# Tagged Questions

**10**

votes

**4**answers

515 views

### Boolean Valued Models of PA

O.K, a massively naive question. I've never really studied any non-standard models of PA before. I was just wondering if there's ever been any attempt to use the kind of Boolean valued model theory ...

**-2**

votes

**1**answer

340 views

### What is the Complete Set of Shortest Axioms of Classical Conditional-Negation Propositional Calculus? [closed]

Suppose that we only have propositional variables and connectives. Suppose our rules of inference are detachment {C$\alpha$$\beta$, $\alpha$} $\vdash$ $\beta$, and uniform substitution. Suppose that ...

**0**

votes

**1**answer

102 views

### Models of BL$\forall$

What results are known about the construction of models for a theory $T$ of the logic BL$\forall$ for languages of higher cardinality? The construction for the countable case relies on
1) The fact ...

**5**

votes

**2**answers

394 views

### complexity of proof of p(n) grows greater with n if for all x P(x) is unprovable?

Is it true that if "for all x P(x)" is unprovable in pA then the complexity of the proof of P(n) becomes greater as n grows bigger?

**9**

votes

**1**answer

332 views

### Sets computable from enough hints

Is there a non-computable set $X\subset\omega$ such that, for some $Y\subset\omega$, any infinite subset or cosubset (=subset of the complement) of $Y$ computes $X$?
More generally, call a set $X$ ...

**3**

votes

**1**answer

234 views

### On fast-growing hierarchy

Is there exists a recursively enumerable set of computable total fast-growing functions $(\mathbb N \rightarrow \mathbb N)$ such, that this set has no upper boundary in the set of all such functions ...

**1**

vote

**0**answers

85 views

### Real algebraic groups and pseudo-finiteness

What is the relationship between pseudo-finite groups and real algebraic groups?
Could you provide an example of a pseudo-finite real algebraic group and of a non pseudo-finite one, if any?
Thank ...

**1**

vote

**2**answers

269 views

### An interpretation of not-Con(PA)

Edit After Andreas Blass answer below and comments below the original post I have changed it to accommodate posters' remarks. I hope it is clear and makes more sense now.
Let $\mathrm{PA}$ be the ...

**6**

votes

**1**answer

290 views

### higher-order reflection

In the first-order context, "reflection" of a formula $\varphi(x)$ below $\kappa$ refers to the the following situation:
There are many ordinals $\alpha<\kappa$ such that for all $a \in ...

**8**

votes

**1**answer

354 views

### Groups and pregeometries

Definition.
For an infinite structure $\mathcal{A}$ and $cl : P(dom(\mathcal{A})) \longrightarrow P(dom(\mathcal{A}))$ , we say
that $(\mathcal{A}, cl)$ is a structure carrying an $\omega$-homogeneous ...

**7**

votes

**1**answer

408 views

### Can $V\neq\text{HOD}$ if every $\Sigma_2$-definable set has an ordinal-definable element?

This question arises from an issue arising in user38200's recent question concerning models of set theory in which every definable set has a definable element. In my answer to that question, with ...

**6**

votes

**1**answer

360 views

### Different approaches to forcing

There are many different approaches to the forcing method, and I am looking for all known such approaches. So my question is:
Question 1. Which different approaches to set theoretic forcing are ...

**6**

votes

**0**answers

234 views

### Silver's unpublished work on reverse Easton iteration

Silver was the first person who used the method of reverse Easton iterations in connection with large cardinals, and used it to force the failure of $GCH$ at some measurable cardinal.
At most papers ...

**5**

votes

**3**answers

296 views

### Does this property of a first-order structure imply categoricity?

Let $\mathfrak{A}$ be a first-order structure over a relational language and let $\kappa$ be an infinite cardinal. Lets say that $\mathfrak{A}$ has the $\kappa$-property if for every structure ...

**6**

votes

**1**answer

344 views

### A “good scale” that is not really a scale

I don't know much about singular cardinal combinatorics, so I apologize in advance if I write something that is wrong or looks funny. First let me recall some basic definitions.
Let $\lambda$ be a ...

**16**

votes

**1**answer

633 views

### Three old questions on the Sacks forcing

I came across the two following Qs in 1970.
Find reals $a,b$ such that $a$ is Sacks over $L[b]$ and vice versa $b$ is Sacks over $L[a]$. Note that a Sacks $\times$ Sacks generic pair definitely does ...

**15**

votes

**2**answers

635 views

### Are the quaternions not uncountably categorical?

Boris Zilber has argued that the field of the complex numbers is "logically perfect". For one thing, the theory of an algebraically closed field of characteristic zero is uncountably categorical: it ...

**13**

votes

**3**answers

658 views

### Can Suslin (or Aronszajn) lines ever be orderings of abelian groups?

I am interested in realizing linear orders as orderings of abelian groups. In particular, can Suslin lines (and other classes of line) be realised in this way?
Let $\mathcal{C}$ be a class of ...

**-3**

votes

**1**answer

280 views

### When are two algorithms essentially the same? [closed]

Inspired by Blass/Dershowitz/Gurevich's paper When are two algorithms the same? (which was referenced in another context here) I tried to boil down the question to the following situation:
Consider ...

**5**

votes

**0**answers

156 views

### A variant of Chang's model with choice

Let $M_n$, $n < \omega$, be a models of $ZFC$ with the same ordinals, closed under countable sequences. Let $\alpha_n$ be an ordinal which is a regular cardinal in $M_n$.
Question 1: Is it ...

**6**

votes

**2**answers

148 views

### Algorithm for determining when polynomial iteration is bounded?

Let $f: \mathbb{Q}\to \mathbb{Q}$ be a polynomial map with rational coefficients. Let $p\in \mathbb{Q}^n$. Is there a known algorithm that given this data determines whether or not the iterates ...

**9**

votes

**0**answers

170 views

### Which forcing types preserve the axiom of determinacy?

Do we have some rudimentary understanding of some properties that a forcing can have in order to guarantee that it doesn't violate the axiom of determinacy?
To be more specific, in Which forcings ...

**4**

votes

**1**answer

253 views

### “Is it possible to give a restricted set-theoretical definition of addition of natural numbers in terms of successor?” [Tarski]

In his paper "Restricted set-theoretical defintions in arithmetic" Raphael Robinson cites a problem posed by Tarski:
Is it possible to give a restricted set-theoretical
definition of addition of ...

**4**

votes

**1**answer

118 views

### When are the congruence lattices nicer?

This is a purely idle question, but one I'm increasingly interested the more thought I put into it:
For $\mathcal{A}$ a universal algebra (that is, nonempty set together with some named functions), a ...

**12**

votes

**1**answer

604 views

### Time in Girard's Geometry of Interaction

Jean-Yves Girard writes at the end of his paper
"Towards a Geometry of Interaction", page 105, that we have three intuitions about the nature of time:
time is logic modulo the order of rules,
time ...

**7**

votes

**1**answer

166 views

### Slim Kurepa tree at a singular strong limit cardinal of uncountable cofinality

For a strong limit cardinal $\kappa$ the notion of $\kappa$-Kurepa tree is trivial: the full binary tree is a $\kappa$-Kurepa tree. Accordingly, we consider the following strengthening:
A slim ...

**-4**

votes

**1**answer

229 views

### An algorithm and symbolic manipulation for IF-THEN-ELSE [closed]

CONCLUSION (so far) Look at the parentheses theorem and at the comments below the question(s) :-) As for now, only Dan Peterson has truly addressed the issue.
Q1 Does there exists an ...

**10**

votes

**0**answers

242 views

### Is there a “hereditary” construction for $L$?

Recall that $L$, Godel's constructible universe is constructed by defining the following hierarchy:
$L_0=\varnothing$, for a limit ordinal $\delta$, $L_\delta=\bigcup_{\alpha<\delta}L_\alpha$, and ...

**4**

votes

**0**answers

213 views

### Examples of unproven but likely true existential sentence (in the sense of incompleteness)

Some examples of universal statements that are unproven but likely true include the Riemann hypothesis (all non-trivial zeros of the zeta function have real part 1/2) and the Goldbach conjecture (all ...

**0**

votes

**1**answer

139 views

### Definability of arithmetic functions and relations

Motivation: Many "weak" arithmetic functions and/or relations ("relations" for short) are equivalent with relations explicitly definable by relations which were recursively defined by them beforehand ...

**12**

votes

**0**answers

368 views

### Ideas behind Gitik's solution of PCF conjecture

Recently Moti Gitik has refuted Shelah's PCF conjecture (see Short extenders forcings II ) by proving the following theorem:
Theorem. Assuming the consistency of infinitely many strong cardinals, one ...

**6**

votes

**6**answers

1k views

### How short can we state the Axiom of Choice?

How short can we state a principle which is equivalent with the Axiom of Choice under $ZF$? The principle should be a sentence in the language of set theory with only $\in$ and$=$ as extralogical ...

**8**

votes

**1**answer

288 views

### Saturated Ultrapowers

I posted it on MSE and didn't receive any comments or answers, so I thought I would post it here.
(See http://math.stackexchange.com/questions/895549/keisler-order-saturated-ultrapowers)
Keisler's ...

**8**

votes

**2**answers

340 views

### Ultracoproducts and Cartesian products

Let $X$ be a metrizable compact topological space, let $\mathcal U$ be an ultrafilter, and denote by $X^{\mathcal U}$ the ultracopower of $X$ with respect to $\mathcal U$.
As a C$^*$-algebraist, I ...

**4**

votes

**2**answers

251 views

### What is the status of the extreme value theorem in forms of constructive mathematics, such as Smooth Infinitesimal Analysis?

In certain intuitionistic frameworks the extreme value theorem cannot be proved. Depending on the exact framework, counterexamples can be constructed as well; see for example pp. 294-295 in
...

**5**

votes

**2**answers

735 views

### Salvaging Leibnizian formalism?

Can one justify Leibniz's formalism in a suitable algebraic or topological context?
We have published some papers recently where we argue that Leibniz's formalism for the calculus wasn't ...

**7**

votes

**1**answer

250 views

### Inaccessible becomes successor of singular

Is it possible, starting from any large cardinal assumption, to find a countably closed forcing $\mathbb{P}$ such that for some inaccessible $\kappa$, $\Vdash_\mathbb{P} "\kappa = \lambda^+$ and ...

**1**

vote

**1**answer

332 views

### Is there any current development of a first order formalization of metamathematics?

I hope that this post isn't off topic, but I already asked math.stackexchange about first order formalizations of first order logic. There are provability logics and extensions in modal logic's that ...

**4**

votes

**2**answers

209 views

### Question about higher inductive types and computational rules

I have been trying to make my way through the homotopy type theory book, slowly but surely, and I just finished reading this introductory series of 3 articles on hott on ScienceForAll.
...

**-4**

votes

**1**answer

211 views

### Universal quantifier in Russell's Theory of descriptions - Who is the UNIVERSE? [closed]

To moderators: Please don't delete or migrate this thread: It's by no means PURE PHILOSOPHICAL, but mostly a mathematical logic question!
In Russell's 1905 paper "on denoting" in which he introduces ...

**1**

vote

**0**answers

69 views

### LTL - Büchi-automaton Translation [closed]

I need some help in Generalized Büchi automaton ..
I do understand the translation of a LTL-formula ϕ into Generalized Büchi automaton A= (Q, Δ, I, F), with F= {F1,...,Fn}
My problem is F ..
I know ...

**6**

votes

**1**answer

177 views

### continuum many mutually generic filters

Given a countable model $M$ of set theory and an atomless, separative partial order $\mathbb{P} \in M$, can we construct (in the real universe) $2^\omega$ many pairwise mutually $\mathbb{P}$-generic ...

**5**

votes

**0**answers

110 views

### A question about ordinal definable sets of real numbers revisited [duplicate]

Citing (almost)
A question about ordinal definable real numbers
If ZFC (Zermelo-Fraenkel set theory with the Axiom of Choice) is consistent, does it remain consistent when the following statement is ...

**8**

votes

**0**answers

118 views

### Can $Ded(\kappa)$ be a supremum?

Definition If there is a dense linear order w/o endpoints of size $\lambda$ with a dense subset of size $\kappa$ then write $D(\kappa,\lambda)$. $Ded(\kappa)=\sup_\lambda \{D(\kappa,\lambda)\}$.
It ...

**4**

votes

**1**answer

201 views

### The word problem of the free left distributive algebra on one generator

A left distributive algebra is a set $A$ together with a binary operation, $\cdot$, satisfying $a\cdot(b\cdot c)=(a\cdot b)\cdot(a\cdot c)$.
One important example of left distributive algebras arises ...

**8**

votes

**1**answer

366 views

### The (un)decidability of Robinson-Arithmetic-without-Multiplication?

I asked this over at math.stackexchange, and though a number of people were interested enough to vote up the question, I didn't get an answer -- which makes me wonder whether it isn't quite so ...

**7**

votes

**1**answer

271 views

### The definition of < in Robinson's Q

I recently had to explain how the basic axioms in Simpson's Subsystems of Second Order Arithmetic were interpretable in Robinson's Q. Most of the axioms are actually the same, except that Simpson ...

**2**

votes

**1**answer

161 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 ...

**5**

votes

**0**answers

142 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

330 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 ...