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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10
votes
1answer
371 views

Extending an infinite simple group

Maybe the question does not fit here. Yesterday in my logic course, I presented a nice example about an application of model theory to group theory. The example is due to Hodges and as following: For ...
7
votes
0answers
208 views

A strengthening of Chang's conjectures

In the Handbook of Set Theory, Foreman has (essentially) the following proposition (3.9): Suppose $\kappa_n>...>\kappa_0$ and $\lambda_n>...>\lambda_0$ are regular cardinals and ...
2
votes
1answer
194 views

Is Extensionality needed for the incompleteness of very weak set theories?

$ST$ is the weak set theory built upon identity theory and containing the axiom for empty set, the axiom for adjunction and the axiom for extensionality. It is known that $ST$ interprets ...
6
votes
0answers
166 views

PCF conjecture and fixed points of the $\aleph$-function

Recently Moti Gitik refuted Shelah's PCF conjecture, by producing a countable set $a$ of regular cardinals with $|pcf(a)| \geq \aleph_1.$ See his papers Short extenders forcings I and Short ...
-2
votes
1answer
106 views

A question on a presumably admissible/derivable rule for (usual) formal systems [closed]

Suppose formal system $X$ has the rule $\vdash\beta\Rightarrow\hspace{2pt}\vdash\gamma$. Is the rule $\vdash\alpha\wedge\beta\Rightarrow\hspace{2pt}\vdash\alpha\wedge\gamma$ (usually) just admissible ...
12
votes
0answers
182 views

Is the game Hanabi NEXPTIME-complete?

The game Hanabi is a cooperative, hidden-information game. You can read the rules elsewhere, but broadly speaking the players are attempting to cooperatively build a fireworks display by playing cards ...
2
votes
0answers
108 views

A question regarding an analogue of the Kleene $T$-predicate for Koepke's ordinal computability

Does Koepke's notion of ordinal computability admit an analogue of the Kleene $T$-predicate? If so, is the existence of such a $T$-predicate independent of $ZFC$? Also, if one assumes the existence ...
13
votes
0answers
140 views

Complexity classes for BSS machines

Given a first-order structure $\mathcal{S}$, a Blum-Shub-Smale machine on $\mathcal{S}$ is essentially a Turing machine where Cells on the tape can hold arbitrary elements of $\mathcal{S}$. The ...
1
vote
0answers
185 views

How may we define a bijection from $\wp(\mathbb{Q})$ to $\mathbb{R}$ in $ZFC$? [closed]

Some expressed difficulty understanding that there are more members in $\mathbb{R}$ than in $\mathbb{Q}$ according to classical set theories because there between all real numbers is a rational ...
10
votes
0answers
154 views

Can we find minimal-diameter metrics without computability?

A beautiful argument by Nabutovsky and Weinberger (see http://math.uchicago.edu/~shmuel/fractal.ps) shows that, if $M$ is any smooth compact manifold of dimension $\ge 5$, then the diameter functional ...
13
votes
2answers
422 views

Logical complexity of algebraically closed fields

One can define fields using a finite list of axioms that quantify over the field itself. However, the obvious way to define algebraically closed fields involves either an infinite list of axioms, or ...
7
votes
3answers
154 views

Ordinal-indexed transitive antichain of sets with urelements

Operate in ZFC. Can we find a function-class $\phi$ whose domain is the class of ordinals such that the following properties hold? If $x \in \phi(\alpha)$, then either $x \in \mathbb{N}$ or there ...
2
votes
0answers
67 views

Kripke semantics for fuzzy logics

I am interested in Fuzzy logic. I have a problem about Gödel Logic, I'm studying Kripke semantics for fuzzy logics and have found the necessary and sufficient conditions on Kripke frames for ...
35
votes
4answers
2k views

Hilbert's (cancelled) 24th problem

Hilbert's 23 problems, ten of which were presented at the 1900 ICM in Paris, are too famous for any mathematician to not know. If one reads the descriptions of the problems in Hilbert's paper, one ...
26
votes
0answers
562 views

Where do uncountable models collapse to?

Suppose $T$ is a complete first-order theory (in an finite, or at worst countable, language). Given any model $\mathcal{M}\models T$ of cardinality $\kappa$, we can ask whether $\mathcal{M}$ can be ...
12
votes
2answers
397 views

Woodin on Posner-Robinson for the hyperjump and sharp

The Posner-Robinson theorem states that, if $X$ is noncomputable, there is some $G$ such that $X\oplus G=G'$; that is, even though genuine jump inversion only works above $0'$, every (nontrivial) $X$ ...
11
votes
1answer
469 views

Can you have many independent reals?

Working in $\sf ZFC$, is it provable, or at least consistent (say, over $L$), that you have $\aleph_1$ forcings, $\Bbb P_\alpha$ such that: $\Bbb P_\alpha$ is c.c.c. $\Bbb P_\alpha$ adds a real ...
3
votes
0answers
385 views

“Nicely” strong measure zero sets

This question is essentially an expanded version of the unanswered half of Two strengthenings of "strong measure zero". A set $X$ of reals is strong measure zero if, for any $f: ...
7
votes
1answer
338 views

Is every ordinal potentially definable?

It is easy to see that, if $V\models\alpha>\omega_1^{CK}$, then $\alpha$ is not recursive in any forcing extension of $V$. The argument goes as follows: The relation "$\Phi_e=r$" is $\Pi^0_2$. ...
11
votes
0answers
312 views

Large cardinals arising from alternate set theories

My question is whether there are model-theoretic large cardinal axioms associated with $NFU$ - or more generally, with set theories other than $ZFC$. Large cardinal properties generally come in one ...
7
votes
1answer
228 views

Proof-theoretic ordinals after liberalizing induction to $RCA_0$

This is kind of a follow-up to this question. For a class $\Gamma$ of second-order formulas (here either $\Sigma_n^0$ or $\Sigma_n^1$), let $X\Gamma$ be a formal theory consisting of $RCA_0$ together ...
12
votes
1answer
244 views

Forcings that are not equivalent to Levy collapse

Assume GCH and that $\kappa$ is a regular uncountable cardinal. Let $\mathbb{P}$ be a separative, $<\kappa$-directed closed, nowhere trivial, $\kappa^+$-cc poset of size $\kappa^+$. Must ...
5
votes
2answers
184 views

Lefschetz Principle for semisimplicity

I think I can prove the following using the compactness of first order logic and I am wondering what a purely algebraic proof would look like. Let $R$ be a unital ring (not necessarily ...
3
votes
0answers
99 views

A question regarding forcing in $NGBC^{-f}$+$BAFA$

Suppose one has a model $M$$\vDash$$NGBC^{-f}$+BAFA. Does there exist a (class) forcing extension $M[G]$$\vDash$$NGBC^{-f}$+$BAFA$ that has a submodel $N$$\vDash$$NGB^{-f}$+$BAFA$+$\lnot$$AC$? Can ...
3
votes
1answer
196 views

Compactness of Lukasiewicz Logic

I'm interested in Fuzzy logic. I have read that the compactness theorem holds for predicate Lukasiewicz logic, with semantics over $[0,1]$. However I found the following question on mathoverflow ...
4
votes
2answers
139 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. ...
6
votes
1answer
92 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
0answers
73 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 ...
9
votes
0answers
157 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 ...
13
votes
1answer
230 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 ...
9
votes
1answer
120 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 ...
10
votes
0answers
154 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 ...
28
votes
7answers
2k 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 ...
10
votes
1answer
422 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 ...
8
votes
0answers
219 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.) ...
21
votes
2answers
748 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 ...
16
votes
1answer
486 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 ...
14
votes
2answers
555 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
0answers
240 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 $V$. ...
4
votes
1answer
107 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 ...
7
votes
1answer
342 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 ...
2
votes
1answer
105 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 ...
6
votes
1answer
152 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 ...
8
votes
2answers
377 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
0answers
546 views

Relative null-ness

Here, "measure" always means Lebesgue measure on $\mathbb{R}$. This question is partly motivated by my answer ...
4
votes
1answer
126 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 ...
2
votes
2answers
202 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
2answers
117 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?
8
votes
0answers
107 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 ...
6
votes
0answers
241 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, ...