All Questions
Tagged with model-theory abstract-algebra
143
questions
2
votes
0
answers
83
views
Comparing mathematical objects by the "rigidity" of their definitions
A loose interest of mine recently has been ordering mathematical objects by how "combinatorial" their study is, in broad terms.
I consider the study of a mathematical object more “...
- 497
1
vote
1
answer
74
views
Power functions in an o-minimal expansion of an ordered field
Let $\mathfrak{R}$ be a fixed o-minimal expansion of an ordered field $(R,<,+,\cdot,-,0,1)$ , A power function is a definable endomorphism of the multiplicative group $(Pos(R) , \cdot ,1)$ .
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- 535
2
votes
1
answer
91
views
Confusion about model-theoretic argument applied to algebraically/real closed fields
I'm taking a first course in Model Theory. There's a kind of argument that is repeatedly used by my lecturer but I just can't get my head around it.
Example
Hilbert's 17th problem: Every positive ...
0
votes
1
answer
132
views
How to prepare myself for postgraduate study of mathematical logic [closed]
I am a year 2 undergraduate student in mathematics, and I have been quite interested in topics like the foundation of mathematics, mathematical logic, set theory, etc. I have studied the basics of ...
- 123
5
votes
1
answer
91
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Transfinite Construction in Differential Fields proof
I'm learning differential fields theory and given my background in model theory I found this book. On p. 203 I find this:
LEMMA 4.7.6. Let $K$ be a differential field, let $P \in K\{Y\} \neq$ be ...
- 109
2
votes
1
answer
213
views
How to express in a first-order language that a graph is $k$-colorable? [closed]
do you think it is possible to express via first-order formulas that an undirected graph is $k$-colorable?
I am thinking about a structure $A=\{V, E \}$ that if it satisfies that, than it is $k$-...
user1140231
2
votes
3
answers
329
views
What are some examples of non isomorphic countable algebraically closed fields of characteristic zero? Or they don´t exist?
I was reading some model theory, and saw stated that $ACF_p$ (the first order theory of algebraically closed fields of characteristic $p$) is $\kappa$-categorical for all $\kappa > \aleph_0$. Is ...
- 581
4
votes
1
answer
212
views
Ultraproduct of polynomial rings
There are several instances of families $ (K_i)_{i \in I}$ and $(L_i)_{i \in I}$ of fields $K_i$ and $L_i$ such that the ultraproducts $\prod_{i \in I} K_i/\mathcal U$ and $\prod_{i \in I} L_i/\...
- 1,780
2
votes
0
answers
51
views
Model companionship result for integral domains as axiomatized in signature with additive and multiplicative inverses
Consider the theory of integral domains in signature $\{+,\cdot,-,0,1\}$ where $+,\cdot$ stand for addition and multiplication and $-$ for additive inverse. Then its model companion is the theory of ...
3
votes
2
answers
157
views
If F is a compact topological field, then F is finite.
Im trying to see why every compact topological field must be finite.
Assuming the topological space is not the trivial topology.
Also: Does compact imply limit point compact in a topological field??
- 171
7
votes
1
answer
205
views
Isomorphism Classes of Real Closed Subfields of $\Bbb C$
The real line $\Bbb R$ is a maximal real closed subfield of the complex plane $\Bbb C$. How many such maximal real closed subfields exist(up to isomorphism)? Is there a way to see that there must be ...
- 171
0
votes
1
answer
45
views
If M holds DCC then Th(M) is not $\omega$-stable
I have some questions regarding the proof that if M is a group and Th(M) is $\omega$-stable then there is no infinite, strictly decreasing sequence of definable subgroups, $R_0\subsetneq R_1\subsetneq ...
user957499
0
votes
1
answer
84
views
How are weak direct powers ultrapowers?
From “ULTRAPRODUCTS AND ELEMENTARY CLASSES” by Keisler:
The definition of ultraproduct, and more generally of reduced product, which we shall adopt here was first given by FRAYNE, SCOTT, and TARSKI ...
- 5,965
2
votes
1
answer
97
views
Definition of ring homomorphism in Bosch's book
In Bosch's Algebraic Geometry and Commutative Algebra, I see that the definition of a ring homomorphism is a map that preserves the two operations and the unity; it isn't mentioned that it must ...
- 1,649
1
vote
0
answers
62
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Is there a term for two theories that have the same set of observable consequences?
I'll express the relationship I have in mind in something like model theory, but there might be analogous relationships in abstract algebras or category theory. I'm looking for a name for the ...
- 555