**4**

votes

**0**answers

184 views

### A question on the size of an admissible ordinal

Let $\mathbf{L}_{\varsigma}$ be the level of ordinal $\varsigma$ of Gödel's constructible universe $\mathbf{L}$. Let $\Sigma_{3}$-KP be Kripke-Platek set theory with infinity and ...

**-5**

votes

**1**answer

171 views

### An axiomatic system with a set of constants that form a complete ordered field [closed]

I am developing a ZFC axiomatic system where together with the empty set, there is a singular (and huge) set of constants that are themselves sets and form a complete ordered field (cof) these ...

**7**

votes

**2**answers

258 views

### Peano arithmetic vs. fast-growing hierarchy with pathological fundamental sequences

Fundamental sequence for a countable limit ordinal $\alpha$ is an increasing sequence $\{\alpha[i]\}$ of ordinals of length $\omega$ such that $\lim_{i\rightarrow\omega}\alpha[i]=\alpha$. There are ...

**12**

votes

**1**answer

445 views

### Is there a nonstandard model of arithmetic having precisely one inductive truth predicate?

$\newcommand\Tr{\text{Tr}}$My question is whether there can be a nonstandard model of PA having a unique inductive truth predicate.
Background. If $\mathcal{N}=\langle N,+,\cdot,0,1,<\rangle$ is ...

**0**

votes

**1**answer

227 views

### How to formalize “Is there a proof for every instance of the halting problem?”? [closed]

In a previous question that I asked here it turned out that for every instance of the halting problem, being the matter whether a certain computer program halts or runs forever, there exists a ...

**0**

votes

**1**answer

199 views

### Is there a consistent theory for each instance of the halting problem?

I got a bit confused by a discussion about the provability of the Goldbach conjecture and the seemingly different opinions about this subject. Since I understand computer science better, I will ask my ...

**5**

votes

**3**answers

351 views

### Extensionality in HoTT versus extensionality in internal language of a category

What's the extension of judgmental identity in HoTT (homotopy type theory)?
The Martin-Löf intensional dependent type theory with identity types is called (definitionally) extensional if the ...

**10**

votes

**3**answers

732 views

### Natural examples of Reverse Mathematics outside classical analysis?

Harvey Friedman at the 1974 ICM motivated Reverse Mathematics by the
following statement:
When the theorem is proved from the right axioms, the axioms can be proved
from the theorem.
Reverse ...

**1**

vote

**1**answer

186 views

### Self-similarity for simple algebraic structures [closed]

I'm doing this thread because I have some ideas about how to define self-similarity in algebra, but I don't know if this is known at all. Any critics, comments and references are more than welcomed. ...

**11**

votes

**3**answers

1k views

### Difference between ZFC and ZF+GCH

I hear that the axiom of choice (AC) derives from
The generalized continuum hypothesis(GCH).
And also hear that both AC and GCH are independent of
Zermelo–Fraenkel set theory(ZF).
So, I'm just ...

**3**

votes

**2**answers

285 views

### A Question related to the Formula Hierarchy

Let large Latin symbols as $X$ and $Y$ denote sets of natural numbers and small symbols as $n$ and $n´$ denote natural numbers and small Greek letters stand for formulas.
Suppose $\alpha$ is ...

**1**

vote

**1**answer

204 views

### Concept of synchronizability

This thread is about the concept of synchronizability. It's a concept I tried to formalize in its most general sense but without success. The goal of this thread is therefore to try to formalize it in ...

**9**

votes

**2**answers

815 views

### Decidability of decidability

The questions I'm going to ask are non formal because they concern decidability of decidability, and I couldn't find any references on that after some quick searches. I hope that this thread is still ...

**2**

votes

**2**answers

250 views

### System of boolean equations, Satisfiability

Are there any methods to "solve" large systems of boolean equations?
$$x_{i1}\vee x_{i2}\vee x_{i3} = b_i, \quad\text{for}\quad i=1,\dots,N,$$
where $x_i, b_i \in\{0, 1\}$
For example
$$x_{1}\vee ...

**7**

votes

**1**answer

595 views

### Does Nelson try to prove PA inconsistent directly?

Edward Nelson is known for his serious attempts to show that Peano axioms, and sometimes even weaker theories, are inconsistent. I wasn't able to find Nelson's papers anywhere, so I wanted to ask a ...

**10**

votes

**1**answer

383 views

### Is there an $L$ like inner model for $\sf Z$?

Godel proved the consistency of the axiom of choice with the axioms of $\sf ZF$ by showing that given any model of $\sf ZF$, there is a definable class which satisfies $\sf ZFC$.
The proof uses a lot ...

**9**

votes

**0**answers

253 views

### Reinhardt cardinals and iterability

Work in $ZF$. Let $j:V\to V$ be a non-trivial elementary embedding which is iterable, so that we can iterate it and form models $M_\alpha, \alpha\in ON,$ with $M_0=V,$ and elementary embeddings ...

**7**

votes

**0**answers

164 views

### Countable choice in $L(\mathbb{R}^*_G)$

Let $\lambda$ be a singular strong limit cardinal and let $G \subset \text{Col}(\omega,\mathord{<}\lambda)$ be a $V$-generic filter. Let $\mathbb{R}^*_G = \bigcup_{\alpha < \lambda} ...

**6**

votes

**1**answer

182 views

### O-minimal Theories with Non-Dense Order Type

I asked this question on MSE, but I haven't received any comments or responses (also, it has a very low view count), so I thought I would also ask it here.
In this paper, Knight, Pillay, and ...

**7**

votes

**1**answer

268 views

### On $V$-decisive and weakly homogeneous forcings

Suppose that $\Bbb P$ is a forcing in $V$, we say that $\Bbb P$ is $V$-decisive if whenever $\varphi(x_1,\ldots,x_n)$ is a statement in the language of forcing, and $u_1,\ldots,u_n\in V$ then $1_{\Bbb ...

**6**

votes

**0**answers

217 views

### Canonical functions in set theory and their applications

Given regular cardinal $\kappa>\omega,$ we can define the canonical functions $f_\alpha: \kappa\to \kappa,$ for $\alpha<\kappa^+.$
Some of their properties are presented in Chapter 22 of the ...

**10**

votes

**3**answers

462 views

### The continuum hypothesis for packing shapes without overlapping

Consider the finite cross $C$ (=union of line segments $\overline{(0, -1)(0, 1)}$ and $\overline{(-1, 0)(1, 0)}$) and the unit half-circle $H$. It is easy to see that we may pack continuum-many ...

**14**

votes

**1**answer

404 views

### Is the regularity of finitely generated rings decidable?

Q: Is there an algorithm to decide whether a given finitely generated (over $\mathbb{Z}$) commutative ring is regular?
I mean by regular that the localization at every prime ideal is a regular local ...

**10**

votes

**2**answers

427 views

### Is every order type of a PA model the \omega of some ZFC model?

Let $N$ be a model of first-order Peano arithmetic, and let $\sigma$ be its order-type. Does it follow that there is a (non-transitive, expect when $M$ is the standard model) $ZFC$-model $M$ such that ...

**2**

votes

**2**answers

102 views

### proof that “small” sets in an extension by iterated forcing already appear in an earlier stage

In Kunen's book (introduction to independence proofs, ) the following lemma is proved (chapter 8, lemma 5.14):
Assume that in M, $\alpha$ is a limit ordinal,
$( ( \mathbb{P}_\xi : \xi \leq \alpha) , ...

**13**

votes

**4**answers

1k views

### Is it possible to formulate the axiom of choice as the existence of a survival strategy?

Consider the following situation:
There is an infinite set $G$ of giraffes.
A lion comes and announces a set $C$ of all possible colours and an infinite cardinal $\kappa$.
The hungry lion ...

**3**

votes

**1**answer

200 views

### How to change the successor of a singular with a Woodin?

I'm looking for references on how to change the successor of a singular cardinal from "more or less" minimal assumptions. If possible, then without adding bounded subsets to the singular either.
In ...

**5**

votes

**1**answer

179 views

### Consistency of Weak Diamond with a Weak Version of Martin's Axiom

If $S \subset \omega_1$ is stationary, then the weak diamond principle $\Phi(S)$ states that for any $F: 2^{<\omega_1} \to 2$, there is a $g: \omega_1 \to 2$ such that for all $f: \omega_1 \to 2$, ...

**4**

votes

**1**answer

349 views

### Forcing is intuitionistic

The main idea of why it´s necessary a generic filter $G$ to extend a countable transitive $\epsilon$-interpretation (not necessarily a model) $M$ is given by the condition (for which $G$ being a ...

**-1**

votes

**3**answers

182 views

### About “absolute proof” of Arithmetic consistency [closed]

Ok so as Godel's theorems states, you cant prove that Peano arithmetic is consistent by using only the axioms within Peano model. You need to use axioms or rules of inferations beyond the model you ...

**9**

votes

**4**answers

498 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

329 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

101 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

390 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

326 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

221 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

79 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

256 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

278 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 ...

**6**

votes

**1**answer

339 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 ...

**6**

votes

**1**answer

399 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

352 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

231 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

287 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

339 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

594 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 ...

**14**

votes

**2**answers

622 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 ...

**12**

votes

**2**answers

581 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

273 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

150 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 ...