**6**

votes

**1**answer

223 views

### Fields of characteristic zero via ultraproducts

Is every noncountable field of characteristic zero the ultraproduct (using a non principal ultrafilter over the set of prime numbers) of fields of positive characteristic?

**1**

vote

**2**answers

120 views

### Quantifying simplicity, in the case of trigonometric and exponential functions

The pair of identities the sine and cosine of a sum of two terms as functions of the sines and cosines of the terms separately is not as simple as the identity that expresses the exponential of a sum ...

**4**

votes

**0**answers

119 views

### The theory of two finite linear orders

My colleague Matthias Baaz is looking for a reference for the following question (or possibly theorem):
Let T be the "theory of pairs of finite linear orders". That is, consider all finite ...

**8**

votes

**3**answers

273 views

### Who proved “sets in every generic are already in the ground model?”

Suppose $\mathbb{P}$ is a notion of forcing in the ground model $V$, and $X$ is a set which is in $V[G]$ for every $\mathbb{P}$-generic filter $G$. Then $X\in V$ already, by a fairly simple (if ...

**6**

votes

**1**answer

216 views

### Robinson Arithmetic and Composite Numbers

Define a number $n$ to be composite if it can be written as $a\cdot b$ for some $a,b$ where $a,b\neq 1$.
Define $p$ to be prime if $p=a\cdot b$ implies $a=1$ or $b=1$.
The theorem that every ...

**4**

votes

**0**answers

78 views

### Stabilization of recursive approximation in $PA^-+I\Sigma_1^0$

Over any model M of $PA^-+I\Sigma_1^0$. Suppose $A\in [T]$ where $T$ is a $\Delta_2^0$-tree and $A$ is one isolated path. Further, $A$ is regular, i.e. $\forall n A\upharpoonright n$ has a code in ...

**1**

vote

**2**answers

313 views

### Bishop's paradox of the countability of sequences

In 'Foundations of Constructive Analysis', in the notes at the end of the first chapter, Bishop poses an apparent paradox as an exercise for the reader:
Since every sequence of rational numbers ...

**2**

votes

**3**answers

327 views

### Show that Z2 is not conservative over PA

It is well-known that $\mathsf{ACA}_0$ is a conservative extension of PA. I assume this theorem gets a lot of attention because $\mathsf{Z}_2$ is not conservative over PA. Thus there ought to be ...

**2**

votes

**0**answers

121 views

### Algebras admitting quantifier elimination

I apologize if this question is meaningless or trivial:
What are examples of Algebras admitting quantifier elimination? Especially are there Groups admitting quantifier elimination?
I need to say ...

**4**

votes

**4**answers

547 views

### “Introduction to mathematical logic” book from a formalist perspective

I'm looking for books that introduce the reader to mathematical logic assuming the perspective of a formalist.
I've found that many books are more or less written for the platonist - like Kunen's ...

**4**

votes

**1**answer

92 views

### Models of intuitionistic linear logic that reflect the resource interpretation

I am interested in models of intuitionistic linear logic, that is, the logic that you get if you take classical linear logic and restrict the set of operators to $\otimes$, $1$, $\multimap$, $\times$, ...

**24**

votes

**1**answer

2k views

### A preprint of Sela concerning the work of Kharlampovich-Miyasnikov

Yesterday, Z. Sela published a preprint in arXiv which claims that the solution of Olga Kharlampovich and Alexi Miyasnikov for the Tarski problem on decidablity of the first order theories of free ...

**21**

votes

**4**answers

841 views

### What is the definition of a large cardinal axiom?

In different books one can find different implicit definitions for a large cardinal axiom.
My question is that which one of these definitions are more popular or standard amongst set theorists?
Any ...

**5**

votes

**3**answers

550 views

### PFA: A New Godel's Program & A New Large Cardinal Ladder (Updated)

We know $PFA$ implies $2^{\aleph_0}=\aleph_2$.
Q1. What does $PFA$ say about other values of continuum function? Does proper forcing axiom carry any further information about values of continuum ...

**7**

votes

**1**answer

227 views

### Are superstrong stronger than strongly compact cardinals? (or vice versa)

In the last part of Kanamori's excellent "The Higher Infinite" there is a small diagram about the strength and consistency strength of some major large cardinal axioms.
Below supercompact cardinals ...

**8**

votes

**1**answer

238 views

### What is the proof-theoretic ordinal of PA + Con(PA), PA + Con(PA + Con(PA)) etc., and why?

I seem to remember having read that the proof-theoretic ordinal (sup of ordinals the theory can prove well-ordered) of $\mathsf{PA} + \mathsf{Con}(\mathsf{PA})$ is the same as that of $\mathsf{PA}$, ...

**4**

votes

**1**answer

171 views

### Why the axiomatic rank of the variety of groups is equal to three?

I am thankful of Anton Klyachko who introduced axiomatic rank to me: the axiomatic rank of a variety is the minimum number of variables which we need to define that variety by identities.
It seems ...

**8**

votes

**1**answer

206 views

### Genericity by names

If $P$ is a notion of forcing in $M$, then $G$ is a $P$-generic filter over $M$ if $G\subseteq P$ is a filter, and for every $D\in M$ which is a dense subset of $P$, $G\cap D\neq\varnothing$.
...

**3**

votes

**1**answer

309 views

### Fundamental Problems in Mathematics that, without Computer Sciences, would not be resolved? [closed]

Could you please give examples of fundamental questions in mathematics (let us say, pure mathematics) which were resolved fundamentally by the use of computers? More precisely, are there examples that ...

**2**

votes

**1**answer

217 views

### Elementary proof of bounds on factor polynomials

The question Getting a bound on the coefficients of the factor polynomial got very nice answers on Gelfond's theorem. But for work on proof theory of arithmetic I want a proof in arithmetic. The ...

**1**

vote

**0**answers

108 views

### Dedekind reals in heyting valued models

Let $V^{H}$ be a Heyting valued model of intuitionistic set theory. What conditions does $H$ have to satisfy in order for the following claim to hold? (where $\| \phi(u) \| \in H$ is the truth value ...

**3**

votes

**0**answers

120 views

### Why is adopting Russell's Axiom of Reducibility as strong as eliminating the Ramified Hierarchy?

In order to respond to concerns of impredicativity, Bertrand Russell developed a system of ramified second-order logic, which is like regular second-order logic except the comprehension schema is ...

**7**

votes

**0**answers

271 views

### “Hard” separation results in reverse mathematics (or similar)

This is a fairly broad question. In particular, I specify 5 questions (Q1, Q2.1, Q2.2, Q3, Q4) which for me all fall under one umbrella. Since this is unreasonably broad, I'm really interested in an ...

**21**

votes

**1**answer

372 views

### When does the choice of the generic matter?

It is a somewhat curious phenomenon that, in forcing arguments, one usually doesn't care about any particular properties of the generic filter being used (this isn't strictly true; there are cases ...

**0**

votes

**3**answers

128 views

### Negated varieties and their relatively free algebras

During the past days, I asked some questions in order to gain a clear understanding of the notion of "free algebras". I suppose that the question below is the most clear image of the concept I have ...

**6**

votes

**1**answer

88 views

### Is 0' of PA degree relative to a non-low set?

Definitions:
A set $X$ is of PA degree relative to a set $Y$ if every infinite $Y$-computable binary tree has an infinite $X$-computable path.
A set $X$ is low if $X'$ is computable from ...

**4**

votes

**3**answers

242 views

### The existence of an algebra whose set of identities and first order theory are equivalent

Is there an algebra $A$ (for example a group) such that $Th(A)$ is logically equivalent to $id(A)$? In other words, is there an algebra $A$ such that
$$
Mod(Th(A))=Var(A)?
$$
Clearly finite algebras ...

**2**

votes

**0**answers

91 views

### What are natural examples of non-relativizable proofs? [duplicate]

As I understand it, a proof that P=NP or P≠NP would need to be non-relativizable (as in recursion theory oracles).
Virtually all proofs seem to be relativizable, though.
What are good examples of ...

**0**

votes

**1**answer

145 views

### relatively free groups in $Var(S_3)$

Suppose $S_3$ is the symmetric group of order 6. Which elements of the variety $Var(S_3)$ are relatively free?
This question is related to my previous question
Relatively free algebras in a variety ...

**25**

votes

**3**answers

936 views

### Latest stand of core model theory?

What is the "strongest" core model to this day? In particular, how far are we from a core model for supercompact cardinals? There are rumors of some notes from a workshop in 2004:
...

**10**

votes

**1**answer

273 views

### Harvey Friedman's strict reverse mathematics vs. Cook-Nguyen's V$^0$

Harvey Friedman posted several manuscripts [1] proposing a program for "strict" reverse mathematics, in the sense that the base theory should be mathematically natural and coding-free.
In them he ...

**3**

votes

**2**answers

107 views

### Isomorphism of a chain of structures

Consider an elementary chain of models of some first-order theory $T$:
$$ (M_\alpha)_{\alpha < \kappa}, M_\alpha \prec M_\beta \; {\rm for} \; \alpha < \beta .$$
Let also $(N_\alpha)$ be ...

**0**

votes

**0**answers

97 views

### What references cover finitary systems of Ramified Analysis with transfinite levels?

The ramified theory of types, invented by Bertrand Russell, is a way of dealing with impredicativity by breaking the comprehension schema of second-order logic into levels. The comprehension schema ...

**11**

votes

**2**answers

456 views

### Can we force the existence of this function?

I want to see if it is possible to force the existence of a function
$F:\aleph_2 \times \aleph_2\rightarrow \aleph_1$ such that:
a) $F(a,b)=F(b,a)$, for all $a,b\in \aleph_2$ and
b) for all ...

**3**

votes

**1**answer

66 views

### Jump of strongly hyperhyperimmune degrees and DNR relative to 0'

A function f is diagonaly non-recursive (DNR) if for every Turing index $e$, $f(e) \neq \Phi_e(e)$.
A set is strongly hyperhyperimmune if there is no r.e. set of disjoint r.e. set intersecting it.
...

**3**

votes

**0**answers

171 views

### Infinite blue eyed islanders puzzle

Can the well known blue eyed islanders puzzle be extended to an infinite number of islanders?
In that puzzle, a set of $k$ islanders, each with either blue eyes or non-blue eyes, each knows the color ...

**-4**

votes

**1**answer

231 views

### Is there a precise definition of “mathematical formula”? [closed]

In the Wikipedia article for Formula (which has no references), it is claimed that:
"The informal use of the term formula in science refers to the general construct of a relationship between given ...

**2**

votes

**4**answers

167 views

### Seemingly ill-founded recursion and the recursion theorem

The following line well-defines a family of subsets $\{S_i\}_{i\in\mathbb N}$ of $\mathbb N$:
$n\in S_i$ iff $n=2i$ or $\exists j<i$ such that $n\in S_j$.
The following line does not:
...

**2**

votes

**1**answer

140 views

### why the difference between terms and propositional variables?

Reading some old logic texts (written around 1930) I noticed that these texts make no difference between propositional variables and terms.
They do make difference between identity and truthvalue
...

**22**

votes

**10**answers

2k views

### Can We Decide Whether Small Computer Programs Halt?

The undecidability of the halting problem states that there is no general procedure for deciding whether an arbitrary sufficiently complex computer program will halt or not.
Are there some large $n$ ...

**2**

votes

**0**answers

90 views

### Stability of analytic Zariski structures

Noetherian Zariski structures are introduced by Hrushovski and Zilber.(1996)
An analytic Zariski structure is a generalization of Noetherian Zariski structures, introduced by Zilber and Peatfield.
...

**10**

votes

**1**answer

218 views

### Is “approximate categoricity” absolute?

Let $T$ be a countable first-order theory, and assume that $T$ has exactly one atomic model up to isomorphism in every uncountable cardinality. (By "atomic" I mean a model which omits the ...

**5**

votes

**1**answer

259 views

### Is this system identical to S4.4?

Consider the normal modal logic system $\mathbf{TAR1}$ given by $\mathbf{T}$ plus the following axiom:
$$\mathrm{AR1}: \lozenge \square p \rightarrow (\square p \lor \square (p \rightarrow \square ...

**7**

votes

**1**answer

236 views

### Reducibility of polynomials maps

Motivated by this question.
Let $f \in \mathbb{Q}[x]$ or$f \in \mathbb{Z}[x]$ .
Consider the sequence $f(x),f(f(x)), \ldots f^n(x)$.
If some $f^k(x)$ is reducible, the rest iterates will be ...

**4**

votes

**0**answers

108 views

### cut-elimination for infinitary logic

Takeuti (1987, 223) deduces a cut-elimination theorem for infinitary logic from the corresponding soundness-and-completeness theorems. However, is there a way to adapt the basic Gentzen-style ...

**2**

votes

**0**answers

94 views

### Peano (Dedekind) categoricity

What is the smallest fragment of second order logic such that $Th(\mathbb{N})$ in that logic is categorical (only one model, namely natural numbers, up to isomorphism). For example, can we do this in ...

**1**

vote

**0**answers

191 views

### What is the role of the (formalized) omega rule in Ramified Analysis?

In the 1960's, Feferman and Schutte did groundbreaking proof-theoretic work to find out the strength of predicative systems of second-order arithmetic. They used the ramified theory of types, a ...

**3**

votes

**2**answers

218 views

### What is the proof-theoretic ordinal of Hyperarithmetical Comprehension?

As I discuss in my answer here, it seems to me that Solomon Feferman shows, on pages 10-11 of his seminal 1964 paper "Systems of Predicative Analysis", that if you consider predicative second-order ...

**6**

votes

**1**answer

222 views

### Recursive ordinals and the minimal standard model of ZF

Does the minimal standard model of ZF contain all recursive ordinals or is it limited (probably by the proof theoretic ordinal of ZF as I suspect but cannot prove)?
Paul J. Cohen's definition of the ...

**0**

votes

**0**answers

72 views

### Uncountable models of set theory [duplicate]

Assume we have a countable transitive model of set theory which satisfies some set of first-order axioms $T$.
Is it possible to get from this a model $\mathcal{M}$ of set theory which satisfies the ...