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Results tagged with definability
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user 36385
definability by formulas in first-order logic, e.g. as explained at https://en.wikipedia.org/wiki/Definable_set, or as in J. Robinson's first-order definition of the integers in the field of rationals
6
votes
Is there a simple instance of intransitivity for implicit definability?
Indeed, the set of primes is not implicitly definable over $(\mathbb{N},S)$. I gave a proof of this in an answer to the question of Geoffrey Irving. See https://mathoverflow.net/a/426382/36385 .
- 5,821
7
votes
Is there a simple instance of intransitivity for implicit definability?
Here is an example with a relatively simple and elementary proof.
Lets say that a set $A$ of natural numbers is universal if for any natural $n$ and a set $a\subseteq n$ there exists $m$ such that $\f …
- 5,821
21
votes
Accepted
Is the set of primes implicitly definable from successor?
The set of primes is not implicitly definable in $(\mathbb{N},S)$. This is immediately implied by following:
Theorem. A unary predicate $P$ on $\mathbb{N}$ is implicitly definable in $(\mathbb{N},S)$ …
- 5,821