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Results tagged with lo.logic
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user 83901
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, theories of truth, truth revision, consistency.
5
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0
answers
138
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If $\omega_1$ is not inaccessible in $L$, how hard can it be to find a non-measurable $\Sigm...
In his wonderfully titled paper Can you take Solovay's inaccessible away? Shelah showed that if every $\mathbf{\Sigma}^1_3$ set of reals is Lebesgue measurable, then $\omega_1$ is an inaccessible card …
- 13.2k
2
votes
0
answers
139
views
Can a path in Kleene's $\mathcal{O}$ enumerate all of the computable reals via uniform diago...
It's a well-known fact that there are computable diagonalization functions on Baire space $\mathbb{B} = \mathbb{N}^\mathbb{N}$ (i.e., functions which take a sequence $(r_i)_{i\in \mathbb{N}}$ of eleme …
- 13.2k
10
votes
1
answer
242
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How much of second-order arithmetic do you need for $\mathbf{\Sigma}^1_1$-determinacy to giv...
This is in some sense a follow-up to this question.
The answer there says that over $\mathsf{Z}_2$ (second-order arithmetic), (boldface) $\mathbf{\Sigma}^1_1$-determinacy is enough to entail the exist …
- 7,545
18
votes
2
answers
1k
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What notable theorems cannot be automatically proven without choice using Shoenfield absolut...
There have been a couple of recent questions, here and here, regarding the role of the axiom of choice in real-analytic results with applicability to general relativity. This lead me to look at some …
- 4,426
23
votes
1
answer
485
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Can a non-standard model $M$ of $\mathsf{ZF}$ contain an internally infinite set that is ext...
Cardinalities in non-standard models of $\mathsf{ZF}$ are not generally reflected externally, but certain internal facts about cardinalities are always externally reflected (e.g., any internally infin …
- 13.2k
2
votes
0
answers
166
views
Which first-order theories have full indiscernible extraction?
Stable theories have the following useful property, which I will state in a sub-optimal way for simplicity's sake:
Fact 1. If $T$ is $\lambda$-stable for some $\lambda \geq |T|^+$, then for any set o …
CommunityBot
- 1
10
votes
0
answers
164
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How nice can sets of reals be under $\mathsf{ZF} + \mathsf{BPI}$?
It's well known that the full axiom of choice is not needed to prove the existence of non-measurable subsets of $\mathbb{R}$. In particular, the Boolean prime ideal theorem ($\mathsf{BPI}$) is suffici …
- 13.2k
6
votes
0
answers
82
views
Is there an unstable NSOP theory in which all invariant types are coheirs?
There are some first-order theories $T$ with the property that for any $M \models T$, the only $M$-invariant global types are those that are coheirs over $M$ (i.e., finitely satisfiable in $M$). Two e …
- 13.2k
4
votes
0
answers
82
views
When do Borel propositional theories have topologically tame truth assignments?
Let $(P_r)_{r \in \mathbb{R}}$ be an $\mathbb{R}$-indexed family of propositional variables. Let $\mathcal{L}$ be the collection of all propositional sentences formed from the variables $(P_r)_{r \in …
- 20.7k
6
votes
1
answer
219
views
How often are forcing extensions of countable computably saturated models of $\mathsf{ZFC}$ ...
Recall that given a finite language $\mathcal{L}$, we say that an $\mathcal{L}$-structure is computably saturated (or recursively saturated) if for any computable set $\Sigma(\bar{x},y)$ of $\mathcal{ …
- 17.7k
6
votes
0
answers
99
views
Does stable embeddedness improve two-cardinal behavior?
Let $T$ be a first-order theory with two designated unary predicates $P$ and $Q$. We'll say that $P$ is bounded over $Q$ if for every $\kappa$, there is a $\lambda$ such that if $M \models T$ and $|Q^ …
- 13.2k
5
votes
1
answer
275
views
Which arithmetical sentences have no counterexamples in the sense of Kreisel?
It is a well-known fact that given a first-order sentence $\psi$ in prenex normal form $\forall x_1 \exists y_1 \forall x_2 \exists y_2 \dots \forall x_n \exists y_n \theta(x_1,\dots,x_n,y_1,\dots,y_n …
- 5,831
7
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204
views
Which first-order theories admit a compact-like superstructure?
Positive set theory is an approach to rectifying Russel's paradox by restricting the syntactic form of formulas for which we allow comprehension. It can be motivated by the construction of certain top …
- 20.7k
7
votes
0
answers
151
views
Is there a complete characterization of hyperimaginaries in $\mathsf{DLO}$?
Recall that in a first-order theory $T$, a hyperimaginary is an equivalence class of some type-definable equivalence relation $E(x,y)$ (with $x$ a possibly infinite tuple of parameters). The $E$-equiv …
- 13.2k
6
votes
1
answer
500
views
Upward Löwenheim–Skolem theorem for well-ordered models with/without measurable cardinals
Consider a complete first order theory $T$ whose language contains a binary predicate $\leq$. Assume that $T$ has an uncountable model that is well-ordered by $\leq$ so that this question isn't stupid …
- 236k