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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.
23
votes
1
answer
485
views
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
18
votes
2
answers
1k
views
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 …
- 13.2k
16
votes
1
answer
738
views
Is there a pair of non-isomorphic structures each of which is isomorphic to an ultrapower of...
Does there exist a pair of non-isomorphic structures $\mathfrak{A}$ and $\mathfrak{B}$ as well as sets $I$ and $J$ and ultrafilters $\mathcal{U}$ on $I$ and $\mathcal{F}$ on $J$ such that $\mathfra …
- 13.2k
13
votes
1
answer
422
views
What is the proof-theoretic ordinal of bare $\mathsf{NFU}$?
On the Stanford Encyclopedia of Philosophy article on alternative axiomatic set theories, it is stated without reference that bare $\mathsf{NFU}$ (i.e., $\mathsf{NFU}$ without the axiom of infinity) i …
- 13.2k
13
votes
1
answer
496
views
Does the statement 'there exists a first-order theory $T$ with no saturated models' have any...
Exercises 6, 7, and 8 in section 10.4 of Hodges' big model theory textbook contain an outline of a proof of the consistency of the following statement
There exists a countable first-order theory $ …
- 13.2k
12
votes
0
answers
241
views
Is there a characterization of the class of first-order formulas that are closed in every co...
Fix a relational language $\mathcal{L}$. (I don't think relational really matters that much but I don't want to worry about it.) A topological $\mathcal{L}$-structure is an $\mathcal{L}$-structure $M$ …
- 13.2k
10
votes
0
answers
235
views
Is there an expansion of $(\mathbb{N},+,<)$ by a pairing function that is still NIP?
I was told once that there is a theory consisting of just a pairing function that is stable, although I cannot find a reference for it. This motivated my question, which is essentially the title, alth …
- 13.2k
10
votes
0
answers
164
views
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
10
votes
1
answer
242
views
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 …
- 13.2k
8
votes
1
answer
322
views
Is there a stable structure on $[0,1]$ that approximates every continuous function?
The $n$-dimensional form of the Weierstrass approximation theorem is the statement that polynomial functions are dense under the $\ell_\infty$-norm in the space of continuous functions on $[0,1]^n$ fo …
- 13.2k
7
votes
1
answer
141
views
Is there a pseudofinite group with a quantifier-free instance of the order property?
Recall that a group $G$ is pseudofinite if every first-order sentence $\varphi$ (in the language of groups) satisfied in $G$ is also satisfied in some finite group. Also recall that an instance of the …
- 13.2k
7
votes
0
answers
232
views
How much is known about the consistency strength of toposes and topos-like categories?
It's a well-known fact that the theory of a well-pointed topos with a natural numbers object (NNO) has the same consistency strength as MacLane set theory (also known as bounded Zermelo). There are ma …
- 13.2k
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
7
votes
0
answers
104
views
Can superstability of a countable theory be characterized in terms of not 'weakly trace inte...
The concept that I want to think about in this question is very similar to trace definability, introduced here and also independently by Guingona, but is somewhat different from it. There's a very goo …
- 13.2k
7
votes
0
answers
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 …
- 13.2k