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A model of a theory is a structure (e.g. an interpretation) that satisfies the sentences of that theory. Wikipedia

Let $A$ be a set of sentences in some language that has only one extra-logical primitive symbol $R$.

Let's say that theory $T$ describes a model $A$ if and only if

  • $T$ proves the existence of a pair $\langle M, [R] \rangle $, where $ [R] $ is a set of all ordered pairs $\langle a,b \rangle $ where $a,b \in M $ and $ a \ R \ b$.
  • if we relativise all sentences of $A$ to $M$ (i.e. bound all of their quantifiers $\in M$), and replace each atomic sentence $a \ R \ b$ in them by $\exists r \in [R] \ (r=\langle a,b \rangle)$.
  • then all the resulting sentences hold in $T$.

We'd say by then that $T$ describes [not prove] that $A$ is satisfied in $\langle M, [R]\rangle$.

However a theory might describe a model of all its axioms, in the exact above sense, but might still not be able to express it in a single sentence!

To clarify: we can have a theory $T$ which prove the existence of sets $M$ and $[R]$ obeying all conditions depicted above, and such that $A$ is the set of all axioms of $T$ itself (even if $A$ is finite); and at the same time $T$ cannot prove the single sentence:

$\forall \phi [(T\vdash \phi) \to \langle M,[R]\rangle \models \phi)]$

and so cannot prove the single sentence:

$\exists M \exists [R] (\langle M,[R]\rangle \models T)$.

So even though $T$ is describing a model of itself, still $T$ cannot prove the sentence "there exists a model of $T$", and so doesn't prove its own consistency.

This mean that in principle we can extend theory $T$ by enlarging its language, adding axioms and inference rules, in such a manner that we have more expressive theories $T_1, ..,T_n$, where each $T_{i+1}$ extends $T_i$, and at the same time each $T_i$ can describe a model of $T_{i+1}$ in the above spoken sense. We can even have $T$ describing a model of $T_n$, i.e. without necessarily proving the sentence "there exists a model of $T$". All of this without implying that $T$ is inconsistent, nor needing to weaken $T$ below expressing basic arithmetic facts.

In some semantic sense description of a model of a theory [in the above particular sense], is next to saying that it is consistent, actually it is next to saying that it is possibly true of something! And constructively speaking that what matters.

Although this in no way touches Godel's incompleteness theorems, but it in some sense develops a parallel approach available to formalism through "describing" models by going higher and higher up in description steps of more complex structures, without affecting the consistency of the base theory!

The only theory that I know of it being able to describe a model of its finitely many axioms in the above sense is New foundations "NF"

Question: Are there other known examples of such theories?

Question 2: had there been formalist approaches along that line?

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  • $\begingroup$ The sentences in this post are hard to follow. Shorter sentences would help. $\endgroup$
    – user44143
    Dec 27, 2019 at 16:45
  • $\begingroup$ OK I'll try. Thanks $\endgroup$ Dec 27, 2019 at 17:50
  • $\begingroup$ I made the changes. That's the best I can do. I hope its clear now. $\endgroup$ Dec 27, 2019 at 18:14
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    $\begingroup$ In what sense are interpretations structures? The latter are (semantic) objects, the former essentially live at the level of syntax. $\endgroup$ Dec 27, 2019 at 18:26
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    $\begingroup$ Awful, that sentence in the wikipedia article is terribly misleading. $\endgroup$ Dec 27, 2019 at 20:56

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