All Questions
Tagged with model-theory proof-theory
29
questions
8
votes
0
answers
134
views
Is there a substructure-preservation result for FOL in finite model theory?
It's well-known$^*$ that the Los-Tarski theorem ("Every substructure-preserved sentence is equivalent to a $\forall^*$-sentence") fails for $\mathsf{FOL}$ in the finite setting: we can find ...
- 20.7k
1
vote
0
answers
98
views
Effortless automated proofs for "simple" formulae?
From small cases to all of them. This is in the spirit of 15 theorem see https://en.wikipedia.org/wiki/15_and_290_theorems
EXAMPLE : Suppose you have the following problem: P(a)
For any fixed non ...
- 1,306
4
votes
0
answers
159
views
In finite model theory, is "invariant FOL with $\varepsilon$-operator" unavoidably second-order?
Throughout, all structures are finite.
Say that a class of finite structures $\mathbb{K}$ is $\mathsf{FOL}_\varepsilon^\text{inv}$-elementary iff it is the class of finite models of a sentence in the ...
- 20.7k
1
vote
1
answer
187
views
Gödel coding and the function $z(x)$
The function $z(x)$ that associates to each formula $\alpha$ of $P$ its Gödel number $z(\alpha)$ is external to the system. How then can expressions in which $z(x)$ be involved be expressed in $P$? ...
- 265
4
votes
0
answers
152
views
Can this theory of dyadic rationals prove that multiplying by three is the same as summing thrice?
(This question arose from a discussion with Jade Vanadium about a theory of dyadic rationals.)
Let $T$ be the 2-sorted FOL theory with sorts $ℕ,ℚ$ and constant-symbols $0,1$ and binary function-...
- 2,848
15
votes
5
answers
3k
views
How is it possible for PA+¬Con(PA) to be consistent?
I'm having some trouble understanding how a certain first-order theory isn't just straight-up inconsistent.
Let $PA$ be the axioms of (first-order) Peano arithmetic and let $C$ be the following ...
- 167
7
votes
0
answers
108
views
How tightly are decidability and "induction-completeness" linked?
It is known that there are a number of expansions of the structure $\mathfrak{N}:=(\mathbb{N};+)$ which are decidable (= have computable theories); one such example is the expansion by a predicate ...
- 20.7k
5
votes
2
answers
273
views
Is the usual enumeration of $\mathsf{PA}$ "minimal for consistency strength"?
This question is about a technical imprecision which is easily avoidable but whose details I'd like to understand better. When we refer to "the consistency strength of $\mathsf{PA}$" (say) ...
- 20.7k
8
votes
1
answer
324
views
Modal logic of "mostly-satisfiability"
For $n\in\omega+1$ let $\mathsf{ZFC}_n$ be $\mathsf{ZC}$ + $\{\Sigma_k$-$\mathsf{Rep}: k<n\}$. Let $\widehat{\mathsf{ZFC}}$ be the strongest consistent theory $\mathsf{ZFC}_n$ (so if $\mathsf{ZFC}$ ...
- 20.7k
14
votes
1
answer
668
views
Are there different "levels" of self-referentiality in arithmetic?
Below, all sentences/formulas are first-order and in the language of arithmetic. For simplicity, we conflate numbers and numerals, and conflate sentences/formulas and their Godel numbers.
Given a ...
- 20.7k
2
votes
1
answer
153
views
Is adding all sentences true of terms in skolemized theory conservative?
Suppose I have a (incomplete) theory $T$ (e.g. PA) which I skolemize to get a theory $T_S$ in the expanded language. I now build $T'$ by adding to $T_S$ any sentence $(\forall x)\phi(x)$ where I can ...
- 2,653
16
votes
3
answers
1k
views
Is there a semantics for intuitionistic logic that is meta-theoretically "self-hosting"?
One can study the standard semantics of classical propositional logic within classical logic set theory, so we can say that the semantics of classical logic is meta-theoretically "self-hosting&...
- 163
0
votes
0
answers
149
views
What are the handy, go-to methods of proving consistency of a proof system?
Suppose you face a proof system portraying some notion or knowledge that you haven't encountered, or others haven't studied before. What would be your first attempts to examine the consistency of the ...
- 505
9
votes
1
answer
624
views
Is $\mathsf{R}$ axiomatizable by finitely many schemes?
Recall that $\mathsf{R}$ is the theory of arithmetic consisting of the quantifier-free theory of $(\mathbb{N};+,\times,0,1,<)$ together with, for each $k\in\mathbb{N}$, the sentence $$\forall x[(\...
- 20.7k
5
votes
0
answers
241
views
Does $\mathsf{Q}$ have any interesting provably recursive functions?
This question was asked and bountied at MSE without success.
For an appropriate theory $T$, say that an $n$-ary $T$-provably recursive function is a $\Sigma_1$ formula $\varphi$ with $n+1$ free ...
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