All Questions
6 questions
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
4
votes
0
answers
162
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-...
- 2,912
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
10
votes
1
answer
630
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
6
votes
2
answers
436
views
Interpreting proper elementarily equivalent end extensions?
Is there a tuple of parameter-free formulas $\Phi$ and a nonstandard $M\models PA$ such that $\Phi^M\models PA$, the induced $M$-definable initial segment embedding $j_\Phi^M:M\rightarrow\Phi^M$ is ...
- 20.7k
8
votes
2
answers
560
views
Models of PRA/EFA with induction on $X$ but not $\omega^X$
As I currently understand it, induction on formulas containing $N+1$ first-order quantifiers is required to prove the well-ordering of the ordinal $(\omega \uparrow\uparrow N) < \epsilon_0$, that ...