# Tagged Questions

**4**

votes

**2**answers

225 views

### What “force” us to accept large cardinal axioms?

Large cardinal axioms are not provable using usual mathematical tools (developed in $\text{ZFC}$).
Their non-existence is consistent with axioms of usual mathematics.
It is provable that some of ...

**2**

votes

**1**answer

114 views

### Proof of existence of recursively inaccessible and Mahlo ordinals

As in title - I'm looking for a proof of the existence of a countable recursively inaccessible or recursively Mahlo ordinals, especially the first one. When looking for it in all the papers I stumbled ...

**5**

votes

**0**answers

371 views

### Different approaches to the multiverse of sets

There are some different approaches to the multiverse of sets, in particular:
1) The approach by Woodin,
2) The approach by Sy Friedman, ...,
3) The approach by Hamkins.
I wonder to know if ...

**9**

votes

**1**answer

303 views

### Applications of SCH outside of set theory

Recall that the Singular Cardinals Hypothesis (SCH) says that if $\kappa$ is a singular cardinal and $2^{cf(\kappa)}<\kappa,$ then $\kappa^{cf(\kappa)}=\kappa^+.$ Clearly it has many applications ...

**4**

votes

**3**answers

327 views

### Systematic brute-force searches for counterexamples

This is getting nowhere on math.stackexchange.com, so I'm putting it here.
Gödel's completeness theorem says that for every statement in first-order predicate calculus with equality, there is either ...

**6**

votes

**1**answer

171 views

### Does the totality of Ackermann's function prove the consistency of $\Sigma_1$-induction?

It is well known that Ackermann's function is not primitive recursive. Therefore, the theories of primitive recursive arithmetic (PRA) and of $\Sigma_1$-induction ($I\Sigma_1$) cannot prove the ...

**4**

votes

**0**answers

82 views

### Relation between fundamental theorem of Ehrenfeucht-Fraisse games and notions of bisimulation and simulation

Vijay D alluded to the relation between the fundamental theorem of Ehrenfeucht-Fraisse games and notions of bisimulation and simulation in response to the whats-a-magical-theorem-in-logic question.
...

**10**

votes

**0**answers

124 views

### Maximality of linear orders in the Keisler order on theories

Recently Malliaris and Shelah (see their preprint http://math.uchicago.edu/~mem/Malliaris-Shelah-CST.pdf) have shown that theories with $SOP_2$ are maximal in the Keisler order. A preceding result of ...

**5**

votes

**1**answer

106 views

### Attribution of an equivalence of the existence of omega-models of RCA0

There are many well-known equivalences in reverse mathematics between statements of the form "Every set is contained a countable coded $\omega$-model of $T$" and $S$, where $S, T$ are subsystems of ...

**7**

votes

**3**answers

517 views

### Consistency of Analysis (second order arithmetic)

Is there a proof of the consistency of Analysis (second order arithmetic), which is similar to Gentzen's proof of the consistency of arithmetic?
Update:
Which (different) methods can be used to ...

**3**

votes

**1**answer

199 views

### Colorful model theory

There are a number of concepts in model theory - often situated around Hrushovski's amalgamation method (see for instance http://math.univ-lyon1.fr/~wagner/nijmegen.pdf) - which are colorfully named:
...

**4**

votes

**1**answer

200 views

### Perfect set property implies $\omega_1$ is a limit cardinal in $L$

Specker proved in 1957 that if in $V$ every set of real numbers has the perfect set property, than in $L$, $\omega_1^V$ is actually a limit cardinal.
The original proof is in German, and I've been ...

**4**

votes

**1**answer

178 views

### Is Van der Waerden's function elementary

Van der Waerden's function was proved to have elementary upper bound on growth rate.
Is the Van der Waerden's function itself elementary in the sense of Kalmar?

**9**

votes

**1**answer

174 views

### Obtaining a lightface pointclass from a boldface one

Define a pointclass to be:
boldface inductive-like if it is $\mathbb{R}$-parameterized, has the scale property, and is closed under $\wedge$, $\vee$, $\forall^\mathbb{R}$, $\exists^\mathbb{R}$, and ...

**-2**

votes

**1**answer

177 views

### Natural constructions (not depending on parameters) [closed]

Consider graph clusterings as a prototypical example of (logical) constructions.
Let a clustering of a graph $(V,E)$ be any covering of $V$, i.e. a set $C$ with $\bigcup C = V$.
I am looking for a ...

**4**

votes

**2**answers

266 views

### Forcing for Arbitrary First Order Theories

Forcing is a relative model construction method for models of $ZF$ as a particular first order theory using models of another first order theory (forcing companion) that in this case is the theory of ...

**2**

votes

**0**answers

150 views

### How many Dedekind-finite sets can $\mathbb{R}$ be partitioned into?

Building off Asaf Karagila's answer to my previous question (Can $\mathbb{R}$ be partitioned into dedekind-finite sets?) on partitioning $\mathbb{R}$ into strictly Dedekind-finite sets:
(1) What ...

**8**

votes

**0**answers

230 views

### From Frege to Gödel - German equivalent?

I know this question does not quite fit here, but I felt it could best be answered here. I recently stumbled upon the book From Frege to Gödel, which is a sourcebook containing some of the most ...

**2**

votes

**0**answers

86 views

### Comparing two non-deterministic Turing equivalents as basis for Logic, request for references

I am designing a logic, that is simpler than FOL + PA. And I like to know if there already exists something in this direction.
First of all a non-deterministic Turing equivalent is defined by ...

**6**

votes

**0**answers

141 views

### “Fraïssé limits” without amalgamation

All structures are countable with countable signature.
Given a structure $\mathcal{A}$, the age of $\mathcal{A}$, $Age(\mathcal{A})$, is the set of structures isomorphic to finitely-generated ...

**8**

votes

**3**answers

275 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 ...

**21**

votes

**4**answers

849 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 ...

**7**

votes

**1**answer

228 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 ...

**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 ...

**-4**

votes

**1**answer

232 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

**1**answer

193 views

### Forcing Language

I asked the following question (slightly paraphrased) about a week ago in Stack exchange but no one knew the answer to the particular question. I was hoping someone here might be able to help me.
...

**6**

votes

**4**answers

1k views

### Interactions of number theoretic conjectures and other fields of mathematics

There are many interesting open conjectures in number theory. My question is not about partial results or possible ways to prove them. It is about their interactions with the other fields of ...

**4**

votes

**1**answer

244 views

### Consistency Strength of the Failure of Square on Singular Cardinals

Q1. What is the consistency strength of the failure of square on singular cardinals?
Q2. What are known as partial results in this direction?

**6**

votes

**2**answers

223 views

### Possible Choices for Cofinality of $\aleph_n$ without Choice

$\text{ZFC}$ proves that each $\aleph_{n}$ for $n\in \omega$ is a regular cardinal. But it seems without the Axiom of Choice there are many consistent possible choices for cofinality of such ...

**11**

votes

**3**answers

554 views

### The interplay between certain aspects of interpretability, model theory and category theory

I have some questions about the interplay of interpretability, model theory and category theory. Since I had difficulties in finding literature or other helpful information about this topic, it would ...

**6**

votes

**1**answer

521 views

### A Hot Betting On HOD

Remark: This question is based on an open question at the end of a paper by Hamkins, Kirmayer, and Perlmutter: "Generalizations of the Kunen Inconistency".
$HOD$ as an inner model of $ZFC$ lies ...

**5**

votes

**2**answers

256 views

### Can we force with Fraisse filters to solve Vaught's conjecture?

Around the classic Fraisse amalgamation theorem in model theory we have the following notions:
Definition (1): If $M$ be an $\mathcal{L}$-structure then define:
$age(M):=\lbrace N~|~N~\text{is ...

**1**

vote

**0**answers

204 views

### Logical and alphabetological variant?

The notion of alphabetical variant is well known, so that a formula $x=x$ is an alphabetical variant of $y=y$ if $x$ and $y$ are distinct variables.
One may want to consider the set term $\{x:x \neq ...

**14**

votes

**6**answers

1k views

### The origins of forcing in mathematical logic and other branches of mathematics

As everyone knows, forcing was created by Cohen to answer questions in set theory.
Question 1. What are the first applications of set theoretic forcing in other branches of mathematical logic, like ...

**2**

votes

**1**answer

86 views

### Predicates encoding functions in Grzegorczyk-hierarchy

A $2$-ary predicate $R$ in Grzegroczyk-hierarchy is binary-valued function $R\colon\mathbb{N}^2\to\{0,1\}$. We say $R$ encodes function $f:\mathbb{N}\to\mathbb{N}$ if $R(x,y)$ is true if and only if ...

**2**

votes

**0**answers

346 views

### Deducing Skolem's nonstandard integers from downward Lowenheim-Skolem?

If one has a nonstandard model $\mathcal{N}$ of PA and adjoins to the first-order theory the countable list of axioms $1<H,\, 2<H,\, 3<H, \ldots$ (satisfied in $\mathcal{N}$) for all the ...

**9**

votes

**1**answer

256 views

### Universal $(\Sigma^2_1)^{\text{Hom}_{\mathord{<}\lambda}}$ set

Does anyone know of a reference for the fact that if $\lambda$ is a limit of Woodin cardinals, then the pointclass $(\Sigma^2_1)^{\text{Hom}_{\mathord{<}\lambda}}$ is $\omega$-parameterized? By ...

**9**

votes

**1**answer

411 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 (normed)
probability measure ...

**4**

votes

**1**answer

183 views

### Who first proved there's an $\omega$-model of $\mathsf{WKL}_0$ in which all sets are low?

I am trying to pin down: who first proved that $\mathsf{WKL}_0$ has an $\omega$-model in which every set is of low degree? As shown in Simpson's Subsystems of Second Order Arithmetic (Theorem ...

**5**

votes

**1**answer

524 views

### What are current trends/questions in algebraic logic?

What are current trends/questions in algebraic logic?I mean the research developed by Paul Halmos.
And anyone could give some reference for overview of it's history?
Also any overview of it's ...

**10**

votes

**2**answers

502 views

### What are Moschovakis cardinals?

The question is exactly that of the title: what are Moschovakis cardinals?
Background. In a recent answer to the question, "Are there examples of statements that have been proven whose consistency ...

**7**

votes

**1**answer

130 views

### definability of Morley rank in the theory of compact complex spaces

Is there a counter-example showing that Morley rank in the theory of compact complex spaces (as defined by Zilber, Pillay, Moosa, ...) is not definable in families?
Given the existence of such a ...

**7**

votes

**1**answer

222 views

### Iterating definability

An odd -- probably basic -- question about model theory:
For $\mathcal{M}$ a structure in a (first-order) signature $\Sigma$, let $\mathcal{M}'$ be the structure in signature $\Sigma\sqcup\lbrace ...

**5**

votes

**0**answers

140 views

### Feasible Type Theories

I am looking for references about efficient type theories,
efficiency in the sense of computational complexity,
and type theory in the sense of Martin-Lof's type theories.
Has there been any studies ...

**14**

votes

**1**answer

505 views

### Goodstein's theorem without transfinite induction

Goodstein's theorem is an example of a theorem that is not provable from first order arithmetic. All proofs of the theorem seem to deploy transfinite induction and I've wondered if one could prove the ...

**4**

votes

**1**answer

125 views

### Amalgamation of two ccc algebras may collapse the continuum

The claim that appears in the title of this question is mentioned in the paper "On Shelah's amalgamation" by Judah and Roslanowski. I'd really like to see a proof of this fact, but unfortunately I ...

**7**

votes

**2**answers

404 views

### Deriving Konig's Lemma directly from Infinite Ramsey's Theorem for triples

Let KL denote König's Lemma (for trees over $\mathbb{N}$), and RT(3) denote the
Infinite Ramsey Theorem for triples over $\mathbb{N}$ (notation as in Simpson's
book Subsystems of second order ...

**9**

votes

**4**answers

470 views

### dense orders are saturated

In a FOM msg of Fri May 21 19:59:44 EDT 1999,
http://www.cs.nyu.edu/pipermail/fom/1999-May/003149.html
Simpson gave a short proof of the following theorem:
Theorem. Let F be a real closed ordered ...

**9**

votes

**2**answers

376 views

### Reverse mathematics below RCA

I'm sure this is a fairly basic question, but I can't seem to find a solid answer:
My primary question is: Is there a reasonably nice subsystem of second-order arithmetic corresponding essentially to ...

**20**

votes

**0**answers

776 views

### Godel on recursion-theoretic hierarchies

At the end of his excellent article, "The Emergence of Descriptive Set Theory" (http://math.bu.edu/people/aki/2.pdf), Kanamori writes:
"Another mathematical eternal return: Toward the end of his ...