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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-2
votes
0answers
54 views

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

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
237 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
187 views

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

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
0answers
53 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
107 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 ...
5
votes
1answer
320 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
191 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
210 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
257 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
177 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
196 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
111 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
123 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
150 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
405 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
438 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
184 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
148 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
240 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
192 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
99 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
73 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
154 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
141 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
721 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
507 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
206 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
133 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
200 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
173 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
303 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
94 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
136 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
301 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
378 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
429 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
231 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
298 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
386 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 ...
2
votes
1answer
95 views

Total formulae in a theory equivalent to $\Delta_0$ formulae in the theory?

Let a formula $\phi$ of the language of first-order Peano arithmetic be total in a theory Th that extends PA iff, for any $k_1, \dots, k_n \in \omega$, Th $\vdash \phi(\bar k_1, \dots, \bar k_n)$ or ...
4
votes
2answers
262 views

Limits of determinacy on reals

For $X\subseteq\mathbb{R}^\omega$, say that $X$ is determined if the associated game on $\mathbb{R}$ of length $\omega$ (players I and II alternate playing reals, player I wins iff the sequence built ...
0
votes
0answers
40 views

representing quasicrystal as tilings and appearing frequencies of each tile

Quasicrystal can be fully represented either using projection method or tilings with constraints. For the latter, is there some sort of study on the "appearing frequency" of each tile or even ...
6
votes
1answer
209 views

Are two forms of the Dual Schroeder-Bernstein property equivalent?

We know the Shroeder-Bernstein (SB) theorem can be proved in ZF, while the Dual Schroeder-Bernstein (DSB) can be proved in ZF+AC but not in ZF. Define as ISB the property that whenever there are both ...
4
votes
1answer
410 views

Did Brouwer evade uncountability?

I have the distinct memory of having often heard and read that intuitionism was inter alia geared to avoid Cantor's uncountable sets, and it may be that this was Brouwer's plan. But are there accounts ...
-2
votes
1answer
138 views

Does PA prove a sentence asserting that all of I-sigma(n) theories are consistent? [closed]

We know that PA proves consistency of $I\Sigma_{n}$ for any $n$. But does PA prove the sentence: $\forall n (con(I\Sigma_{n}))$?
1
vote
3answers
574 views

Brouwer vs. Cantor

Brouwer criticises Cantor e.g. in Intuitionistiche Mengenlehre. Is there a link or reference to some streamlined modern account of Brouwer's ideas?
11
votes
5answers
767 views

Does k(X) have a k-basis for every set X, without AC?

This question is inspired by Pace Nielsen's recent question Does a left basis imply a right basis, without AC?. For any field $k$, the field $k(x)$ of rational functions in one variable has an ...
11
votes
1answer
213 views

counterexample regarding quotient algebras in forcing

Suppose $A$ and $B$ are complete subalgebras of a complete boolean algebra $C$. Let $G \subseteq A$ be generic. In the extension $V[G]$, we can define the quotient algebras $B/G$ and $C/G$ in the ...
22
votes
2answers
692 views

Forcing as a replacement of induction and diagonal arguments

Let me give some examples motivating the question. The use of forcing instead of induction: For this consider Cantor's theorem: Theorem 1. Any two countable dense linear orders $I, J$ without end ...