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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-5
votes
1answer
134 views

New hilbert system and theorem [closed]

Let we have following axioms and modus ponens : $$(A1):(B ⇒ (C ⇒B ))$$ $$(A2):((B ⇒ (C ⇒D )) ⇒ ((B ⇒C ) ⇒ (B ⇒D )))$$ $$(A3):( ( B ⇒C) ⇒(¬C ) ⇒ (¬B ))$$ now can we prove following theorem ? $\vdash ...
4
votes
1answer
298 views

Show that the positive existential theory is undecidable

To show that the positive existential theory of $\mathbb{C}[t, e^{\lambda t} \mid \lambda \in \mathbb{C}]$ in the language $\{+, \cdot , ' , 0 , 1, t\}$ is undecidable we have to prove the following: $...
6
votes
0answers
98 views

Finitely presented algebras with isomorphic semilattices of congruences

Let $\mathbb{T}$ be a finitary algebraic theory. For each $\mathbb{T}$-algebra $A$, let $Q (A)$ be the join semilattice of finitely generated congruences on $A$. There is an evident pushforward ...
3
votes
1answer
246 views

How can the critical point of an elementary embedding be omega_1?

I've seen an example of an elementary embedding such that $\omega_1$ is the critical point. I was wondering what's wrong with the following proof that this cannot be: Let $\phi(x_1,x_2)$ be the ...
2
votes
0answers
114 views

Is a finitely generated residually free group “almost LERF”?

Let $G$ be a finitely generated residually free group. (i.e. for each $1 \neq g \in G$ there exists a homomorphism $\tau \colon G \to F$ such that $F$ is a free group, and $\tau(g) \neq 1$.) Let ...
19
votes
3answers
663 views

Is being rational decidable?

Given a real number uniquely defined by a finite system of equations and inequalities with rational coefficients involving the standard elementary functions only. Is it decidable whether this number ...
14
votes
1answer
368 views

Joyal's construction of the spectrum of a commutative ring

I am trying to understand bits and pieces of Lawvere's article Continuously Variable Sets; Algebraic Geometry = Geometric Logic. I'm not doing very well. I know this is a lot to ask, but basically, I ...
2
votes
0answers
70 views

A Question on Provability Logic and Co-Necessitation

The provability logic $GL$ has the characteristic axioms: $K\hspace{15pt}\Box(\alpha\rightarrow \beta)\rightarrow(\Box\alpha\rightarrow\Box\beta)$ $L\hspace{15pt}\Box(\Box \alpha\rightarrow \alpha)\...
3
votes
3answers
234 views

Semantic reflection

Let $\ulcorner \cdot \urcorner$ be a fixed encoding of formulas by numbers, e.g. let $\ulcorner \varphi \urcorner$ denote the Godel number of $\varphi$. Let $T$ be a first-order arithmetic theory, e....
7
votes
0answers
230 views

Proving Richardson's theorem for constants

(I asked this a little over 3 months ago on math.SE, and when I initially re-asked here, no one had responded there. $\:$ After I re-asked here, Eric Towers responded there, since I had forgotten to ...
4
votes
1answer
199 views

What is the right notion of generalized element of a category?

I've been working out how the internal language of a category C extends to taking the category itself as a type. The most obvious way to interpret $X : \mathbf{C}$ is, of course, that $X$ is an ...
18
votes
2answers
549 views

Removing large cardinals from an uncountable transitive model

The usual way of removing large cardinals from a given model of set theory is to cut off the model below the least large cardinal of interest. But this method may have dramatic effects on the external ...
6
votes
2answers
254 views

The role of the index set in the product of uncountably many topological spaces

Let $‎\langle‎ ‎‎X_i,\mathcal{T}_i \rangle_{i\in I}$ be a family of topological spaces. Consider $X=\prod_{i\in I} X_i$ with product topology. Question. Is there a topological property that holds in $...
3
votes
2answers
235 views

Do limit groups satisfy Howson's theorem?

Let $G$ be a limit group, and let $A,B \leq G$ be finitely generated subgroups generating $G$ (i.e. $\langle A \cup B \rangle = G$). Must $A \cap B$ be finitely generated? Recall that a limit ...
2
votes
1answer
226 views

Effectively non-recursiveness of some sets

A set $A$ is completely productive if there exists a computable function $f$ such that for every $e$, $f(e)\in (A-W_e)\cup (W_e-A)$‎. ‎A set is effectively non-recursive if it is r.e‎. ‎and its ...
10
votes
1answer
206 views

Is there a (first-order) sentence which admits $(\aleph_2,\aleph_0)$ iff a Kurepa tree exists?

In Chang and Keisler's Model Theory I came across the following theorem (Theorem 7.2.13): Theorem There exists a (first-order) sentence $\sigma$ such that for all infinite cardinals $\alpha$, $\sigma$...
7
votes
1answer
136 views

Consistency Strength of “HC is elementary in V[G]”

Let $P$ be the Levy-collapse of the ordinals, so $P$ is a class forcing notion that makes every ordinal countable. Note that since $P$ is weakly homogeneous, for any formula $\phi(\overline{a})$ ...
5
votes
0answers
130 views

Sacks minimality without choice

The usual argument for the minimality result for Sacks forcing uses choice. Theorem (Sacks): Let $s \subseteq \mathbb S_\kappa$ be generic for the forcing to add a Sacks subset to $\kappa$, where $\...
11
votes
1answer
414 views

Is it consistent with ZFC that no nontrivial forcing notion has automatic mutual genericity?

A nontrivial forcing notion $\newcommand\Q{\mathbb{Q}}\Q$ exhibits automatic mutual genericity, if whenever $G,H\subseteq\Q$ are distinct $V$-generic filters (existing, say, in some forcing extension ...
7
votes
2answers
213 views

How to extend Morley's omitting type theorem to uncountable languages?

In his 1965 paper Omitting Classes of Elements (found in The Theory of Models: Proceedings of the 1963 International Symposium at Berkeley, published by North-Holland Publ. Co., Amsterdam (1965)), ...
7
votes
0answers
151 views

$V$ as a $HOD$ of its class generic extension

By an old result of Roguski, The theory of the class $HOD$, any model $V$ of $ZFC$ has a class generic extension $V[G]$ such that $HOD$ of $V[G]$ equals $V$. This result is also stated and generalized ...
9
votes
0answers
227 views

(A little bit) Beyond the E-recursive

The E-recursive functions are a particular generalization of classical recursion theory to the entire set-theoretic universe, $V$. They are defined via a schemes: see http://www.math.harvard.edu/~...
10
votes
1answer
372 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
196 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
168 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
187 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 ...
14
votes
0answers
142 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
187 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 number....
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
426 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
155 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
69 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
575 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
400 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
473 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
391 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: \omega\...
7
votes
1answer
341 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
315 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
253 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
252 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 $\mathbb{...
5
votes
2answers
188 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
204 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
142 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
94 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 ...