**2**

votes

**1**answer

63 views

### What can be achieved by liberalizing induction for $RCA_0$?

$RCA_0$ has $\Delta_0$-comprehension and $\Sigma_1$ induction. Let $X\Sigma_{n}$ be $RCA_0$ plus $\Sigma_n$-induction and let $X\Sigma_{\omega}$-induction be $RCA_0$ plus the full induction schema.
...

**5**

votes

**1**answer

56 views

### Does $WKL_0$ provide more comprehension than $RCA_0$?

$WKL_0$ extends $RCA_0$ with the statement that any infinite subset of the infinite binary tree has an infinite branch. Does $WKL_0$ Prove that there are sets which are not proven to exist by the ...

**2**

votes

**0**answers

60 views

### Which self-reference restrictions can be weakened in probabilstic logic?

This work suggests that there is some generalization of Truth in terms of probability, which can be definable within the logic itself.
Is where any other thorems on self-reference restrictions, which ...

**7**

votes

**0**answers

119 views

### From interpolation to separation

Lusin's separation theorem states that, if $A$ and $B$ are disjoint analytic subsets of a Polish space, then there is a Borel set $X$ separating them ($A\subseteq X$, $B\cap X=\emptyset$). Craig's ...

**-4**

votes

**0**answers

36 views

### Proof for sequent (Logic) [on hold]

I'm currently trying to prove the following:
http://i.imgur.com/dACOdOS.png
And this is what I've come up with so far:
And now I'm stuck. Perhaps the next step might be to eliminate for all x?
...

**8**

votes

**0**answers

66 views

### Near model completeness

A theory $T$ is called near model complete if every formula is equivalent to a Boolean combination of existential formulas mod $T$. I wonder whether there is an equivalent "semantic" definition of ...

**7**

votes

**0**answers

33 views

### Does non-stablity imply that there is a difference between non-forking and coheir extension

Fix some theory $T$.
Let $p$ be a type over some Model M and let $q$ be some global extension of $p$.
Note:
The number of global coheirs of $p$ is bounded by the number of ultrafilters on $M$.
Also ...

**8**

votes

**0**answers

130 views

### Homogeneity of a variant of Prikry forcing

Prikry forcing is easily seen to be cone homogeneous (for any $p, q \in \mathbb{P}$, there are $p' \leq p, q' \leq q$ and an isomorphism $\Phi: \mathbb{P}/p' \simeq \mathbb{P}/q'$); in particular for ...

**18**

votes

**5**answers

1k views

### Why should we believe in the axiom of regularity?

Today I started reading Maddy's Believing the axioms. As I knew beforehand, it includes some discussion of ZFC axioms. However, I really hoped for a more extensive discussion of axiom of ...

**8**

votes

**1**answer

341 views

### Cardinality of definable sets of reals

Throughout this question we assume ZFC.
If CH holds, then the following is obvious:
(S) Every definable infinite subset of $\mathbb R$ has size either $\aleph_0$ or $2^{\aleph_0}$.
(It's true ...

**7**

votes

**0**answers

190 views

### What is the name for a Banach space property closed under ultraproducts?

In Banach space theory, a super-property is a property of a Banach space that is preserved under ultrapowers. (Update (2015-09-28): The property must also be closed under isometric embeddings.) ...

**20**

votes

**2**answers

622 views

### Antirandom reals

This is a crossposting of http://math.stackexchange.com/questions/1446602/anti-random-reals, which has not gotten any answers; after thinking about the problem, I've become more convinced that it ...

**14**

votes

**1**answer

351 views

### Two strengthenings of “strong measure zero”

A set $X\subseteq\mathbb{R}$ is strong measure zero if, for every sequence $(\epsilon_i)_{i\in\mathbb{N}}$ of positive reals, there is a sequence $(I_i)_{i\in\mathbb{N}}$ of open intervals covering ...

**13**

votes

**2**answers

503 views

### Who needs RCS iterations?

According to this paper of Chaz Schlindwein, any countable support iteration of semi-proper forcings is semi-proper. This seems like a breakthrough simplification, and I wonder why it is not more ...

**10**

votes

**0**answers

224 views

### When does $HOD^{V[G]} \subseteq V$?

Assume that $\mathbb{P}\in HOD$ is non-trivial. It is well-known that if $\mathbb{P}$
satisfies some homogeneity properties, then $HOD^{V[G]} \subseteq V$, where $G$ is $\mathbb{P}$-generic over $V$.
...

**3**

votes

**1**answer

78 views

### Is DNC/DNR stronger than “prompt” non-computability?

We propose a (probably not new) definition. Let $\varphi_e$ be an effective enumeration of the partial computable functions.
A total function $f$ is promptly non-computable (PNC) [or promptly ...

**6**

votes

**1**answer

285 views

### Is the axiom schema of replacement used in algebraic number theory (or more generally outside logic)

Here's a precise question. Does Wiles' proof of FLT run just fine in the set theory that logicians would perhaps call "Zermelo + choice" -- i.e. drop the axiom schema of replacement but assume the ...

**1**

vote

**1**answer

80 views

### What are the adequacy conditions for Rosser Provability?

Famously, Rosser introduced a provability predicate $\pi[A]$ that holds iff $\exists x(xP[A]\wedge\forall y(y\le x\to\lnot yP[\lnot A]))$.
Supposing $PA$ is consistent, what are the adequacy ...

**5**

votes

**1**answer

134 views

### Different ways of making $HOD$ far from $V$

There are different criteria for building a model $V$ of $ZFC$ which is far from its
$HOD$, for example:
$(A)$ Cardinality criteria: For this in a joint work with James Cummings and Sy Friedman, we ...

**7**

votes

**2**answers

328 views

### Some “axiom of choice” and “dependent choice” issues

I am probably about to ask some fairly basic questions, and yet I have found it quite hard to find the answers to these.
If I understand correctly, mathematicians tend to be quite happy working with ...

**21**

votes

**0**answers

510 views

### Relative null-ness

Here, "measure" always means Lebesgue measure on $\mathbb{R}$. This question is partly motivated by my answer ...

**2**

votes

**1**answer

97 views

### Analogy of $\omega$-models in constructive mathematics

I apologize that this question is a bit vague, however that is partially the point.
In subsystems of second order arithmetic, one considers $\omega$-models, these are models of $\mathsf{RCA}_0$ whose ...

**1**

vote

**1**answer

101 views

### type theory that does not treat the terms of $\mathrm{Prop}$ as types

In type theory there is a type $\mathrm{Prop}$ that contains every proposition, so $p\colon\mathrm{Prop}$ (in words, "$p$ is of type $\mathrm{Prop}$") where $p$ is a proposition. In all type theories ...

**1**

vote

**0**answers

58 views

### Common solutions for a set of inequalities [migrated]

First, please tolerate my stupidity. I'm a Software Engineering guy with minimal background in maths.
Suppose I have a set of first-order formulas over the theory of linear arithmetic: $\varphi_1, ...

**0**

votes

**2**answers

95 views

### classical typed higher order logic natural deduction

Has somebody worked out a typed higher order logic? I mean something like type theory but not with this intuitionistic touch.
Is there a natural deduction system for this logic?

**6**

votes

**0**answers

84 views

### Two cardinal obstructions

Given a theory $T$ and a formula $\phi(x)$ we say that they admit a $(\kappa, \lambda)$ model if there is a model $M$ such that $|M| = \kappa$ and $|\phi(M)| = \lambda$.
In all examples that I know ...

**5**

votes

**0**answers

226 views

### The least admissible above a dominating real

Let $\mathbb{P}$ be the usual forcing which adds a dominating real: conditions in $\mathbb{P}$ are pairs $(p, f)$ with $p:\omega\rightarrow\omega$ finite partial and $f:\omega\rightarrow\omega$ total, ...

**10**

votes

**1**answer

223 views

### Partition relation at successor cardinal

Is it consistent that there are regular cardinals $\kappa < \lambda$, such that $\lambda$ is a successor cardinal and for every coloring $d\colon[\lambda]^2\to\kappa$ there is some ...

**2**

votes

**1**answer

120 views

### Does totally proper forcing imply countable distributivity?

For a suitable model $M$ for $Q$ and a condition $q \in Q$ we say that $q$ is $(M,Q)$-generic if whenever $r \leqslant q$, $D \in M$ dense, $D \subset Q$, $r$ is compatible with an element of $D \cap ...

**5**

votes

**1**answer

180 views

### Fuzzy logic of Godel

In Gödel logic, is conjunction definable from implication, negation , and disjunction?
We know that conjunction in that logic is not definable from negation and implication.

**5**

votes

**0**answers

123 views

### How “small” can an ordinal be made by forcing?

I know that forcing essentially does not change the ordinals, but by small I mean in comparison with other ordinals whose definition might not be stable under forcing, like the smallest uncountable ...

**8**

votes

**0**answers

203 views

### On an unpublished result of Magidor

In 1970th, Magidor proved the following important results:
(1) Assuming the existence of a supercompact cardinal, it is consistent that $\aleph_\omega$
is strong limit and ...

**2**

votes

**1**answer

170 views

### Notation: $Sigma$ and $Pi$ of intersections

In Jech - Set Theory, the proof of Theorem 31.7, I came along some notations I wish to understand correctly.
For a countable elementary substructure $M \prec H_\lambda$ and $A \in M$ and a generic ...

**2**

votes

**2**answers

130 views

### Precise interpretability strength of $\mathcal P_{DF}(\omega)$ and $L_{\omega_1^{CK}}$

I am curious about the relationship between the definable power set of $\omega$ and the $\omega_1^{CK}$th level of the constructible sets $L$.
In short, $\omega_1^{CK}$ is the least nonrecursive ...

**3**

votes

**1**answer

239 views

### On generic forcing conditions

Let $P$ be a forcing poset, and $Q \in V^P$ a forcing poset in $V^P$. Let $M \prec H(\lambda)$ ($\lambda$ sufficiently large) countable with $P,Q \in M$.
What I want to know is if then the following ...

**4**

votes

**1**answer

297 views

### What are key $\Sigma^0_2$ or $\Pi^0_3$ theorems?

I am researching a logical system that is limited to $\Pi^0_2$ sentences and I am busy to prove that FOL + PA is a conservative extension of that system. Meaning that with $\Sigma^0_n$ sentences (that ...

**3**

votes

**0**answers

94 views

### Cardinality based results in Topological Vector Spaces?

Given a topological vector space $V$, let its density be the smallest cardinal $A$ such that a set of cardinality $A$ is dense in $V$. Naively, it seems one of two things happen:
TVS's $V$ of ...

**3**

votes

**1**answer

183 views

### May open sentences be eliminated?

Saul Kripke famously invoked a free logic to avoid validating the Barcan Formula and its converse. In that context he adduced a generality interpretation of free variables. The converse of the Barcan ...

**3**

votes

**1**answer

92 views

### Understanding Corollary 3, Sec. 5.6, of Papadimitriou's Computational Complexity

I am struggling to understand Corollary 3 from Section 5.6 of Papadimitriou's Complexity Theory book (Addison-Wesley, 1993). It got me completely confused... If anyone out there has read it and ...

**2**

votes

**0**answers

123 views

### RCS iteration such that the RCS limit is semi-proper

For a countable ordinal $\eta$ and an ordinal $\gamma$ let $\langle P_\alpha, \dot{Q}_\alpha \colon \alpha < \gamma \rangle$ be an RCS iteration with RCS limit $P_\gamma$, such that
...

**6**

votes

**2**answers

231 views

### A question regarding strong cardinals and measure sequence

Let $E$ be a $(\kappa, \lambda)$-extender such that $j: V\to M\simeq Ult(V,E)$ is the corresponding elementary embedding with critical point $\kappa$, $M\supset V_{\kappa+2}$, $M^\kappa\subset M$. Let ...

**5**

votes

**2**answers

250 views

### Is every non-empty $\Delta_0$ set provably the range of some primitive recursive function?

Suppose $A(x)$ is a $\Delta_0$ formula defining a non-empty set of natural numbers. It's an easy theorem that there is a primitive recursive function $f:\mathbb{N} \rightarrow \mathbb{N}$ such that ...

**4**

votes

**1**answer

289 views

### “set of all irreducible representations of a group”, set-theoretic issues [closed]

I am working on a problem related to representations of the Weil group of a local field $\mathcal{W}_F$. In many articles one introduces the set $\hat{\mathcal{W}}_F$ of all equivalence classes of ...

**8**

votes

**0**answers

237 views

### A Banach-Tarski game

This is partially inspired by the question http://math.stackexchange.com/questions/1383397/cutting-a-banach-tarski-cake, which I find intriguing if unclearly written.
A paradoxical family of subsets ...

**3**

votes

**3**answers

682 views

### In which sense “closure” is a closure?

In predicate and first-order logic, if $\phi$ is a sentence, then $\forall X . \phi$ is said to be the (universal) closure of $\phi$. Is the use of the word "closure" incidental, or is there a ...

**2**

votes

**0**answers

121 views

### How distributive are the bad Laver tables?

Suppose that $n\in\omega\setminus\{0\}$. Then define $(S_{n},*)$ to be the algebra where $S_{n}=\{1,...,n\}$ and $*$ is the unique operation on $S_{n}$ where
$n*x=x$
$x*1=x+1\,\text{mod}\, n$ and
if ...

**6**

votes

**1**answer

190 views

### Reverse of a termspace forcing fact

Suppose $\kappa$ is an inaccessible cardinal. Consider the termspace forcing for adding a Cohen subset of $\kappa$ after $\mathbb P= Col(\omega,<\kappa)$. Members are Levy names for countable ...

**1**

vote

**0**answers

98 views

### Oracle queries asked in parallel

Definition: Assume that $\phi(q)$ is of the form $\exists y \leq 2^{p(n)} \varphi(q,y)$, where $p$ is a polynomial and $n = |q|$ (i.e. $n$ is the length of the binary representation of $q$). Then a ...

**2**

votes

**1**answer

155 views

### A question on the name of a property

What is the name of the property that a system $T$ has if $\vdash_{T}\exists x F(x)$ only if there is a term $a$ so that $\vdash_{T} F(a)$?
If I recall correctly Heyting Arithmetics has the ...

**13**

votes

**1**answer

604 views

### Does “cardinal arithmetic is well-defined” imply axiom of choice?

Let me quickly explain what I mean with my question.
Let $(\kappa_i)_{i\in I}$ be a collection of cardinal numbers, indexed by elements of some set $I$. We can try to define $\sum\limits_{i\in ...