17
votes
Is there a minimal (least?) countably saturated real-closed field?
In Solution d'un problème d'Erdös, Gillman et Henriksen et application à l'étude des homomorphismes de $\mathcal{C}(K)$, Acta Math. Acad. Sci. Hungar. 30 (1977), no.1-2, 113–127 (EuDML), Jean Esterle ...
- 6,190
17
votes
Accepted
Is the field of constructible numbers known to be decidable?
See
Barry Mazur, Karl Rubin, Alexandra Shlapentokh, Defining $\mathbb{Z}$ using unit groups, Acta Arithmetica (Published online: 27 June 2024) DOI: 10.4064/aa230505-6-6
One of the corollaries of our ...
- 186
16
votes
Accepted
Rigid non-archimedean real closed fields
Charles Steinhorn and I have answered this question positively by constructing a rigid non-archimedean real closed field of transcendence degree 2. Our preprint is now posted on arxiv.
https://arxiv....
- 3,460
5
votes
Accepted
Can we see quantifier elimination by comparing semirings?
No, for example consider $T=\mathsf{Th}(\mathbb{N};=,0,1,+)$, i.e. Presburger arithmetic in non-extended signature.
Quantifier elimination does not hold for this $T$: this would require to extend the ...
- 5,546
4
votes
Mostowski's absoluteness theorem and proving that theories extending $0^\#$ have incomparable minimal transitive models
This answer by Farmer S to another question, which Farmer S linked in a comment to this question, proves that if $\alpha$ is the least ordinal that is the height of a transitive model of $\text{ZFC}+0^...
4
votes
Accepted
Two equivalent statements about formulas projected onto an Ultrafilter
First, since I found your notation confusing, I hope you don't mind if I rewrite your Question 1 in more standard notation.
Fix a language $L$. Let $I$ be a non-empty set, and let $(\varphi_i)_{i\in I}...
- 4,742
4
votes
Example of applying real quantifier elimination algorithm for polynomials
Although this question is nearly 5 years old, I'd like to write a proper answer to this in case it helps anyone.
First, I'd suggest not to study Tarski's construction if you are interested in ...
- 41
3
votes
Chevalley's theorem on valuation spectra
This is expanded in Huber's seminar notes on the subject as Satz 1.1.21.:
A set $\{v\in \mathrm{Spv}: \mathrm{Frac}(A/\mathrm{supp}(v)),A(v)\models \phi\}$ where $\phi$ is a quantifier-free formula ...
- 2,374
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