes
3answers
125 views

Application of Fraïssé construction in set theory

As you know Fraïssé limit construction and its generalization, Hrushovski's construction, have many applications in model theory to build models with interesting property. Now I would like to know ...
3
votes
3answers
321 views

Category of Gödel Codings? [Reference Request]

Consider computation with the integers $\mathbb{Q}$. The traditional theory of recursive functions on $\mathbb{N}$ applies to $\mathbb{Q}$ by the identification of $\frac{a}{b} \in \mathbb{Q}$ with ...
4
votes
0answers
181 views

Quasi-disjoint subsets of an infinite set and $\neg \mathsf{AC}$

Is it consistent with $\mathsf{ZF}$ (without $\mathsf{AC}$) that there is an infinite set $X$ and a subset $S\subseteq\mathcal P(X)$ of the same cardinality as $\mathcal P(X)$ with the property that ...
11
votes
2answers
1k views

Do the real numbers “know” that they are countable in a larger model?

(This was first posted to math.stackexchange but had no answers there after several days): Let ${\mathbb R}$ be the set of real numbers in whatever is your favorite model of $ZFC$. Then (by Levy ...
2
votes
0answers
53 views

Potentiality classes and Borel reductions

In a 1998 paper by Hjorth, Kechris, and Louveau, there was a definition given of a "potentiality class." That is, given an invariant equivalence relation $E$ on a standard Borel space $X$, we say $E$ ...
4
votes
1answer
148 views

Uncountably categorical theories which are interpretable in a strongly minimal

Definition: Let $\lambda$ be a cardinal. An $\mathcal{L}$-theory $T$ is called $\lambda$-categorical whenever every two models of $T$ of cardinality $\lambda$ are isomorphic. Definition: An ...
6
votes
1answer
405 views

Adding sets not containing arithmetic progressions of length three by forcing

Consider the following forcing notion: conditions in $\mathbb{P}$ are pairs $(s, N),$ where: 1) $s\in 2^{<\omega}$, 2) $N\in \mathbb{N}$, 3) (by identifying $s$ with a subset of $lh(s)$) $s$ ...
5
votes
0answers
147 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 ...
9
votes
2answers
413 views

Non-Forcing and Independence

I asked this question about two weeks ago on MSE and haven't gotten an answer, so I thought I would post the question here. Do there exists sentences which are independent of ZFC, cannot be shown to ...
0
votes
1answer
83 views

Completeness of a set of propositional formulas [closed]

A set $\sum$ of formulas in propositional logic is complete if for each propositional formula $\phi$ either $\sum \vdash \phi$ or $\sum \vdash \neg \phi$. Clearly every inconsistent set of formulas ...
-2
votes
0answers
72 views

Prove the contrapositive: (p => q) => (~q => ~p) using only FITCH rules of inference [closed]

I stumbled upon this problem when taking Stanford's Intro to Logic Course. Wikipedia shows some proofs to the contraposition ( http://en.wikipedia.org/wiki/Contraposition ) but I couldn't fit any of ...
11
votes
1answer
295 views

Is factorial definable using a $\Delta_0$ formula?

The factorial function is primitive recursive, and therefore definable by a $\Sigma_1$ formula. Is it also definable by a $\Delta_0$ formula (i.e. bounded quantifiers)? If not, why?
1
vote
1answer
207 views

Confusion with proof about a fact $\mathbb{P}$-name [closed]

Let $\mathbb{P}$ be poset. Let $B$ be a set. We say that a $\mathbb{P}$-name $\dot{b}$ is a nice name for member of $B$ if there is a maximal antichain $A\subseteq\mathbb{P}$ and a function ...
6
votes
1answer
94 views

Chains of forking extension in stable theories

Let $T$ be an stable theory. Further we work in the monster model of $T^{eq}$. We say that a chain of types of the form $$tp(a_1/A_1)\subset tp(a_2/A_2) ... \subset tp(a_n/A_n)$$ is a forking chain ...
3
votes
1answer
120 views

A question on many-one reducibility

Let $\phi_0,\phi_1,\phi_2,\ldots$ be an acceptable programming system. For each $x\in\mathbb{N}$, let $W_x$ the domain of $\phi_x$, and let $K=\{x\in\mathbb{N}:W_x\neq\emptyset\}$. Is there a ...
6
votes
1answer
348 views

Alternate proof of Morley's theorem?

I'm trying to understand the result given in the first box at slide 45 of this talk. Specifically: 1) What is the source cited? I have not been able to find any article by Keisler, Chudnovsky and/or ...
5
votes
2answers
212 views

Primitive Recursive Arithmetic via Universal Algebra

From the wikipedia article on Primitive Recursive Arithmetic (see http://en.wikipedia.org/wiki/Primitive_recursive_arithmetic): "Primitive recursive arithmetic, or PRA, is a quantifier-free ...
7
votes
1answer
215 views

Is $ACA_0$ + `True Arithmetic exists' interpretable in $ACA$?

Maybe someone here can help me with a question concerning second-order arithmetic. Consider the system $ACA_T := ACA_0 + \exists X \forall x (x \in X \leftrightarrow T(x))$, where $T(x)$ is a ...
6
votes
1answer
269 views

Axiom of choice and the equality between second-order constructible universe and HOD

I try to prove $L_{SO}=\mathrm{HOD}$, where $L_{SO}$ is second-order constructible universe which has similar definition with $L$ but it uses second-order definability rather than the first-order ...
4
votes
1answer
183 views

Explicit bounds for transfer results in algebraic geometry

Assume you have an ideal $I\subseteq\mathbb{Z}[X_1,\ldots,X_n]$ of the polynomial ring in $n$ variables over the integers. For any field $\Bbbk$, I can consider the ideal ...
3
votes
1answer
203 views

Was $\Sigma x$ used as quantifier?

Kurt Gödel in 1931 used $x\Pi a$ where we in contemporary notation would use $(\forall x) A$ or $(x)A$, and $Ex a$ where we would use $(\exists x) A$. I believe that I remember that $\Sigma xA$ has ...
0
votes
0answers
114 views

Reference Request: Category of explicit maps between primitive recursive sets?

[Edited] Let $\mathsf{PR}$ be the category defined as follows: Choose a specific presentation of Primitive Recursive Arithmetic, that is, with a specific set of terms for primitive recursive ...
4
votes
1answer
130 views

presaturated ideals

In this paper, Gitik and Shelah make the following claim (part of Proposition 1.5): Claim (Gitik-Shelah): Suppose $\kappa < \lambda$ are regular, $2^\lambda = \lambda^+$, and $D$ is a normal ...
1
vote
0answers
152 views

Property theories

Property theory is, as I have understood it, first of all characterized by an attempt to approach naive comprehension type-freely and without committing to extensionality. There is e.g. the work of ...
5
votes
2answers
419 views

A specific Model of ZFC

In his paper "Some Second Order Set Theory", Joel Hamkins asked whether there is a model of set theory $V$ that is elementary equivalent to $V[G]$, Whenever $G$ is $V$-generic for the collapse of a ...
10
votes
4answers
456 views

The groupoid of algebraic expressions and proofs

Fix a set of variables $V$, and suppose we're given a presentation of a monosorted algebraic theory, with variable symbols taken from $V$. For the sake of example, suppose the presentation consists of ...
5
votes
1answer
192 views

$\text{Cont}(X,X)$ and $\neg\mathsf{GCH}$

For a topological space $(X,\tau)$ let $\text{Cont}(X,X)$ denote the set of continuous functions $f:X\to X$. Is it consistent that there a space $(X,\tau)$ such that $$|X| < |\text{Cont}(X,X)| < ...
2
votes
2answers
152 views

Computable Categories in the most direct sense?

While there is a lot of work in category related to notions of realizability and computability, etc... I've failed to find work on categories that are computable in the sense of having object and ...
4
votes
1answer
246 views

stationary tower forcing

It is known that if $\delta$ is a Woodin cardinal and $\kappa < \delta$, then the stationary tower forcing $\mathbb Q^\kappa_{<\delta}$ preserves cardinals up to $\kappa$ and forces $\delta = ...
6
votes
1answer
198 views

What is known about the boundary between Richardson's theorem and the Tarski-Seidenberg theorem?

Tarski proved that equalities and inequalities in can be decided over $\mathbb{R}[x].$ Richardson proved that adding composition with the sine and exponential functions caused the problem to become ...
3
votes
1answer
100 views

Is $PA+ \neg R_{PA}$ $\Pi_1$- conservative over $PA$?

I think it is well-known that $PA+\neg con(PA)$ is $\Pi_1$-conservative over $PA$ (for proof see Smorynski's article, 'the incompleteness theorems', in handbook of mathematical logic). What can we ...
3
votes
0answers
78 views

Logical framework for type theories like ML and CIC

I'm looking for a logical framework in which it is possible to easily present both intensional and extensional theories of dependent types with a partially ordered set of universes à la Russell ...
6
votes
0answers
156 views

Singular Jonsson cardinals

Is the consistency of the following well-known: $(*)$: There exists a singular cardinal $\kappa$ such that : (1) $\kappa$ is a Jonsson cardinal, (2) $\kappa$ is not a fixed point of the ...
7
votes
1answer
142 views

$2$-uniformization versus $\omega$-uniformization of ladder systems

Let $S \subseteq \omega_1$ be a stationary set of limit ordinals and let $L = \langle L_\alpha \;|\; \alpha\in S\rangle$ be a ladder system. We say that $L$ has $\kappa$-uniformization if for every ...
18
votes
1answer
737 views

What do you do if you believe a problem is undecidable?

While the title of this question is subjective, I hope to make what I'm looking for quite concrete. The first, and main question is this: If you believe that a problem you are working on is formally ...
12
votes
1answer
513 views

Ordinary mathematics in Chang's model

This question is prompted by a paper by Andre Kornell that just appeared on the arXiv. A large portion of the paper is devoted to showing that a surprising amount of ordinary mathematics can be ...
12
votes
0answers
209 views

The topos for forcing in computability theory

My understanding is that forcing (such as Cohen forcing) can be described via a topos. For example this nlab article on forcing describes forcing as a "the topos of sheaves on a suitable site." My ...
2
votes
1answer
135 views

Two questions on substitutability

(1) The condition that a term $a$ be substitutable for another term in an expression can be given a recursive definition. Who first developed such a definition? (2) One sometimes see the phrase "$a$ ...
9
votes
0answers
207 views

Proving regularity properties from forcing axioms

It's well known that PFA implies projective determinacy. It's also well known that PD implies that all projective sets are Lebesgue measurable, have the Baire property, etc. Is there a direct proof ...
0
votes
1answer
174 views

Is there a nontrivial maximally recursive function? [closed]

Say that a (recursive) function $f:\Bbb N\rightarrow\Bbb N$ is maximally recursive if, for all $n\in\Bbb N$, the value $f(n+1)$ can be calculated only by first knowing $f(n)$. A rather trivial example ...
8
votes
0answers
308 views

What is known about the reverse mathematics of algebraic number fields?

I know work on the reverse mathematics of countable algebraic field extensions including Galois theory, notably including Dorais, Hirst, and Shafer http://arxiv.org/pdf/1209.4944v2.pdf. But algebraic ...
2
votes
0answers
96 views

May $\Sigma_3$-collection hold below $\Sigma_3$-admissible ordinals for Gödel's L?

Suppose you have a system X=$KP$ + infinity plus $\Sigma_{3}$-collection and $\Delta_{2}$-specification. May $L_{\delta}\vDash X$ for some $\delta$ smaller than all $\Sigma_{3}$ admissible ordinals?
1
vote
1answer
142 views

Propositional logic: Minimal set of formulas, which is consistent and complete

Hello together, I have a rather basic issue on propositional logic: first, consider an arbitrary set of formulas $T$ that is consistent and complete, i.e., for every propositional formula $\varphi$, ...
9
votes
1answer
302 views

singularize the least inaccessible?

Is it consistent that there is some partial order $\mathbb P$ and some inaccessible cardinal $\kappa$, which is the least inaccessible, such that $\mathbb P$ forces $\kappa$ to be singular while ...
8
votes
1answer
381 views

Adding a real with infinite conditions

Consider the forcing $\Bbb P$ whose conditions are partial functions $p\colon\omega\to2$ with $\operatorname{dom}(p)$ a co-infinite subset of $\omega$, ordered by reverse inclusion. Does $\Bbb P$ ...
1
vote
1answer
81 views

On whether a formula of KP is $\Pi_3$

In the context of KP, is the formula $\forall w(w\in x \leftrightarrow\forall y\exists z F(w,y,z))$ $\Pi_3$ when $F(w,y,x)$ is $\Delta_0$?
5
votes
1answer
434 views

Consequences of ZF+“all subsets of reals are Lebesgue measurable”

(I'm not sure if this is entirely suitable here so feel free to close it if it's not.) The statement "there is a Lebesgue measure on $\mathbb{R}$($2^\omega$)" means: there is a total $\sigma$-additive ...
7
votes
2answers
233 views

Decidability of diophantine equation in a theory

Given a theory $T \subseteq \operatorname{Th}(\mathbb{N})$, define the decision problem $D_T$ as follows: Given a polynomial $p$ with integer coefficients and variables $\bar{x}$, decide whether ...
10
votes
2answers
300 views

Is there a compendium of the consistency strength between the most important formal theories?

Preliminar Notions: A formal system is a tuple $(\Sigma,G,A,R)$ where $\Sigma$ is an alphabet (set of symbols), $G$ is a formal grammar on $\Sigma$ that generates a formal language $L$ (set of well ...
11
votes
1answer
402 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 ...