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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142
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0answers
12k views

Ultrafilters and automorphisms of the complex field

It is well-known that it is consistent with $ZF$ that the only automorphisms of the complex field $\mathbb{C}$ are the identity map and complex conjugation. For example, we have that $|Aut(\mathbb{C})|...
30
votes
0answers
1k views

Peano Arithmetic and the Field of Rationals

In 1949 Julia Robinson showed the undecidability of the first order theory of the field of rationals by demonstrating that the set of natural numbers $\Bbb{N}$ is first order definable in $(\Bbb{Q}, +,...
29
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0answers
1k views

Concerning the various proofs from the axiom of choice that R^3 admits of surprising geometrical decompositions into circles, skew lines and so on: can we prove in any instance that there are no Borel such decompositions? Or that AC is required?

This question follows up on a comment I made on Joseph O'Rourke's recent question, one of several questions here on mathoverflow concerning surprising geometric partitions of space using the axiom of ...
28
votes
0answers
973 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 ...
26
votes
0answers
587 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 ...
23
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466 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 ...
22
votes
0answers
907 views

Supercompact and Reinhardt cardinals without choice

A friend of mine and I ran into the following question while reading about proper forcing, and have been unable to resolve it: Definition. A cardinal $\kappa$ is supercompact if for all ordinals $\...
21
votes
0answers
551 views

Relative null-ness

Here, "measure" always means Lebesgue measure on $\mathbb{R}$. This question is partly motivated by my answer http://math.stackexchange.com/questions/1444498/is-there-a-categorizaiton-system-for-null-...
21
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0answers
423 views

Decidability of equality of elementary expressions

In the following definition the term expression is to be understood as a finite tree built from formal symbols without any predefined meaning assigned to them. Define the set $\mathcal{E}$ of ...
19
votes
0answers
458 views

Density of first-order definable sets in a directed union of finite groups

This is a generalization of the following question by John Wiltshire-Gordon. Consider an inductive family of finite groups: $$ G_0 \hookrightarrow G_1 \hookrightarrow \ldots \hookrightarrow G_i \...
17
votes
0answers
301 views

Can Gentzen-style proofs give omega-consistency and beyond?

In 1936, Gentzen famously showed that Primitive Recursive Arithmetic, plus the assumption that the ordinal $\epsilon_0$ is well-founded, is able to prove Con(PA). But of course, Con(PA) doesn't yet ...
17
votes
0answers
458 views

Souslin trees on the first inaccessible cardinal

This may be well-known or simply deducible from the existing theorems, but I didn't find an answer in my set theory books: Is there a model of $ZFC$ in which there are no $\kappa$-Souslin trees where ...
17
votes
0answers
850 views

Hrushovski's Construction

Zilber expressed a conjecture for $\aleph_{1}$- categorical theories (In the 80s). Zilber's Conjecture: The geometry of any $\aleph_{1}$- categorical structure is one of the following: (a) Trivial (...
17
votes
0answers
397 views

Are there lightweight foundations for arbitrarily extendable objects?

My experience with foundations is rather scant, but I've run into some types of objects that seem to resist the sort of set-theoretic encoding schemes via Kurowski tuples that are rather common for ...
17
votes
0answers
599 views

Antichains of Cardinals in ZF Without Choice…

With the Axiom of Choice, the cardinals form a nice linearly ordered "set". In the absence of the Axiom of Choice, the cardinals form a partially ordered "set". Broadly, I am wondering what ...
16
votes
0answers
718 views

Souslin trees and weakly compact cardinals

In Souslin trees on the first inaccessible cardinal it is asked if it is consistent that there are no $\kappa-$Souslin trees at the least inaccessible cardinal $\kappa$. In this question I would like ...
14
votes
0answers
391 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 ...
14
votes
0answers
374 views

What is the Cantor-Bendixson rank of the space of first order theories?

Let $L$ be the language $\{R\}$ containing a single binary relation symbol. Consider the space $S_0(L)$ of complete, first-order $L$-theories. This is a seperable, compact Hausdorff space; what is its ...
14
votes
0answers
145 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 ...
14
votes
0answers
528 views

Has anyone read/debunked Yessenin-Volpin–Hennix “Beware of the Gödel-Wette paradox”?

A student recently asked me about the status of a 2001 arXiv post, Beware of the Gödel-Wette paradox!, by Alexander Yessenin-Volpin (aka Esenin-Volpin and several other transliterations) and Catherine ...
14
votes
0answers
600 views

Minimal resources for Undecidability of First-Order Logic: the number of variables

It is well-known that First-Order Logic (FO) with a full vocabulary (i.e., a countable numbers of unary predicate symbols, a countable number of binary predicate symbols, etc.) is undecidable. And it ...
13
votes
0answers
289 views

Adding a saturated ideal

Is it consistent that there is no $\omega_2$-saturated ideal on $\omega_1$, but one is introduced by an $\omega_2$-closed forcing? Some motivation: If $\delta$ is a Woodin cardinal, then it remains ...
13
votes
0answers
448 views

Ideas behind Gitik's solution of PCF conjecture

Recently Moti Gitik has refuted Shelah's PCF conjecture (see Short extenders forcings II ) by proving the following theorem: Theorem. Assuming the consistency of infinitely many strong cardinals, one ...
13
votes
0answers
442 views

To what extent does (co)homology of groups made discrete depend on set theory?

There's a well-known paper by Milnor, "On the homology of Lie groups made discrete," that discusses the relation between the homology of a Lie group $G$ and the underlying discrete group $G^\delta$. ...
12
votes
0answers
249 views

If $U,D$ are $\kappa$-complete nonprincipal ultrafilters on $\kappa$ and $j_U(U) = j_D(D)$, is $U=D$?

Here, $\, j_U, \, j_D$ are the canonical elementary embeddings induced by $U,D$ respectively. I note that it is consistent with the existence of a measurable that the answer be yes: it is true in the ...
12
votes
0answers
286 views

Is the universality of the surreal number line a weak global choice principle?

I'd like to consider the principle asserting that the surreal number line is universal for all class linear orders, or in other words, that every linear order (including proper-class-sized) linear ...
12
votes
0answers
193 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 ...
12
votes
0answers
330 views

The topos for forcing in computability theory

My understanding is that forcing (such as Cohen forcing) can be described via a topos. For example this nlab article on forcing describes forcing as a "the topos of sheaves on a suitable site." My ...
12
votes
0answers
343 views

Do all linear orders in this class have computable copies?

This is a question which has been bothering me now for quite some time. I've talked to a number of people about it, and we've shown that a few basic ideas can't work, but other than that haven't made ...
12
votes
0answers
695 views

Can a model of $V\neq L$ contain a class giving the $L$-ordering on all its sets?

This question is inspired by the excellent question by Douglas Ulrich When is $L$-Rank definable in inner models of $V=L$? Suppose $M \in L$ is a countable model of $ZFC$, and furthermore suppose $M \...
12
votes
0answers
313 views

How to measure the strength of Zermelo over bounded Zermelo?

Bounded Zermelo is Zermelo set theory with only bounded separation. It has the same strength as simple type theory or MacLane set theory or ETCS. It is a finitely axiomatized fragment of Zermelo, so ...
12
votes
0answers
286 views

What is the simplest known arithmetic definition of exponentiation?

For $n$ a natural number, let $E_n$ denote the set of bounded arithmetic formulas consisting of n alternating blocks of bounded quantifiers starting with an existential quantifier, followed by a ...
11
votes
0answers
320 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 ...
11
votes
0answers
203 views

Generic $\mathbf{\Sigma}_3^1$-absoluteness for class forcings

In the paper "Generic Absoluteness" by Bagaria and Friedman (http://www.logic.univie.ac.at/~sdf/papers/bagfried.pdf) it is shown that in ZFC generic $\mathbf{\Sigma_3^1}$-absoluteness is false for ...
11
votes
0answers
312 views

On sentences true in all finite groups (revisited)

Let $w$ be a group word with variables $\bar x, \bar y$, where $\bar x=(x_1,\dots ,x_m)$ and $\bar y=(y_1,\dots ,y_n).$ I am interested in the following questions. (1) Is the sentence $(\forall\bar ...
11
votes
0answers
393 views

Decidability of $x^3+y^3+z^3 = c$

I wondering if it is known whether the following problem is algorithmically decidable or undecidable by Turing machines: given an integer c, determine if there are integers $(x,y,z)$ such that $x^3+y^...
11
votes
0answers
497 views

What is known about the reverse mathematics of algebraic number fields?

I know work on the reverse mathematics of countable algebraic field extensions including Galois theory, notably including Dorais, Hirst, and Shafer http://arxiv.org/pdf/1209.4944v2.pdf. But algebraic ...
11
votes
0answers
167 views

Savings property: A transformation which turns nonnegative martingales into uniformly integrable ones

Background I work in a subfield of computability theory called algorithmic randomness. We have been using martingales as long as probability theory (going back to work of von Mises). However, since ...
11
votes
0answers
616 views

Can one exhibit an explicit Kuratowski infinite set without invoking Replacement?

The customary formulation of the Axiom of Infinity within Zermelo-Fraenkel set theory asserts the existence of an inductive set: a set $ I$ with $\varnothing\in I$ such that $x\in I$ implies $x\...
11
votes
0answers
502 views

Diagonal lemma from recursion theorem?

Does Gödel's diagonal lemma follow from Kleene's recursion theorem? I believe the converse is true, by an argument like the following. Let e ↦ θe be a bijection between ω and ...
11
votes
0answers
1k views

Reverse mathematics strength of identically zero polynomials are the zero polynomial

According to wikipedia, the statement "every polynomial over a countable field that is not the zero polynomial has only finitely many roots" is equivalent to RCA0 over RCA0* (which is called ERCA-0 in ...
10
votes
0answers
187 views

Example of $\aleph_1$-categorical linear order

Is it possible to have an $L_{\omega_1,\omega}$-sentence $\phi$ in a vocabulary that includes $<$ that satisfies the following? $<$ is a linear order on a definable subset; $\phi$ is $\aleph_1$...
10
votes
0answers
295 views

A question concerning model theory of groups

Several days ago, Professor Martin Bridson gave a very nice talk in my department. Some questions concerning his talk came into my brain Since I am neither a model theorist nor a algebraist, I am not ...
10
votes
0answers
229 views

A variant of strong ideals, is it consistent?

Is it consistent relative to large cardinals that there is a precipitous ideal on $\omega_1$ forcing a generic elementary embedding $j : V \to M \subseteq V[G]$, such that $j(\omega_1) = \omega_n^V$ ...
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 ...
10
votes
0answers
155 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 ...
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 $V$. ...
10
votes
0answers
316 views

c.c.c forcing notions and adding minimal generic reals

Is the following statement consistent: ``There is no non-trivial c.c.c forcing notion adding a minimal generic real''? The question is related to Prikry's question: Is it consistent that any ...
10
votes
0answers
268 views

Is there a “hereditary” construction for $L$?

Recall that $L$, Godel's constructible universe is constructed by defining the following hierarchy: $L_0=\varnothing$, for a limit ordinal $\delta$, $L_\delta=\bigcup_{\alpha<\delta}L_\alpha$, and ...
10
votes
0answers
205 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 ...