All Questions
7 questions
12
votes
0
answers
249
views
+50
Is there a decidable theory of arithmetic with a non-collapsing quantifier hierarchy?
This question is very close to this old MSE question of mine, which is still unanswered.
Is there an (ideally reasonably-natural!) expansion of the structure $(\mathbb{N};+)$ in a finite language ...
- 20.7k
7
votes
0
answers
110
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
4
votes
1
answer
439
views
Alternative proof of Tennenbaum's theorem
The standard proof of Tennenbaums's theorem uses the existence of recursively enumerable inseparable sets and is presented e.g. in Kaye [1, 2], Smith [3].
In the following, $\mathcal{M}$ will always ...
- 211
16
votes
1
answer
831
views
Can there be computable non-standard models of PA in a weaker sense?
By Tennenbaum's theorem, in the usual sense of computability for models,
neither addition nor multiplication can be computable in a countable non-standard model of PA.
Weak version:
Can addition or ...
- 48.8k
5
votes
0
answers
318
views
$\Sigma_n$-complete sets in the Levy hierarchy
Recall that a set $A \subseteq \mathbb N$ is (many-one, Turing) $\Sigma_n$-complete if it's $\Sigma_n$ and any other $\Sigma_n$ set (many-one, Turing) reduces to it. This definition actually makes ...
- 63.9k
7
votes
1
answer
597
views
Can an uncountable model of Peano Arithmetic be recursive?
Can an uncountable model of Peano Arithmetic be recursive?
What does it mean for an uncountable model to be recursive? Well, we represent the elements of the model using real numbers instead of ...
- 4,727
3
votes
3
answers
683
views
Turing Functional and $\Sigma_1^0$-formulas in models of fragments of PA
In models of PA with restricted induction power (for example, only $I\Sigma_n$ is present), the failure of higher induction scheme is characterised by the existence of definable cuts (like $\Sigma_2$ ...
- 229