Intereting Posts

Proving Thomae's function is nowhere differentiable.
Borel Measures and Bounded Variation
Kindle as a Tool for Mathematicians?
Functional Analysis – Banach-Steinhaus theorem
In how many ways can a number be expressed as a sum of consecutive numbers?
How to evaluate $\int_0^x\vartheta_3(0,t)\ dt$?
Replicating a cosine graph with sine, given transformations?
Term independent of x and y in this expansion
I almost quit self-studying mathematics, but should I continue?
Calculate $1\times 3\times 5\times \cdots \times 2013$ last three digits.
Getting a Distributive-Lattice from Poset or, equivalently, a Greedoid from a Poset
definite integral without using complex line integral
Non-associative, non-commutative binary operation with a identity
Can morphisms in the category Set be partial functions?
One-point compactification of manifold

Usually, in model theory, one presupposes that structures (models) are non-empty. I don’t like this (related: What's the deal with empty models in first-order logic?). So let us explicitly permit empty structures.

My requests are:

- Can you give a complete calculus of first-order logic that works for empty structures too? By “working for empty structures too”, I mean that if the demanded calculus proves a sentence, then this sentence should hold in all structures, also in the empty structures.
- Can you give a completeness proof for this calculus?

You may wonder: *why don’t you look in the standard logic texts, where a complete calculus + proof of the completeness should be given?*

But my problem is: In every logic text which allows empty models, they do not give a complete calculus and proof the completeness. Instead, they prove the compactness theorem and other standard results with purely model theoretic methods, rather than proof theoretic methods. Examples of such texts:

- Do we know if there exist true mathematical statements that can not be proven?
- Show that $A \setminus ( B \setminus C ) \equiv ( A \setminus B) \cup ( A \cap C )$
- Can a basis for $\mathbb{R}$ be Borel?
- What's a non-standard model of Tarskian Euclidean geometry?
- What is the formal definition of a variable?
- Understanding an Easy Relative Consistency Proof

- http://math.uga.edu/~pete/modeltheory2010FULL.pdf
- http://www.math.uni-hamburg.de/home/geschke/teaching/ModelTheory.pdf

- Aftermath of the incompletness theorem proof
- Albert, Bernard and Cheryl popular question (Please comment on my theory)
- Why is this true? $(\exists x)(P(x) \Rightarrow (\forall y) P(y))$
- What is the difference between $\omega$ and $\mathbb{N}$?
- Classifying Types of Paradoxes: Liar's Paradox, Et Alia
- Non-standard models of arithmetic for Dummies (2)
- Why is mathematical induction a valid proof technique?
- Recommendation on a rigorous and deep introductory logic textbook
- 'Algebraic' way to prove the boolean identity $a + \overline{a}*b = a + b$
- axioms of equality

There is a full treatment of empty-domain models in MENDELSON: Introduction to Mathematical Logic. You will see there why they are a special case of First Order Logic.

- Why does the amoeba shrink to its skeleton when we go to infinity?
- Finding counterexamples: bijective continuous functions that are not homeomorphisms
- Euler characteristic of a covering space
- Cellular Boundary Formula
- A natural number multiplied by some integer results in a number with only ones and zeros
- Geodesic equations and christoffel symbols
- What software is used to draw undirected graphs?
- Principal part of Laurent expansion.
- Extending a “linear” map to $\mathrm{span}(S)$
- Directly indecomposable rings
- Strongly complete profinite group
- Examples on product topology $ \gg $ box topology?
- Show the negative-definiteness of a squared Riemannian metric
- Repeated Factorials and Repeated Square Rooting
- evaluation of $\lim_{n\rightarrow \infty}\frac{1}{2n}\cdot \ln \binom{2n}{n}$