All Questions
Tagged with foundations model-theory
22 questions
5
votes
1
answer
596
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 ...
- 4,897
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 ...
- 9,330
8
votes
0
answers
156
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 $\...
- 3,612
6
votes
0
answers
153
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 ${\equiv}\subseteq X^2\times X^2$ ...
- 41.8k
6
votes
1
answer
282
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}...
- 41.8k
6
votes
1
answer
429
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, ...
- 7,853
3
votes
0
answers
184
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 ...
- 433
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 ...
- 41.8k
3
votes
0
answers
301
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, ...
- 18.7k
6
votes
1
answer
935
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 (...
- 395
14
votes
1
answer
900
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 ...
- 639
4
votes
0
answers
424
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 ...
- 773
39
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 ...
- 750
4
votes
3
answers
915
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
248
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
1
vote
0
answers
257
views
Is there a non-trivial consistency preserving transformation?
In set theory "equiconsistency" (and not "consistency") of the theories is the main part of researches. So we usually try to construct a new model using a given one. In the ...
user36136
15
votes
4
answers
2k
views
Where is the end of universe?
In some sense the empty set ($\emptyset$) and the global set of all sets ($G$) are the ends of the universe of mathematical objects. The world which $ZFC$ describes has an end from the bottom and is ...
user36136
38
votes
4
answers
4k
views
Illustrating Edward Nelson's Worldview with Nonstandard Models of Arithmetic
Mathematician Edward Nelson is known for his extreme views on the foundations of mathematics, variously described as "ultrafintism" or "strict finitism" (Nelson's preferred term), which came into the ...
- 4,587
12
votes
1
answer
1k
views
Elementary Equivalence =? Homotopy Equivalence
One of the most interesting novelties in recent foundational studies is Voevodsky's Homotopical Type Theory project (see here).
Finally homotopy theory ideas have entered in a royal fashion the ...
- 7,853
1
vote
2
answers
832
views
Intension vs. Extension: Coextensive relations in model and set theory
(originally posted at MSE as Same same but different: Coextensive relations in model and set theory, slightly modified)
The official definition of a structure in model theory in its presumably most ...
- 9,720
8
votes
2
answers
863
views
Consistent hierarchy of axiomatic systems
First of all, I am not an expert in model theory. I just want to get my personal view on the foundations of mathematics straight.
I just learned in Sergey Melikhov's answer to another question ...
- 25.5k
5
votes
5
answers
1k
views
Concrete models of abstract structures
Most mathematicians seem to be contented with the fact, that abstract structures cannot be directly modelled as such in a set theory without ur-elements. What seems to me the standard stance: Set ...
- 9,720