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Results tagged with model-theory
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user 43439
Model theory is the branch of mathematical logic which deals with the connection between a formal language and its interpretations, or models.
33
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2
answers
2k
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Quantifier complexity of the definition of continuity of functions
This was previously asked at MSE, but I was told to ask it on MO. Consider the structure $(\mathbb{R};+,-,*,0,1,<)$. We adjoin to it a unary function $f$ defined everywhere on the set of real numbers, …
- 2,013
7
votes
1
answer
450
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Are no infinite subsets of the set of all propositional atoms definable in this structure, e...
I asked this on Math Stack Exchange, but apparently no one paid attention to it. So, I am asking it again, filling in the background necessary to understand it.
Consider a countably infinite set $P$ o …
- 2,013
4
votes
1
answer
273
views
What is an axiomatization of the equality-free theory of antisymmetric relations?
An antisymmetric relation is defined as a binary relation $R$ on a set $S$ such that $(xRy \land yRx) \rightarrow x=y$, for all $x,y$ in $S$. Certainly, they can't be defined in first-order logic with …
- 2,013
11
votes
1
answer
419
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Is there a finitely axiomatizable class of structures whose equality-free theory is not fini...
This was originally an MSE question, but I was told to ask it on MathOverflow. Does there exist a class $C$ of $L$-structures for a finite signature $L$, which is finitely axiomatizable in first-order …
- 2,013
0
votes
1
answer
125
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An infinite Leibnizian structure in a finite language with precisely $n$ definable elements
This question was inspired by Joel David Hamkins's excellent question on Leibnizian structures with no definable elements. Let $n$ be a positive integer. Is there an infinite structure in a finite lan …
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27
votes
6
answers
2k
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Is this theory the complete theory of the real ordered field?
We know that the real ordered field can be characterized up to isomorphism as a complete ordered field. However this is a second order characterization. That raises the following question. Consider th …
- 2,013
13
votes
0
answers
229
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Is there a finite equational basis for the join of the commutative and associative equations?
I asked this on math stack exchange, but I was told to post it on mathoverflow. Consider the lattice of equational theories of a single binary operation $*$. The meet of the theory axiomatized by the …
- 2,013
10
votes
0
answers
411
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Equational theory in the signature (+,*,0,1) of sedenions and beyond
Consider a Cayley-Dickson algebra $(X,+,∗,0,1)$, that is an algebra generated from the reals by the Cayley-Dickson construction. From complexes to quaternions, we lose commutativity of multiplication, …
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8
votes
2
answers
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Is the equational theory of groups axiomatized by the associative law?
Consider the class of groups in the signature {*}. Is the equational theory of that class axiomatized by the associative law? I asked this on math stack exchange but I didn't receive a satisfactory an …
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