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Results tagged with lo.logic
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user 152899
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.
6
votes
Accepted
Are there good mutually interpretable axioms for synthetic Euclidean and hyperbolic geometry?
Here is an expanded version of my previous comments. There are a lot of things to check here and I haven't.
In my opinion the right axioms for Euclidean/Hyperbolic geometry are the Tarski axioms. Tars …
- 1,689
29
votes
2
answers
785
views
Does $\mathrm{SO}(3)$ act faithfully on a countable set?
Let $\mathrm{SO}(3)$ be the group of rotations of $\mathbb{R}^3$ and let $S_\infty$ be the group of all permutations of $\mathbb{N}$. Is $\mathrm{SO}(3)$ isomorphic to a subgroup of $S_\infty$?
This q …
- 1,689
23
votes
Can one show that the real field is not interpretable in the complex field without the axiom...
(1) $\mathbb{C}$ is stable, (2) $\mathbb{R}$ is unstable, and (3) stability is preserved under interpretations. Of course the usual development of stability uses a lot of choice, but (2) and (3) are e …
- 1,689
13
votes
0
answers
436
views
The original Erdős-Volkmann ring problem
The Erdős-Volkmann ring problem and its solution are famous, but the original problem is actually still open. I'll describe this and a related problem from geometric measure theory, I think both of th …
- 1,689
9
votes
1
answer
326
views
Does the random graph interpret the random directed graph?
The random graph is the Fraisse limit of the class of finite graphs, the random directed graph is the Fraisse limit of the class of directed graphs, a directed graph is just a set with a binary relati …
- 1,689
9
votes
Accepted
Is there a complete characterization of ordered fields without definable proper subfields?
This is an interesting question. We know some things about this, but we do not have a characterization of fields with this property. As Wojowu says above the restriction to countable fields doesn't he …
- 1,689
3
votes
Non-set-theoretic consequences of forcing axioms
Farah's proof that all automorphism's of the Calkin algebra are inner under ZFC + Open coloring axiom,. The Calkin algebra is the quotient of the algebra of continuous linear operators on a separable …
- 1,689
5
votes
Applications of Robinson's consistency theorem in algebra?
I don't know of a nice purely algebraic application off the top of my head, I don't even know of many applications to model theory. I was actually pretty happy with I found some application of Craig i …
- 1,689
17
votes
Is (Z,+,0,1,P2,P3) decidable?
Christian Schulz (a grad student at Urbana) and Philipp Hieronymi have recently shown that $(\mathbb{Z},+,<,2^{\mathbb{N}},3^{\mathbb{N}})$ is undecidable. And I believe they prove this for $(\mathbb{ …
- 1,689
5
votes
Why are model theorists so fond of definable groups?
Alex's answer is very good, but there is another reason that is worth mentioning. One of the main goals of model theory is to known when first order theories of interest are interpretable in other the …
- 1,689