All Questions
Tagged with model-theory foundations
22
questions
5
votes
1
answer
534
views
The "first-order theory of the second-order theory of $\mathrm{ZFC}$"
$\newcommand\ZFC{\mathrm{ZFC}}\DeclareMathOperator\Con{Con}$It is often interesting to look at the theory of all first-order statements that are true in some second-order theory, giving us things like ...
- 4,857
7
votes
1
answer
1k
views
Propositional calculus, first order theories, models, completeness
In the usual context of model theory one studies first order theories: the Gödel completeness theorem asserts that $\varphi$ is a theorem of a theory $T$ (i.e. $\varphi$ is provable from the axioms of ...
- 9,240
8
votes
0
answers
145
views
How strong is exponentiation with only open induction? (Or: "how low can we go?")
Do the strongest theories currently known to be unconstrained by Tennenbaum's theorem ($IOpen$ and some modest extensions) remain so when augmented with a definition of exponentiation and axiom $\...
- 3,581
6
votes
0
answers
148
views
Does every Tarski plane embed into a 3-dimensional Tarski space?
By a Tarski space I understand a mathematical structure $(X,B,\equiv)$ consisting of set $X$, a betweenness relation $B\subseteq X^3$ and a congruence relation ${\equiv}\subseteq X^2\times X^2$ ...
- 41.1k
6
votes
1
answer
274
views
The algebraic structure of a line in a (Tarski) plane
By a Tarski plane (resp. plane) I understand a mathematical structure $(X,B,\equiv)$ consisting of a set $X$, a ternary betweenness relation $B\subseteq X^3$ and the 4-ary congruence relation ${\equiv}...
- 41.1k
6
votes
1
answer
408
views
Joyal arithmetic universes and the Box operator
Last month @godelian, alias Christian Espíndola, in a FOM post has mentioned Joyal's proof of Godel's second incompleteness via the so-called Arithmetic Universes, introduced by Joyal around 1973, ...
- 7,693
3
votes
0
answers
180
views
Can we interpret ZFC in GEM?
I have long held similar views as put forth here. As professor Hamkins points out though, $(M,\subseteq)$ is insufficient to model most of mathematics. However, this is unsatisfactory to me, as this ...
- 424
7
votes
2
answers
2k
views
What is a good definition of a mathematical structure?
At the moment I am writing a textbook in Foundations of Mathematics for students and trying to give a precise definition of a mathematical structure, which is the principal notion of structuralist ...
- 41.1k
3
votes
0
answers
289
views
What does second order set theory give us that is new?
There is a natural analogy between the theories PA and ZFC. See the linked question by Gro-Tsen here.
Peano arithmetic (PA) is a first order approximation to the natural numbers. As is well known, ...
- 18.3k
5
votes
1
answer
895
views
Smallest ordinal modelling $\aleph_1$?
Let $X_1$ be the class of all ordinals $\alpha$ such that there exists a transitive model $M$ of ZF(C) such that $M$ thinks that $\alpha$ is $\aleph_1$.
Every class of ordinals has a minimum element (...
- 385
14
votes
1
answer
868
views
A peculiarity of Henkin's 1950 proof of completeness for higher order logic
My question concerns Henkin's original (1950) completeness proof in Completeness in the theory of types for classical higher order logic and type theory relative to so-called general models.
His 1950 ...
- 629
4
votes
0
answers
422
views
What are the requirements of a foundational theory?
There are multiple languages to describe all of mathematics, and there are some equivalences between them, some more successful then others.
My question is can we describe some requirements (in some ...
- 753
38
votes
7
answers
6k
views
Is V, the Universe of Sets, a fixed object?
When I first read Set Theory by Jech, I came under the impression that the Universe of Sets, $V$ was a fixed, well defined object like $\pi$ or the Klein four group. However as I have read on, I am ...
- 690
4
votes
3
answers
873
views
Compactness of existential second order logic and definability of certain quantifiers
It is known (as a slogan) that the "existential fragment of second-order logic (ESO) is compact".
My first question is:
(1) Is ESO compact for:
(a) uncountable languages
(b) languages with ...
- 135
8
votes
0
answers
247
views
Is there a notion analogous to separability but requiring definable rather than countable sets?
Among models of $\lambda$-calculus, some like the Bohm tree model have the property that every element is a directed sup of definable elements, whereas others like the $D_\infty$ and $P(\omega)$ ...
- 211