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Results tagged with lo.logic
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user 11640
first-order and higher-order logic, model theory, set theory, proof theory, computability theory, formal languages, definability, interplay of syntax and semantics, constructive logic, intuitionism, philosophical logic, modal logic, completeness, Gödel incompleteness, decidability, undecidability, theories of truth, truth revision, consistency.
7
votes
1
answer
204
views
Finitistic interpretation of Nelson's internal set theory
What does “standard” in internal set theory really mean?
Is it secretly a way of reconciling conventional mathematics with (ultra)finitism?
Until recently I thought “standard” was just a way of talkin …
- 15.9k
4
votes
Accepted
Set theoretical foundations for derived categories
Fundamentally, working in NBG is not much different from working in ZFC, except that you are allowed one level of freedom to form collections of sets that are not themselves sets.
As such, you still h …
- 15.9k
11
votes
1
answer
414
views
Examples of natural algebraic irreflexive relations
To motivate the question, consider the theory of rings.
Define $x \parallel y$ to mean $\exists w \exists z .((x - y) z = w (x - y) = 1)$, or in words, "$x - y$ is a unit".
Then $\parallel$ is a binar …
- 15.9k
8
votes
Accepted
Multiplicative group of a ring as a morphism of theories
The functor sending a (not necessarily commutative) ring to its group of units is induced by a morphism of cartesian (= finite limit) theories.
More generally, suppose given (small!) cartesian theorie …
- 15.9k
16
votes
2
answers
674
views
How to formulate the univalence axiom without universes?
The standard formulation of the univalence axiom for a universe type $U$ is that, for all $X : U$ and $Y : U$, the canonical map $(X =_U Y) \to (X \simeq Y)$ is an equivalence.
As we (usually) cannot …
- 66.8k
17
votes
Major applications of the internal language of toposes
I don't know if this counts as an application of the internal language or as an avoidance of it, but I think it is worth listing anyway.
In the development of homological algebra and homotopy theory i …
- 15.9k
3
votes
What are some interesting hyperdoctrines that are not classical models?
Every hyperdoctrine is "syntactic", in the sense that given any hyperdoctrine you can construct a theory whose syntactic hyperdoctrine is equivalent to the one you start with.
Thus, hyperdoctrines cor …
- 15.9k
2
votes
0
answers
146
views
Are partial elements necessary in boolean-valued models?
It seems to me that there is a difference in the treatment of "partial" elements in boolean-valued models in set theory vs topos theory: in set theory, one usually only considers "global" elements of …
- 15.9k
34
votes
3
answers
3k
views
What is the theory of local rings and local ring homomorphisms?
It is well-known that the category of local rings and ring homomorphisms admits an axiomatisation in coherent logic. Explicitly, it is the coherent theory over the signature $0, 1, -, +, \times$ with …
- 2,821
23
votes
1
answer
962
views
Are there axioms satisfied in commutative rings and distributive lattices but not satisfied ...
Consider the language of rigs (also called semirings): it has constants $0$ and $1$ and binary operations $+$ and $\times$. The theory of commutative rigs is generated by the usual axioms: $+$ is asso …
- 48.8k
14
votes
1
answer
1k
views
Is it possible for a theorem to be constructive only in a non-constructive metatheory?
There are several theorems in category-theoretic logic which say something like, "any proposition in X logic that is provable in topos logic assuming (the law of excluded middle and) the axiom of choi …
- 55
8
votes
Accepted
Posets (partially ordered sets) in equational logic
No. The category of models of an equational theory (i.e. a variety in the sense of universal algebra) is always a regular category, but the category of posets is not regular.
- 15.9k
7
votes
0
answers
130
views
Finitely presented algebras with isomorphic semilattices of congruences
Let $\mathbb{T}$ be a finitary algebraic theory. For each $\mathbb{T}$-algebra $A$, let $Q (A)$ be the join semilattice of finitely generated congruences on $A$. There is an evident pushforward opera …
- 15.9k
10
votes
Accepted
If two structures are elementarily equivalent, is there a zigzag of elementary embeddings be...
The Keisler–Shelah theorem implies that the following are equivalent:
$M$ and $N$ are elementarily equivalent.
For some set $X$ and some ultrafilter $U$ on $X$, $M^X / U$ and $N^X / U$ are isomorphi …
- 15.9k
6
votes
2
answers
451
views
When are all greater cardinals sharply greater?
Makkai and Paré introduced the following binary relation on regular cardinals: given $\kappa$ and $\lambda$, $\kappa \vartriangleleft \lambda$ (read, $\kappa$ is sharply less than $\lambda$) when $\k …
- 627