1
vote
1answer
29 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
0answers
361 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
1answer
294 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
3answers
323 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
1answer
165 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
0answers
79 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
0answers
122 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
1answer
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
3answers
513 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
1answer
196 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
1answer
199 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
1answer
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
1answer
173 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
1answer
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
2answers
260 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
0answers
149 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
0answers
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
0answers
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
0answers
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
3answers
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 ...
21
votes
4answers
842 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
1answer
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 ...
0
votes
0answers
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
1answer
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
1answer
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
4answers
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
1answer
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
2answers
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
3answers
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
1answer
518 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
2answers
255 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
0answers
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
6answers
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
1answer
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
0answers
345 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
1answer
255 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
1answer
410 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
1answer
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
1answer
523 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
2answers
500 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
1answer
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
1answer
221 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
0answers
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
1answer
504 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
1answer
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
2answers
403 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
4answers
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
2answers
375 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
0answers
770 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 ...
5
votes
1answer
207 views

Consistency of many Erdos cardinals

Does anyone have a reference for the consistency strength of saying that the Erdos cardinal $\kappa(\alpha)$ exists for all $\alpha$? Or of various weakenings about how far such cardinals extend into ...