**6**

votes

**3**answers

321 views

### Does the Feferman-Schutte analysis give a precise characterization of Predicative Second-Order Arithmetic?

A definition is called impredicative if it involves quantification over a domain that contains the thing being defined. For instance, if you define hereditary property to be a property which applies ...

**4**

votes

**3**answers

385 views

### Cohen algebra (generalization)

Let Bor($X$) = class of all borel subsets of $X$. Cohen algebra is defined as Bor(X) modulo the ideal of meager sets.
The Cohen algebra has a combinatorial : it is the unique atomless complete ...

**6**

votes

**1**answer

404 views

### A question on rank-to-rank embeddings

Consider a non-trivial elementary embedding $j:V_\lambda\to V_\lambda$ and, for each $A\subset V_\lambda$, set $j(A)=\bigcup_{\delta<\lambda}j(A\cap V_\delta)$.
In Implications between strong ...

**8**

votes

**1**answer

318 views

### Forcing to “minimally” add new reals.

Suppose that I want to force to add a "single" new subset of $\omega$ and not much else. For example, consider the Cohen forcing consisting of finite partial functions from $\omega$ to 2. The forcing ...

**0**

votes

**0**answers

127 views

### A relation that commutes with logical equivalence? [on hold]

I would like help with proving or disproving a conjecture concerning logical equivalence.
Say that a set $S$ of occurrences of formulas within a formula $\varphi$ is simplifiable (with respect to ...

**9**

votes

**1**answer

264 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 ...

**10**

votes

**2**answers

578 views

### Are all functions in Bishop's constructive mathematics continuous?

When I started to write this question I was really confused, but now I think I am starting to get it. Nonetheless I'd like some confirmation that my understanding is correct.
I have read, heard ...

**8**

votes

**1**answer

357 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 $\Bbb P$ ...

**6**

votes

**2**answers

204 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
...

**9**

votes

**2**answers

225 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 ...

**1**

vote

**1**answer

77 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 $\Delta_0$?

**16**

votes

**7**answers

1k views

### Extensional theorems mostly used intensionally

Some theorems are stated and proved extensionally, but in practice are almost always used intensionally. Let me give an example to make this clear -- integration by parts:
$$ \int_a^b f(x)g'(x)ds = ...

**3**

votes

**0**answers

294 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 ...

**44**

votes

**8**answers

5k views

### Succinctly naming big numbers: ZFC versus Busy-Beaver

Years ago, I wrote an essay called Who Can Name the Bigger Number?, which posed the following challenge:
You have fifteen seconds. Using standard math notation, English words, or both, name a single ...

**11**

votes

**5**answers

756 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 ...

**9**

votes

**0**answers

209 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

**1**answer

78 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

**2**answers

249 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 ...

**5**

votes

**2**answers

410 views

### generalizing the ultrapower

Given an 2-valued measure $\mu$ on a set $I$, and structures $M_i \ (i \in I)$, one can construct the ultrapower $\prod_{i\in I}M_i / U$ (where $U$ is the ultrafilter associated with $\mu$.) One can ...

**0**

votes

**0**answers

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 ...

**5**

votes

**1**answer

199 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 ...

**21**

votes

**2**answers

677 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 ...

**4**

votes

**1**answer

389 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 ...

**1**

vote

**3**answers

566 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?

**-2**

votes

**1**answer

126 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:
$\forall n (con(I\Sigma_{n}))$?

**14**

votes

**3**answers

2k views

### What is the history of the Y-combinator?

Inspired by the comments to this question, I wonder if someone can explain the history of the fixed point combinator (often called the Y combinator) in lambda calculus.
Where did it first appear? ...

**10**

votes

**1**answer

204 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 ...

**7**

votes

**1**answer

541 views

### Application of the Riemann hypothesis and the ABC conjecture to independence results

In Old Home Page of Andreas Weiermann Andreas Weiermann has stated the following:
Quite recently I submitted a preprint about an application of the Riemann hypothesis and the ABC conjecture to ...

**6**

votes

**0**answers

214 views

### A new cardinality living in every forcing extension?

This question is motivated by the papers http://arxiv.org/abs/1405.7456 and http://arxiv.org/abs/1410.1224.
Say that a set $X$ is "generically presentable" over $V\models ZF$ if there is some ...

**5**

votes

**0**answers

228 views

### Cardinal characteristics without choice

(I'm taking my definition of a cardinal characteristic from Blass' excellent article http://www.math.lsa.umich.edu/~ablass/need.pdf, which cites Vojtas/Fremlin/Miller; theirs is more general, but I'm ...

**13**

votes

**0**answers

390 views

### Has anyone read/debunked Yessenin-Volpin–Hennix “Beware of the Gödel-Wette paradox”?

A student recently asked me about the status of a 2001 arXiv post, Beware of the Gödel-Wette paradox!, by Alexander Yessenin-Volpin (aka Esenin-Volpin and several other transliterations) and Catherine ...

**2**

votes

**1**answer

222 views

### Axiomatic ZFC Set Theory [closed]

Can the Axiom Schema of Comprehension be omitted from ZFC since it is implied by the Axiom Schema of Replacement?

**4**

votes

**1**answer

306 views

### On directedness, transitivity and ancestral directedness

Let $\textit{C}$ be the modal logical schema $(\square (\square \alpha \rightarrow \alpha) \wedge \square (\square \lnot\alpha \rightarrow \lnot \alpha))\rightarrow (\square \alpha \vee \square \lnot ...

**3**

votes

**2**answers

1k views

### Why are universal introduction and existential elimination valid inference rules?

I'm studying the inference rules of natural deduction for the first-order logic. I cannot understand why universal introduction and existential elimination are valid rules. The first one says that if ...

**-1**

votes

**1**answer

89 views

### types and elementary extensions [closed]

Let $\mathcal{M}$ and $\mathcal{N}$ be two $\mathcal{L}$-structures and suppose that for n-tupls $\bar{a}\in M^n$ and $\bar{b}\in N^n$,
$tp^\mathcal{M}(\bar{a})=tp^\mathcal{N}(\bar{b})$ where ...

**2**

votes

**3**answers

298 views

### The Set-Theoretic Multiverse and Joint Embeddings

I am curious whether or not the following axiom is independent of Hamkins's axioms for the Set-Theoretic Multiverse. Hamkins's axioms can be found here on pages 1-2 and here on pages 24-26.
Consider ...

**12**

votes

**4**answers

893 views

### Sierpinski's construction of a non-measurable set

In the early 20th century there was a lot of fuss over the axiom of choice implying that there are Lebesgue non-measurable sets of reals. In his book about The Axiom of Choice, Gregory Moore points to ...

**12**

votes

**12**answers

3k views

### The best text to study both incompleteness theorems

Hi!
What text on both incompleteness theorems you would recommend for beginner?
Specifically, I'm looking for the text with the following properties:
1) The proofs should be finitistic, in Godel's ...

**6**

votes

**0**answers

187 views

### Is there a notion analogous to separability but requiring definable rather than countable sets?

Among models of $\lambda$-calculus, some like the Bohm tree model have the property that every element is a directed sup of definable elements, whereas others like the $D_\infty$ and $P(\omega)$ ...

**10**

votes

**2**answers

653 views

### Where is the Erdős–Rado theorem stated in Erdős and Rado's Bull AMS paper?

This may be inappropriate for MO, but here goes: if I have understood the statement of the Erdős–Rado theorem correctly, then it contains as a special case the following result:
if $\mu$ is ...

**4**

votes

**2**answers

229 views

### Is it consistent that $\frak{d} < 2^{\aleph_0}$?

Let $\omega^\omega$ denote the set of all functions $f:\omega\to\omega$. We write $f <^* g$ if there is $N\in\omega$ such that $f(n) < g(n)$ for all $n>N$. A set $D\subseteq \omega^\omega$ is ...

**73**

votes

**9**answers

12k views

### Solutions to the Continuum Hypothesis

Related MO questions: What is the general opinion on the Generalized Continuum Hypothesis? ; Completion of ZFC ; Complete resolutions of GCH
Background
The Continuum Hypothesis (CH) posed by Cantor ...

**5**

votes

**2**answers

233 views

### When does Skolemization require the axiom of choice?

Skolemization is often used for eliminating existential quantifiers, which is often useful for proving theorems, especially in automated resolution theorem proving. Skolemization in first order ...

**8**

votes

**3**answers

619 views

### Does a left basis imply a right basis, without AC?

If $_DV_D$ is a $D$-$D$-bimodule, and we have a $D$-basis for $V_D$, do we still need AC to get a $D$-basis for $_DV$?
(The original question appears below. But this shorter question gets at the ...

**28**

votes

**1**answer

703 views

### Producing finite objects by forcing!

It is a trivial fact that forcing can not produce finite sets of ground model objects. However there are situations,
where we can use forcing to prove the existence of finite objects with some ...

**3**

votes

**3**answers

380 views

### What is the proper name for “compact closed” multiplicative intuitionistic linear logic?

Multiplicative intuitionistic linear logic (MILL) has only multiplicative conjunction $\otimes$ and linear implication $\multimap$ as connectives. It has models in symmetric monoidal closed ...

**5**

votes

**2**answers

158 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 $\otimes$, $1$, $\multimap$, $\times$, ...

**10**

votes

**6**answers

1k views

### Intuitionistic logic as quantization of classical logic?

A classically trained mathematician is more likely to be familiar (at least anecdotally) with an area of mathematical physics such as deformation quantization than with intuitionistic logic. It is ...

**-1**

votes

**1**answer

296 views

### A question on intuitionistic propositional logic [closed]

One week ago, I asked a question on math.stackexchange.com (http://math.stackexchange.com/questions/209120/a-question-on-intuitionistc-propositional-logic). But nobody answered my question. So I ...

**11**

votes

**1**answer

574 views

### Is there a probability theory developed in intuitionistic logic?

Since Boole it is known that probability theory is closely related to logic.
According to the axioms of Kolmogorov, probability theory is formulated with a (normalized)
probability measure ...