Intereting Posts

correcting a mistake in Spivak
Do these series converge to the von Mangoldt function?
Do there exist any odd prime powers that can be represented as $n^4+4^n$?
The First Homology Group is the Abelianization of the Fundamental Group.
Proving that $-a=(-1)\cdot a$
$f,g$ continuous from $X$ to $Y$. if they are agree on a dense set $A$ of $X$ then they agree on $X$
Finding the closure of some subsets of the ordered square
Normal subgroups of free groups: finitely generated $\implies$ finite index.
Intuition behind Variance forumla
Do I understand metric tensor correctly?
$y(x)$ be a continuous solution of the initial value problem $y'+2y=f(x)$ , $y(0)=0$
For a field $K$, is there a way to prove that $K$ is a PID without mentioning Euclidean domain?
Find the CDF with a given random variable
$\mathbb{Z}/(X^2-1)$ is not isomorphic with $\mathbb{Z}\times \mathbb{Z}$
Why does topology rarely come up outside of topology?

In a certain sense, only finite structures are definable up to isomorphism in first order logic. But if we rely on a metatheory containing a sufficient strong set theory (like required for second order logic), would it be possible to also define certain infinite structures up to isomorphism by requiring additional conditions like minimality (i.e. conditions independent of the syntax and axioms)? The set theory would be required here in order to give meaning to the additional conditions (i.e. *minimal* might mean here that no proper subset of the model satisfies the given first order theory).

- Good textbook for learning Sequent Calculus
- “This statement is false.”
- How to avoid perceived circularity when defining a formal language?
- How to show that there is a deduction of $\varphi\leftrightarrow\psi$
- How to find the logical formula for a given truth table?
- Learning how to prove that a function can't proved total?
- First Order Language for vector spaces over fields
- Is there a 'nice' axiomatization in the language of arithmetic of the statements ZF proves about the natural numbers?
- How is exponentiation defined in Peano arithmetic?
- What are the main relationships between exclusive OR / logical biconditional?

If you allow ANY kind of set-theoretic notions, then of course you can describe any model up to isomorphism, even without using any model-theoretic notions, by simply defining the set.

As for your question, there is actually a notion of minimal model, that is, a model which has no proper elementary substructure, but it is not in general unique up to isomorphism. For example, the theory of additive group of integers has $\mathfrak c$ nonisomorphic minimal models. On the other hand, some theories do have a unique minimal model, for example the theory of natural numbers with successor.

There are also other model-theoretic notions which are strong enough to uniquely define a model of a given complete theory, such as:

- a primal model of any theory, which can intuitively be described as one which can be inductively constructed by blocks defined in terms of all the previous ones by a simple formula
- a prime model of a countable theory, which is a model such that it can be embedded as an elementary substructure into any other model of the theory (nb a primal model is always prime)
- a saturated model in a given cardinalty, which can be seen as a very rich model, which has elements having any noncontradictory set of properties you can define without using very many elements of the model (I’m not entirely sure if we need any stronger assumptions if the theory is uncountable, but for countable complete theories a saturated model in a given cardinality is unique)

Note that, however, not every theory has a minimal, prime, or primal model, for example the theory of additive integers has no prime model, and the theory of countably infinitely many independent equivalence relations with two classes has no minimal model.

Assuming existence of a strongly inaccessible cardinal, every theory has a saturated model in a strongly inaccessible cardinality (I know it to be true for countable theories, but it should probably still hold for any theories whose languages are smaller than some strongly inaccessible cardinal), but without it I think there might be theories without any (or at least I haven’t heard of a theorem that would provide them for arbitrary theories without large cardinal assumptions).

As Benedict pointed out, there are also some theories which have a unique model in a given cardinality $\kappa$, which are called $\kappa$-categorical. If a theory is $\kappa$-categorical (and $\kappa\geq \lvert T\rvert$), then the sole model of cardinality $\kappa$ is necessarily saturated, but cardinality is of course a more set-theoretic description.

Another, more algebraic notion which can sometimes be used to uniquely describe a model of some theories is the acl-dimension. Some theories, called strongly minimal theories (examples of which are vector spaces over a fixed field or algebraically closed fields of fixed characteristic) have a well-behaved notion of independence generalizing both linear and algebraic independence, and for those theories a model of the theory is uniquely (up to isomorphism) determined by its dimension.

In yet another direction, you can use infinitary logic. Similarly to how regular first-order logic can uniquely define finite models with a single sentence, if you allow countably infinite conjunctions you can uniquely define countable models with a single sentence (that is Scott’s isomorphism theorem). For stronger infinitary languages, it it also possible for larger cardinalities. However, infinitary logic is quite a bit more messy than the usual first-order logic as far as I know.

Suppose $T$ is a first order theory in a countable language $\mathcal{L}$. $T$ is $\kappa$-categorical iff for an infinite cardinal $\kappa$, $T$ has only one $\kappa$-sized model (up to isomorphism). By Morley’s categoricity theorem, if $T$ is $\delta$-categorical for some $\delta > \omega$, then $T$ is $\gamma$-categorical for all $\gamma > \omega$.

So for theories which are $\kappa$-categorical for uncountable $\kappa$, if you can fix the cardinality of the structures satisfying $T$ in the metatheory to be at least $\omega_1$, then you will have only one model of $T$ up to isomorphism. Similarly if you have an $\omega$-categorical theory and fix the cardinality of $T$-structures as $\aleph_0$.

During reading Ebbinghaus et al., I realized that there is a set of formulas which defines the class of non standard natural numbers. Of course, there is also a set of formulas which defines the class of natural numbers (both standard and non standard). By taking the “class difference” of these two classes, it should be possible to define the “standard natural numbers” up to isomorphism without reference to any underlying “strong” set theory.

- A question on measure space and measurable function
- Translation of Weber's Lehrbuch der Algebra vol 1, 2, 3
- What is the topological dual of a dual space with the weak* topology?
- Minimum size of largest Clique in a Graph
- How to read letters such as $\mathbb A$, $\mathbb B$, etc., or $\mathfrak A$, $\mathfrak B$, etc.?
- Limit of a particular variety of infinite product/series
- Equivalence of $a \rightarrow b$ and $\lnot a \vee b$
- Necessary and Sufficient Condition for a sub-field
- A sum for stirling numbers Pi, e.
- If $G$ is a locally cyclic group , then is $\operatorname{Aut}(G)$ abelian?
- Why is it called Sylvester's Law of Inertia?
- Characterizing units in polynomial rings
- stirling numbers and harmonic number identities
- To what divisors $a$ of $n$ can Euler's Theorem multiplied by $a$ be generalized, i.e. when is $a^{\phi(n)+1}\equiv a \pmod n$?
- How does one get the formula for this bijection from $\mathbb{N}\times\mathbb{N}$ onto $\mathbb{N}$?