All Questions
Tagged with philosophy model-theory
34
questions
4
votes
0
answers
87
views
(When) are recursive "definitions" definitions?
This is a "soft" question, but I'm greatly interested in canvassing opinions on it. I don't know whether there is anything like a consensus on the answer. Under what conditions (if any) are ...
- 93
0
votes
1
answer
287
views
Set theory and model theory: which set is ZFC?
I have yet another post about what is model theory doing and why is it valid; I hope I can be coherent.
(1) https://mathoverflow.net/questions/23060/set-theory-and-model-theory
(2) What exactly is the ...
- 19
7
votes
1
answer
319
views
How should one understand the "universe of sets"?
One way to understand the axioms of $\mathsf{ZFC}$ is to see them as a describing the "universe of sets" $V$, together with the "true membership relation" $\in$. The universe $V$ ...
- 20.7k
5
votes
2
answers
207
views
Introductory text on logic for those interested in the intersection of logic, algebra, and topology?
I'm a current MA student doing research in formal semantics, which is an application of, among other things, logic and model theory to the study of the semantics of natural languages. I'd like to ...
- 380
0
votes
1
answer
170
views
What are the merits of having a "good" proof system?
Background: My understanding is that model-theoretic semantics (MTS) and proof-theoretic semantics (PTS) differ in the following ways. In MTS, you first define the notion of truth in models and then ...
user1048887
8
votes
1
answer
604
views
Do the everyday mathematician and the model theorist mean the same thing by "truth"?
In Terrence Tao's book Analysis I, the axioms of ZFC are considered to be true statements, and every other true statement in the book is proved from these axioms. Model theory is not mentioned.
This ...
- 792
1
vote
1
answer
85
views
Clarification needed in Definability versus Leibnizian structure chapter on Hamkins's book
In Lectures on the Philosophy of Mathematics, chapter 1, 1.10 Structuralism, there is the sentence:
For example, the real ordered field $<\mathbb{R},+,·,<,0,1>$ is
Leibnizian, since for any ...
- 304
2
votes
1
answer
504
views
Concrete and abstract models of axiomatic systems
In order to prove the consistency of an axiomatic system we must come up with a model. Wikipedia gives the following definition for a model of an axiomatic system:
A model for an axiomatic system is ...
- 256
1
vote
0
answers
125
views
Universal quantifier over an uncountable set
To prove that a segment has the same number of points with half a segment one might say that one can find a bijective function mapping every point from the segment to the half segment. Let' say:
$$\...
- 19
2
votes
0
answers
242
views
What's the connection between the homomorphism theorem in Enderton "A mathematical Intro to Logic" and Lyndon’s Positivity Theorem?
In Enderton p. 96, the homomorphism theorem implies (I state only the essential): Let $h:\cal{A} \rightarrow {\cal B}$ be an epimorphism (i.e., a surjective homomorphism). For any sentence $\alpha$ ...
- 644
3
votes
2
answers
364
views
How do models determine truth values if the external theory is incomplete?
I'm currently learning model theory from Chang and Kiesler's Model Theory: 3rd edition. Something about the basic relationship between syntax and semantics is troubling me.
The book describes ...
17
votes
4
answers
1k
views
In classical predicate logics, why is it usually assumed that at least one object exists?
In classical predicate logic it is commonly assumed that the domain of objects is non-empty. This validates inferences such as $$\forall x Fx \models \exists x Fx$$ as well as, if the identity ...
- 402
0
votes
2
answers
259
views
Formal or informal definability of the standard model of natural numbers
I started a discussion in comments to this answer, but it grew beyond what fits comments, so I'm promoting it to this separate question. Here is a recap:
$\Large\color{green}{\unicode{0x2BA9}}\,$ The ...
- 47.3k
4
votes
2
answers
212
views
Is there a modal logic axiom corresponding to the condition that for every world w there is some world v such that vRw? (Reverse Seriality)
Seriality on certain modal frames - that for every world w there is some world v such that wRv - corresponds to the axiom: $\square$P$\rightarrow$$\diamondsuit$P, assuming a standard interpretation of ...
- 143
9
votes
1
answer
734
views
Is an interpretation just a homomorphism between theories?
I don't understand model theory, but I think I've read enough about it that I can pretend to by parroting the language of model theorists. So instead of asking "how does model theory work," I'm going ...
- 5,040