**0**

votes

**0**answers

29 views

### Logic resolution and logic consequence [on hold]

Which of this are false?
a) If some formula H results from premises D, then H could be derived from D with using resolution (reapetedly) rule.
b) If some formula H results from premises D, then we ...

**7**

votes

**1**answer

145 views

### preserving saturated ideals

A reliable source made the following claim:
Suppose CH there is an $\omega_2$-saturated ideal on $\omega_1$. Then this is preserved by $\mathrm{Add}(\omega_1,\omega_2)$.
Question 1: How do you ...

**28**

votes

**3**answers

2k views

### What is the reverse mathematical strength of the fundamental theorem of algebra?

Reverse mathematics (RM) is that area that tries to pin down exactly which axioms are necessary to prove theorems, given some weak base theory. Harvey Friedman has pointed out several times (on the ...

**3**

votes

**0**answers

58 views

### On an automatic translation of typed lambda calculus in untyped lambda calculus

I have a question regarding the "compilation" of typed lambda calculus in untyped lambda calculus.
Take for example the inductive definition of lists, with introduction rules:
and:
We can ...

**1**

vote

**0**answers

60 views

### Type theory: can multiple elimination rules be defined, in principle?

I'd like to ask a question on type theory:
Consider the usual type theoretical definition of the natural numbers. We could give an elimination rule in the form:
or in the form:
I called the ...

**2**

votes

**2**answers

146 views

### Pseudo-decision procedures for first order arithmetic

I was reading this paper http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.117.2911&rep=rep1&type=pdf in which the author describes an algorithm, based on Groebner basis, ...

**5**

votes

**0**answers

71 views

### Lascar strong types in fragments of arithmetic

Are Lascar strong types (definition below) in models of fragments of arithmetic always type definable? (They trivially are, in models of full induction.)
Definition Given a saturated model ${\cal M}$ ...

**4**

votes

**2**answers

759 views

### What lets the Square of Opposition fail in Intuitionistic Logic?

See moderator's note in the comments.
I just came across the following. In intuitionistic logic
and classical logic we have the following consequences:
...

**7**

votes

**1**answer

220 views

### Countable group with uncountable number of subgroups $< 2^{\aleph_0}$

Is it consistent that there is a countable group $G$ such that the cardinality of the set of subgroups of $G$ is uncountable, but strictly less than $2^{\aleph_0}$?

**10**

votes

**0**answers

159 views

### Generic $\mathbf{\Sigma}_3^1$-absoluteness for class forcings

In the paper "Generic Absoluteness" by Bagaria and Friedman (http://www.logic.univie.ac.at/~sdf/papers/bagfried.pdf) it is shown that in ZFC generic $\mathbf{\Sigma_3^1}$-absoluteness is false for ...

**7**

votes

**2**answers

264 views

### Topological tameness beyond the Gandy-Harrington topology

The Gandy-Harrington topology on $\omega^\omega$ is the topology generated by all lightface $\Sigma^1_1$ sets; that is, all sets which are continuous-in-the-usual-sense images of $\omega^\omega$.
...

**-1**

votes

**1**answer

96 views

### Non-standard naturals and goodstein sequences [closed]

By the Kirby–Paris theorem, Goodstein's theorem is independent of Peano arithmetic (PA). Therefore there are non-standard models in which every Goodstein sequence terminates. However, Tennenbaum's ...

**-3**

votes

**0**answers

28 views

### Calculating minimum number of comparisons to ensure 1 and only 1 value set [migrated]

This is a math problem with software limitations, but I felt it more appropriate here than in a software SE.
I am forced to work with a very inflexible tool and must check 5 fields, each of which can ...

**9**

votes

**1**answer

363 views

### Just a little absoluteness might be cheaper?

Absoluteness is a wonderful thing, but expensive consistency-strength wise. My question is, when can we get large amounts of absoluteness in specific situations for much cheaper?
Specifically, fix a ...

**8**

votes

**1**answer

181 views

### Is the Jordan decomposition of a self-adjoint functional constructive?

Let $A$ be an abstract C*-algebra, and let $\varphi\colon A \rightarrow \mathbb C$ be a bounded linear function. Assuming the axiom of choice, there exist unique positive bounded linear functions ...

**9**

votes

**0**answers

175 views

### The Chang model after collapsing an inaccessible limit of Woodins

If $\kappa$ is an inaccessible cardinal and $G \subset \operatorname{Col}(\omega,\mathord{<}\kappa)$ is a $V$-generic filter, then in $V[G]$ the Chang model $L(\text{Ord}^\omega)$ satisfies "every ...

**18**

votes

**3**answers

2k views

### Who needs Replacement anyway?

The set theory ETCS famously comes without the Replacement axiom schema (or an equivalent) that is part of ZFC. One (to me, not apparently useful) set that one cannot build in ETCS is $\coprod_{n\in ...

**4**

votes

**0**answers

173 views

### Primitive Closure Arithmetic

I am researching a system that was designed by myself and what I call PCA (Primitive Closure Arithmetic), because it looks like PRA.
The differences are:
- PRA uses recursive definition with a ...

**0**

votes

**1**answer

151 views

### Definition of “Expected/Unexpected Event”

Background of my question is Martin Gardner's "unexpected hanging" paradoxon, which has once again be the subject of an article in a popular-scientific magazin (this time because this year it has been ...

**11**

votes

**0**answers

184 views

### Adding a saturated ideal

Is it consistent that there is no $\omega_2$-saturated ideal on $\omega_1$, but one is introduced by an $\omega_2$-closed forcing?

**10**

votes

**0**answers

194 views

### c.c.c forcing notions and adding minimal generic reals

Is the following statement consistent:
``There is no non-trivial c.c.c forcing notion adding a minimal generic real''?
The question is related to Prikry's question: Is it consistent that any ...

**7**

votes

**1**answer

242 views

### For a ring $k$ and a set $X$, what are the $k$-algebra homomorphisms $k^X \to k$?

Let $k$ be a commutative ring. Feel free to assume it's a field.
Let $X$ be a set. This question is only interesting when $X$ is infinite.
Write $k^X$ for the $k$-algebra of functions $X \to k$, ...

**11**

votes

**3**answers

417 views

### The size of Lindelof space

Question. Suppose that $X$ is a Lindelof space such
that every point of $X$ is a $G_{\delta}$-point. Then is it true that $|X| ≤ 2^{\omega}$?

**9**

votes

**1**answer

371 views

### Is $\clubsuit_{\omega_1}$ enough to get Suslin tree?

This is problem 15.3 in Arnie Miller's problem list:
(Juhasz) Suppose there exists $\langle A_{\alpha} : \alpha \in L \rangle$, where $L$ is the set of limit ordinals below $\omega_1$ and for each ...

**1**

vote

**0**answers

65 views

### BL Algebras that allow for Compactness to hold

Say we have a model $M$ of a theory $T$ of some core fuzzy logic.
When dealing with compactness, we run in to a situation where the new model being built (by the use of compactness over $M$), will ...

**4**

votes

**1**answer

157 views

### Cohen algebra and $\mathcal P(\omega)/ \mathrm{fin}$

Let $C$ be the Cohen algebra, the boolean completion of the partial order of finite partial functions from $\omega$ to 2, ordered by reverse inclusion. Does there exist an ideal $I$ on $C$ such that ...

**11**

votes

**1**answer

192 views

### Does Peano's existence theorem admits a constructive proof?

$$y(t)=y_0+\int_0^t b(y(s))ds$$ $b\in C(R^d)\cap L^\infty(R^d)$
The classical proof for Peano's existence theorem in ODE need use the Ascoli's theorem, so it's not constructive. When $d=1$, in the ...

**7**

votes

**2**answers

476 views

### What does “simplification of proofs as evaluation of programs” mean?

I am currently going through Philip Walder's "Proposition as Types" and a passage of the introduction has struck me:
for each way to simplify a proof
there is a corresponding way to evaluate a ...

**2**

votes

**0**answers

196 views

### Is there a $\Sigma^0_3$-complete ideal on $\omega$?

In Kechris's book "Classical Descriputive set theory" Chapter 23 (Exercise 23.4), it claim that there is a $\Sigma^0_\xi$-complete ideal on $\omega$ for each $\xi\geq 3$.
There is a candidate ...

**5**

votes

**1**answer

292 views

### Details for Woodin's forcing argument for a saturated ideal from the Levy collapse

Theorem 2.65 in Woodin's book shows that a saturated ideal on $\omega_1$ exists after Levy-collapsing a Woodin cardinal $\delta$ to $\omega_2$. I am confused about the part of the argument where he ...

**8**

votes

**1**answer

260 views

### Can there be a tree of height $\omega_2$ having all levels countable, with no cofinal branch?

For many years I had the idea that if a well-founded tree is both very tall and very narrow, then it must have a cofinal branch. For example, it is a fun exercise to show that any $\omega_1$-tree ...

**10**

votes

**0**answers

286 views

### On sentences true in all finite groups (revisited)

Let $w$ be a group word with variables $\bar x, \bar y$, where
$\bar x=(x_1,\dots ,x_m)$ and
$\bar y=(y_1,\dots ,y_n).$
I am interested in the following questions.
(1) Is the sentence $(\forall\bar ...

**7**

votes

**1**answer

184 views

### Introducing meets while preserving directed closure

A poset $\mathbb{P}$ is called well-met iff every pair of compatible conditions in $\mathbb{P}$ has a greatest lower bound.
Question: Suppose $\mathbb{P}$ is a separative partial order which is ...

**5**

votes

**0**answers

90 views

### O-Minimal sentences in $L_{\omega_1,\omega}$?

Is there any meaningful sense in which we can talk about o-minimal sentences of $L_{\omega_1,\omega}$? I can give a first attempt, easily; given a countable fragment $F$ and a sentence $\Phi$ in that ...

**22**

votes

**1**answer

832 views

### How hard is it to destroy a diamond? (with a real)

If we start with $V\models\lozenge$, it is not hard to force the failure of diamond. You can blow up the continuum, or destroy all the Suslin trees. You can blow up the continuum of $\aleph_1$, and ...

**7**

votes

**3**answers

262 views

### Do non-normal states exist in the Solovay model?

Let H be an infinite dimensional Hilbert space. Then there exist non-normal states on B(H) in ZFC (i.e. states that are not represented by a density operator).
Is this also true in the Solovay model ...

**1**

vote

**2**answers

153 views

### Are monoids with zero and partial homomorphisms related?

Context: Let $\Sigma=\{U,C,A,G\}$ and $L\subset\Sigma^*$, i.e. $L$ is a language over the alphabet $\Sigma$. Let $\Sigma'=\{0,1\}$ and define a homomorphism $f:\Sigma^*\to\Sigma'^*$ by extending $U ...

**38**

votes

**4**answers

1k views

### On sentences true in all finite groups

Let $w$ be a group word with two variables $x$ and $y$.
Is the sentence $(\forall x)(\exists y)w=1$
true in every group if it is true
in every finite group?
The same question about the sentence ...

**4**

votes

**1**answer

163 views

### Induction and nonstandard halting times of standard machines

For a nonstandard model of enough arithmetic - say, $\mathcal{N}\models I\Sigma_1$ - we can define the set of halting times of standard machines relative to $\mathcal{N}$: ...

**4**

votes

**2**answers

128 views

### ${\frak b}$ and ${\frak d}$ defined with $\leq$ instead of $\leq^*$

Let $\omega^\omega$ denote the collection of all functions $f:\omega\to\omega$. For $f,g\in\omega$ we define
$f\leq g$ if $f(n)\leq g(n)$ for all $n\in\omega$;
$f\leq^* g$ if there is $N\in\omega$ ...

**7**

votes

**0**answers

321 views

### Godel's second incompleteness theorem for non-r.e. theories

R. Jeroslow in this paper proves that a non-recursively enumerable theory whose set of theorems is $\Delta_2$-definable may prove consistency of itself
but it can not prove 2-consistency of itself.
...

**5**

votes

**1**answer

319 views

### Elementary equivalence of the direct product and direct sum of groups

It is well-known that the direct product of any family of abelian groups
is an elementary extension of the direct sum of the family
(see e.g. Lemma A.1.6 in the book `Model Theory' by W. Hodges,
...

**5**

votes

**1**answer

110 views

### Minimal degrees of structures

For this question, a structure means a first-order structure in a computable language with domain $\omega$; a copy of a structure $\mathcal{A}$ is a structure $\mathcal{B}\cong\mathcal{A}$.
Given a ...

**6**

votes

**1**answer

276 views

### If two structures are elementarily equivalent, is there a zigzag of elementary embeddings between them?

Fix a first-order signature $\Sigma$. There is an equivalence relation $\sim$ on the class $\Sigma\mathrm{-Str}$ of all $\Sigma$-structures given by $M \sim N$ iff $M$ and $N$ are elementarily ...

**2**

votes

**1**answer

120 views

### Explanation of the definition of Saturated Sets in Lambda Calculus

I have a question on the definition of Saturated Sets, as particular subset of the set of strongly normalizing terms in lambda calculus.
Here is the definition: a set $S$ of strongly normalizing ...

**29**

votes

**0**answers

727 views

### Bidi: A new cardinal characteristic of the continuum?

This question assumes familiarity with combinatorial cardinal
characteristics of the continuum.
Identify an infinite set $a\subseteq\mathbb{N}$ with its increasing
enumeration. Thus, for each natural ...

**1**

vote

**1**answer

288 views

### Is second-order ZFC categorical with regard to its proper class models

Second-order ZFC offers partial categoricity in the sense that, given any two models, one of them must be isomorphic to an initial segment of the other [1]. However, this leaves questions regarding ...

**2**

votes

**1**answer

182 views

### Is there a 'largest' second-order categorical axiomatization of set theory, extended from ZFC2

While it's possible to obtain categorical second-order axiomatizations of set theory by extending ZFC2 with additional axioms (see [1]), these axioms tend to be somewhat arbitrary (e.g. adding an ...

**1**

vote

**1**answer

98 views

### PA proves that functions are total

Is there a total recursive function $f:N \to N$ such that for no $\Sigma_1$ formula $\phi(x,y)$ which defines it (i.e., defines its graph), is it true that PA proves that "$\phi$ defines a total ...

**4**

votes

**0**answers

113 views

### Is the lowenheim-skolem number of nth order logic larger than the corresponding number for 2nd order logic

According to this paper, by Vaananen, the $LS$ number for $2^{nd}$ order logic is given by "the supremum of $Π_{2}$-definable ordinals", where "The Lowenheim-Skolem number $LS(L)$ of $L$ is the ...