All Questions
Tagged with theories-of-arithmetic reference-request
22 questions
7
votes
4
answers
574
views
A conservative extension of Peano Arithmetic
Ulrich Kohlenbach makes the following intriguing comment here:
"In the 70s S. Feferman introduced a mathematically strong system S=restricted(PA^omega)+QF-AC+mu for classical mathematics (and in ...
- 16.6k
5
votes
2
answers
433
views
Models of second-order arithmetic closed under relative constructibility
I know little to nothing about second-order arithmetic and its subsystems. However, I would like to understand when a model of (a subsystem of) second-order arithmetic ($\mathsf{Z}_2$) is downward ...
- 2,286
3
votes
1
answer
135
views
$\Pi^0_1$ sentences modulo "schematic entailment"
Let $\mathfrak{P}$ be the preorder of $\Delta^0_0$ (= only bounded quantifiers) formulas with one free variable in the language of arithmetic, under the relation $\alpha(x)\le\beta(x)$ iff there is a ...
- 20.7k
6
votes
1
answer
232
views
Interpretation of $ZFC^-$ in 2nd order Peano arithmetic
Let $Z_2^-$ be the 2nd order Peano arithmetic without the schema of Countable Choice. It has been known, since 1960s at least, that $ZFC^-$ (without the power set) admits an interpretation in $Z_2$ ...
- 2,281
2
votes
0
answers
79
views
Which sets of natural numbers are "lambda-analytic"?
Begin with a bit of notation. Let $t = t_0, \ldots, t_d$ be a finite sequence of real numbers. Define
$$\lambda^t(x) = x^{t_0} \log(x)^{t_1} \log(\log(x))^{t_2} \cdots.$$
for all real numbers $x \in ...
- 13.3k
68
votes
4
answers
12k
views
Nelson's program to show inconsistency of ZF
At the end of the paper Division by three by Peter G. Doyle and John H. Conway, the authors say:
Not that we believe there really are any such things as infinite sets, or that the Zermelo-Fraenkel ...
- 25.5k
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
4
votes
1
answer
377
views
Does ACA prove categoricity of the reals?
$\def\f#1{\text{#1}}$Does $\f{ACA}$ prove that any two internally complete ordered fields are isomorphic?
Here internal completeness is expressed roughly as "every sequence of reals with an upper ...
- 2,912
7
votes
2
answers
708
views
On a theorem of Zhang Jinwen about models of arithmetic
In the paper ''A Nonstandard Model of Arithmetic Constructed by means of Forcing Method'', Zhang Jinwen states the following in his abstract:
The first nonstandard model of arithmetic was given by ...
- 32.2k
3
votes
0
answers
146
views
Does Robinson arithmetic interpret a Kripke model of the double negation translation of $\mathsf{I}\Delta_0 + \mathrm{Exp}$?
It is a well-known fact that while while Robinson arithmetic can interpret surprisingly strong theories, it cannot interpret $\mathsf{I}\Delta_0 + \mathrm{Exp}$, i.e., Peano arithmetic with induction ...
- 13.2k
4
votes
0
answers
431
views
How can I prove that primitive recursion “preserves” representability in Peano Arithmetic?
I'm working on my thesis about Gödel's Incompleteness Theorems, and at some point I need to prove that the $\textsf{PA}$ system is able to represent all the recursive functions.
By recursive function ...
- 49
10
votes
1
answer
542
views
Looking for “Set theory for a small universe” by Ketonen
In the paper Partition theorems for systems of finite subsets of integers, Pudlák and Rödl show a Ramsey-type result. The main feature of this result is that the sizes of sets in such systems are not ...
- 1,124
13
votes
2
answers
1k
views
nonstandard models and mathematical theorems
Is there a first order statement about the natural numbers (not nonstandard analysis) such that the truth of the statement is easier to see in a nonstandard model? In other words, do nonstandard ...
12
votes
1
answer
1k
views
How to locate the paper that established Robinson Arithmetic?
If I'm not mistaken, it was in his seminal paper “An Essentially Undecidable Axiom System”, published in
Proceedings of the International Congress of Mathematicians (1950), 1952:729–730,
where R.M. ...
- 2,992
4
votes
1
answer
876
views
Derivability conditions for Robinson arithmetic
Two pieces of hearsay I have encountered about Robinson's Q:
Q fails to satisfy the Löb derivability conditions;
Pudlák criticised the Löb derivability conditions and suggested rival, weaker ...
- 2,283
3
votes
3
answers
314
views
Semantic reflection
Let $\ulcorner \cdot \urcorner$ be a fixed encoding of formulas by numbers, e.g.
let $\ulcorner \varphi \urcorner$ denote the Godel number of $\varphi$.
Let $T$ be a first-order arithmetic theory, e....
- 5,502
9
votes
1
answer
1k
views
Does Nelson try to prove PA inconsistent directly?
Edward Nelson is known for his serious attempts to show that Peano axioms, and sometimes even weaker theories, are inconsistent. I wasn't able to find Nelson's papers anywhere, so I wanted to ask a ...
- 28.2k
7
votes
1
answer
705
views
Does the totality of Ackermann's function prove the consistency of $\Sigma_1$-induction?
It is well known that Ackermann's function is not primitive recursive. Therefore, the theories of primitive recursive arithmetic (PRA) and of $\Sigma_1$-induction ($I\Sigma_1$) cannot prove the ...
- 777
7
votes
4
answers
912
views
Reference Request: Non-Standard Models of PA
I am attempting to write an expository paper on non-standard models of PA that is accesible to students taking an introductory graduate course in mathematical logic (covering Godel's incompleteness ...
- 1,441
5
votes
0
answers
287
views
Is the two variable fragment of arithmetic, i.e., theory of ($\mathbb{N}, + ,\times$), decidable?
Any references would be appreciated. Most places only address different vocabularies (e.g. a survey of arithmetical definability by Bes).
3
votes
1
answer
181
views
Least ordinal not embedded in a total order
If $(E,<)$ is a linear order, let $s(E,<)$ denote the least ordinal which doesn't embed in $(E,<)$.
I am trying to prove the following:
If $(M,+,.,0,1)$ is a model of open induction, (or ...
- 2,519
7
votes
1
answer
447
views
The definition of < in Robinson's Q
I recently had to explain how the basic axioms in Simpson's Subsystems of Second Order Arithmetic were interpretable in Robinson's Q. Most of the axioms are actually the same, except that Simpson ...
- 44.4k