All Questions
Tagged with model-theory foundations
61
questions
5
votes
1
answer
138
views
Is $\mathsf{ZFC_2}$ "class categorical"?
Let $\mathsf{ZFC_2}$ denote $\mathsf{ZFC}$ with a second-order replacement axiom. It has been discussed in some other answers that every model of $\mathsf{ZFC_2}$ is isomorphic to $V_\kappa$ for some ...
- 6,672
-3
votes
1
answer
243
views
Why Löwenheim–Skolem theorem asserts the non-existence of such predicates in 1st order logic
Suppose there was a predicate, in the language of 1st order $ \mathsf {PA} $, such that it is only true for standard natural numbers i.e. it accepts ALL and ONLY standard natural number, and it ...
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
4
votes
1
answer
68
views
Does $\omega$-consistency depend on the encoding?
A given theory $T$ can interpret the language of arithmetic in different ways, and therefore it seems to me like it is possible that in one of these ways $T$ would be $\omega$-consistent but not in ...
- 612
1
vote
1
answer
95
views
What recursive extensions are there of axiomatic second-order logic.
There are two semantics used for second-order logic, Henkin semantics and standard semantics. It’s easy to make a recursive deductive system $D$ that is sound and complete with respect to Henkin ...
- 10.3k
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
0
votes
1
answer
74
views
Godelian sentences in other first order languages
I've been asked to teach an intro to logic and computation course. This is not a field that I am particularly familiar with, but I am happy to have the chance to learn it - hopefully - properly.
Since ...
user923098
0
votes
0
answers
149
views
How does formalization work in mathematics?
I would be extremely grateful is someone could review/comment/complement my reasoning and understanding of formalization in mathematics.
Let $T$ be a mathematical theory, say real analysis. $T$ is ...
- 538
1
vote
2
answers
266
views
How to construct ZFC from scratch?
I am interested in understanding the foundations of mathematics. This naturally leads to the study of set-theory and its different axiomatizations. My issue is with the different ways to construct ...
- 538
1
vote
0
answers
77
views
Elementarily equivalent models of arithmetic that are not isomorphic.
The book is Predicate Calculus by Goldrei. Given the hint and other similar exercises, this is the only way I know how to go about this:
Take the set $\text{Th}(\mathcal{N}) \cup \{ \textbf{c} \not = \...
- 613
2
votes
1
answer
105
views
Model Theory in Van Dalen
I am reading Van Dalen's Logic and Structure (5th ed.), and I am confused about the following part (Lemma 4.3.8).
$\mathfrak A$ is isomorphically embedded in $\mathfrak B$ $\iff$ $\mathfrak {\hat B}$ ...
- 65
2
votes
2
answers
476
views
What formula of ZFC defines the set of natural numbers?
Let $\mathsf{ZFC}'$ be the extension of $\mathsf{ZFC}$ containing the constant symbol $\Bbb N$, which we take to represent the natural numbers. In order to say that $\mathsf{ZFC}'$ is a definitional ...
- 5,040
0
votes
2
answers
191
views
An example of a statement that is true within a group, but that is unprovable from the group axioms? [closed]
We know that:
If our language is the group language $L_G = \{ e, \cdot \}$ and our theory is the three group axioms:
$$ (i) \exists e \in G: \forall g \in G: eg = ge = g$$
$$ (ii) \forall g \in G \...
3
votes
0
answers
116
views
How to model structures over a proper class?
I am a novice to mathematical logic and model theory, just starting to learn the basics. I'm confused about the difference between defining theories within ZFC and defining theories directly in logic. ...
- 6,672
1
vote
1
answer
330
views
Can metalogic and model theory be formalized?
All of mathematics formulated using ZFC can be "formalized" in the sense that each statement could be translated into a logical string, and each proof can be translated into a formal proof. ...
- 6,672