first-order and higher-order logic, model theory, set theory, proof theory, computability theory, formal languages, definability, interplay of syntax and semantics, constructive logic, intuitionism, philosophical logic, modal logic, completeness, Gödel incompleteness, decidability, undecidability, ...

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30
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2answers
3k views

Why is this new result such a big deal?

This popular article reports a recent result in reverse mathematics, showing that a certain theorem in Ramsey theory is provable from RCA$_0$, the base theory in SOSOA. Then there are a bunch of ...
7
votes
1answer
193 views

Normal subgroups of Aut(M)

Let $S$ be the set of all finite permutations of $\mathbb{N}$, i.e. they fix all but a finite set, and $A\subset S$ the set of all even permutations. Theorem The normal subgroups of $S_\infty$ are ...
-1
votes
0answers
51 views

Ddifference between deduct and deduce [closed]

In the context of logic, the term "deduction" was used as a way of thinking. I think, apparently, the verb of "deduction" should be "deduct", hence it is natural and reasonable to use the term ...
2
votes
0answers
68 views

Are Braid Groups with Finitely many Generators NIP?

I am curious what braid groups (strings in $\mathbb{R}^3$) are NIP. Consider the following: Let $B_\mathbb{N}$ be braid group with "braids" indexed by the natural numbers (alternatively, the direct ...
1
vote
1answer
146 views

Interpreting peano arithmetic without parameters

I will accept an answer in the form of references to the literature about my question as well as any other information. I am quite ignorant of the area and that will be clear from my question. I ...
8
votes
0answers
153 views

Turing degree of finding independent formulas

In this Paper of D. Isaacson, it is proved that the true arithmetic($Th(\mathbb{N}$)) is the only $\omega$-consistent and complete extension of $Q$ (Robinson's arithmetic). This, together the fact ...
2
votes
1answer
165 views

What is the extension of the truth-table degrees to Baire Space called?

Recall that for sets $A, B \in 2^\omega$ that we say $A \leq_{tt} B$ if there is a total Turing functional $F \colon 2^\omega \to 2^\omega$ such that $F(B)=A$. This is called truth-table ...
4
votes
0answers
92 views

Automorphism group of a structure without the SAP

A few years ago, a number of examples were given of Fraisse structures without the SAP in answer to the question raised in A Fraïssé class without the strong amalgamation property. It is ...
6
votes
0answers
119 views

Examples of analytic $\mathcal{I}$-mad families

If $\mathcal{I}$ is an ideal (proper and containing the finite sets) on $\omega$, call a family of subsets $\mathcal{A}\subseteq[\omega]^\omega$ $\mathcal{I}$-almost disjoint if for all distinct ...
8
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0answers
184 views

Inner models and strongly compact cardinals

The following question is motivated by a result of Magidor that it is consistent that the least strongly compact cardinal is the least measurable cardinal. Question. Assume $\kappa$ is a strongly ...
4
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0answers
138 views

Spectrum Problem for Higher-Order Logic

Definitions. Given a sentence $\varphi$ of $n$th-order logic, we define the spectrum of $\varphi$ to be the set of cardinalities of finite structures that satisfy $\varphi$. A set $X\subseteq\mathbb ...
4
votes
1answer
142 views

Expressive power of $\omega$-order logic

According to the article Second-order and Higher-order Logic from the Stanford Encyclopedia of Philosophy, there is no need to stop at second-order logic; one can keep going. [...] we can allow ...
0
votes
0answers
118 views

local analyticity of volumes of slices of semi-algebraic sets

I would like a reference and/or a simple proof using well-known results of the following, which I think is true. (If it's false, I'd like to know that as well of course -- and ideally a way to modify ...
5
votes
1answer
82 views

Complexity of integer programming with added predicates

A classical theorem in Integer Programming by Lenstra says that any integer system $$A x \le b$$ can be solved in polynomial time, where $A \in \mathbb{Z}^{m \times n}, x \in \mathbb{Z}^n, b \in ...
10
votes
1answer
260 views

Precipitous ideals and GCH

It is well known that ZFC + "There is a measurable cardinal" is equiconsistent with ZFC + "There is a precipitous ideal on $\omega_1$." Is ZFC + "There is a measurable $\kappa$ such that $2^\kappa ...
4
votes
0answers
175 views

Unbounded towers and combinatorial cardinal characteristics of the continuum

Update: Perhaps the question is too difficult. I would appreciate, thus, even just comments or related observations. This question assumes familiarity with combinatorial cardinal characteristics of ...
1
vote
1answer
131 views

$\mathcal A\equiv\mathcal B\implies \mathcal A\cong\mathcal B$ for finite $\mathcal L$-structures where $\mathcal L$ is an infinite signature

Let $\mathcal L$ be an infinite signature and $\mathcal A$, $\mathcal B$ two finite $\mathcal L$-structures such that for each first-order $\mathcal L$-sentence $\varphi$, $$\mathcal ...
4
votes
1answer
214 views

Is it consistent that the gaps between cardinals $\kappa$ and $2^\kappa$ “get larger and larger”?

Is the following statement consistent in $\mathsf{ZFC}$? For every ordinal $\beta$ there is an ordinal $\lambda_0$ such that for all ordinals $\lambda\geq\lambda_0$ we have ...
10
votes
1answer
351 views

What Turing degree is this function?

Over at http://www.scottaaronson.com/blog/?p=2725#comment-1089004 we had a discussion of intermediate Turing degrees. The following function came up: Take Chaitin’s constant, and rearrange its ...
-8
votes
1answer
349 views

Missing Axiom: There are no other axioms. Leads to a proof of CH within ZFC [closed]

I have come to the conclusion that there is often an implied axiom: There are no other axioms. Failure to explicitly state this axiom and to consider its consequences can result in the misleading ...
0
votes
1answer
154 views

Can epsilon-induction be derived from the transitive closure operator?

I was wondering (and could not seem to prove or disprove) if epsilon-induction could be derived from the transitive closure operator for binary relations, if we do not have the Foundation Axiom. The ...
2
votes
2answers
350 views

What are the sense and reference of the propositions $R \notin R$, $R \in R$, where $R=\{x \mid x \notin x\}$ in Frege's Grundgesetze?

In the paper, Aldo Antonelli and Robert May, Frege's new science, Notre Dame J. Form. Log. 41 (2000), no. 3, 242–-270, MR 1943495. the authors give the following quote of Frege, from his paper ...
2
votes
2answers
178 views

Henkin semantics for Second-order Logic

I know that the natural numbers can be categorically characterized in second-order logic with the standard semantics. However, I could not find an example of a non-standard Henkin structure (one that ...
0
votes
0answers
81 views

What is the known weakest axiom system has Löb's derivability conditions?

We know that Peano Arithmetic satisfies Löb's derivability conditions, which is required in the proof of Gödel's 2nd incompleteness theorem. Is this the best result? If not, is there any known weaker ...
14
votes
0answers
345 views

The axiom $I_0$ in the absence of $AC$

It is well-known that if $AC$ holds and if $j: L(V_{\lambda+1}) \to L(V_{\lambda+1})$ is a non-trivial elementary embedding with $crit(j) < \lambda,$ then $\lambda$ has countable cofinality (and in ...
1
vote
1answer
182 views

Non-regular languages fulfilling the Pumping Lemma

Some non-regular languages don't yield to the Pumping Lemma ($L_1=a^nb^mc^m$ should work). But now consider the set of non-regular languages L only over the alphabet {a}. (Like $L_2=a^{n^2}$ or ...
4
votes
2answers
321 views

Embedding property of weakly compact cardinals

One of the characterizations of $\kappa$ being a weakly compact cardinal is being inaccessible, and for every $\kappa$-model $M$, there is a [$\kappa$-model] $N$ and an elementary embedding $j\colon ...
6
votes
1answer
258 views

moving up a consequence of PFA

The Proper Forcing Axiom (PFA) implies that every forcing which adds a subset of $\omega_1$ either adds a real or collapses $\omega_2$. Is it consistent that every forcing which adds a subset of ...
6
votes
0answers
188 views

Nonexistence of generic objects over $L(\mathbb{R})$

A well known result (stated and credited to Todorcevic in "Semiselective Coideals", by Farah, Mathematika, 1997, but with antecedents going back to Mathias) says that, under the appropriate large ...
9
votes
5answers
307 views

Probability theory without deductive closure

Human knowledge is not deductively closed. Uncertainty can arise from that just as much as from lack of brute facts. (When a Harvard graduate was reported to have thought that the earth is farther ...
14
votes
1answer
464 views

Does Con(ZF + Reinhardt) really imply Con(ZFC + I0)?

The question is: if I assert in ZF that there exists a Reinhardt cardinal, do I really get a theory of higher consistency strength than when I assert in ZFC that there exists an I0 cardinal (the ...
4
votes
1answer
263 views

Large cardinals without choice?

For any given extension $T$ of ZFC (or perhaps NBGC or something), we can ask whether there is an extension $T'$ of ZF which does not prove AC such that $Con(T) \leftrightarrow Con(T')$ $Con(T) \to ...
15
votes
4answers
2k views

Languages beyond enumerable

A language is a set of finite-length strings from some finite alphabet $\Sigma$. It is no loss of generality (for my purposes) to take $\Sigma=\{0,1\}$; so a language is a set of bit-strings. ...
4
votes
2answers
234 views

Is injectivity of $2^{(\ldots)}$ weaker than $\mathsf{GCH}$? [duplicate]

The following statement cannot be proven in $\mathsf{ZFC}$: (S) : If $A, B$ are sets with $|A| < |B|$, then $2^{|A|} = |{\cal P}(A)| < |{\cal P}(B)| = 2^{|B|}$. Obviously, ...
4
votes
1answer
209 views

Does “Every infinite set is splittable” imply $\mathsf{AC}$? [duplicate]

We say an infinite set $X$ is splittable if there are $X_1, X_2\subseteq X$ with $X_1\cap X_2 = \emptyset$, $X_1\cup X_2 = X$ and there are bijections $\varphi:X_1\to X_2$ and $\psi:X_1\to X$. Does ...
7
votes
2answers
379 views

Difference between constructive Dedekind and Cauchy reals in computation

If the Axiom of Countable Choice (ACC) $$ \forall n\in \mathbb{N} . \exists x \in X . \varphi [n, x] \implies \exists f: \mathbb{N} \longrightarrow X . \forall n \in \mathbb{N} . \varphi [n, f(n)] $$ ...
6
votes
1answer
352 views

Does the consistency strength hierarchy coincide with the “arithmetic consequence” hierarchy at ZF + Reinhardt?

In these slides (see especially slide 26), Steel emphasizes the phenomenon that for all known "natural" extensions of ZFC, the ordering by consistency strength agrees with the ordering by containment ...
3
votes
1answer
233 views

Representation of meager sets in Cohen extensions

Let $M$ be a transitive model of ZFC and $c\in {}^\omega2$ a Cohen real over $M$. Let $A$ be a meager Borel subset of $^\omega2$ in $M[c]$. I would like to prove that there exists a meager Borel set ...
6
votes
1answer
199 views

Theorem of Bukovsky characterizing ground models

It was mentioned in a talk that Bukovsky proved the following are equivalent for inner models $M \subseteq V$: (1) There is a partial order $\mathbb P \in M$ and a $\mathbb P$-generic filter $G \in ...
9
votes
0answers
451 views

Algebraic proofs of algebraic theorems about algebraically closed fields

It is well-known that the first order theory of algebraically closed fields admits quantifier elimination, whence the theory $ACF_p$ of algebraically closed fields of given characteristic $p$ is ...
3
votes
1answer
193 views

“Lexicographic” ordering on ${\cal P}(\omega)$

For $A\neq B\in {\cal P}(\omega)$ we set $$\mu(A,B) = \min\big((A\setminus B)\cup (B\setminus A)\big).$$ We define $A < B$ if and only if $A \neq B$, and $A = B\cap \mu(A,B)$ (that is $A$ is an ...
1
vote
0answers
39 views

Quantifier elimination of pp-subgroups of modules

This is a model-theoretic questions. Let $M$ be a $R$-module. Our language will be the standard language of modules, i.e. the language of abelian groups together with an unary function symbol for ...
8
votes
1answer
223 views

Must $L_\alpha$ be correct about well-foundedness?

If $R \in L_\alpha$ is a binary relation so that $L_\alpha$ thinks $R$ is well-founded, must $R$ truly be well-founded? (Edit) That is, if $L_\alpha$ thinks that every nonempty subset of the domain of ...
5
votes
1answer
189 views

Easier Girard's paradox in a circular pure type system (PTS)

System U is an inconsistent PTS in that one has a term of type $\bot = \forall p\colon \ast \ldotp p$, and such a term is explicitly constructed in Hurkens' A Simplification of Girard's Paradox. ...
22
votes
0answers
454 views

Can one divide by the cardinal of an amorphous set?

This question arose in a discussion with Peter Doyle. It is provable in ZF that one can divide by any positive finite cardinal $k$: if $X \times \{1,\ldots,k\} \simeq Y \times \{1,\ldots,k\}$ then $X ...
9
votes
1answer
295 views

Two questions about higher Souslin trees

Assume $V=L$ and let $\kappa$ be a Mahlo cardinal. Let $L[G]$ be the generic extension obatined by Mitchell forcing to make $2^{\aleph_0}=\aleph_2=\kappa.$ It is known that in the extension there ...
6
votes
1answer
199 views

Fat stationary sets

Recall a stationary subset $S$ of a regular cardinal $\kappa$ is fat when for every $\alpha < \kappa$, and every club $C$, there is a closed set of order type $\alpha$ contained in $S \cap C$. It ...
2
votes
1answer
149 views

Posets (partially ordered sets) in equational logic

I know about equational logic, cf. https://en.wikipedia.org/wiki/Lattice_(order)#Lattices_as_algebraic_structures, and understood that lattices are expressed equationally, i.e., in terms of equational ...
2
votes
0answers
120 views

When do wide initial segments ruin admissibility?

Suppose $\alpha$ is admissible and $\beta<\alpha$. We know that $L_\alpha$ is an admissible set (by definition), but adding subsets of $\beta$ to $L_\alpha$ might break admissibility: while set ...
15
votes
2answers
628 views

$\mathfrak{ufo}$: An unidentified combinatorial cardinal characteristic of the continuum?

An ultrafilter ornament is a chain of free filters on $\mathbb{N}$ that are not ultrafilters, whose union is an ultrafilter. Let $\mathfrak{ufo}$ be the minimal cardinality of an ultrafilter ...