All Questions
Tagged with or theories-of-arithmetic theories-of-arithmetic
334 questions
5
votes
1
answer
145
views
Mathematical strength of the statement "Heyting Arithmetic admits Markov's rule"
Consider the following theorem about Heyting arithmetic (HA):
For every arithmetical formula $\phi$ whose only free variable is $n$, if $\text{HA} \vdash \forall n. \phi \lor \lnot \phi$ and $\text{...
- 6,403
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
5
votes
1
answer
371
views
Are PA and Counting Theory synonymous\bi-interpretable?
The following question is whether $\sf PA$ is synonymous or even bi-interpretable with a theory about counting objects in finite sets.
Counting Theory:
$\textbf{Logic:}$ Bi-sorted first order logic ...
- 11.3k
1
vote
0
answers
91
views
About synonymy relationships around these two theories?
The following question is about patterns of synonymy relationships around two theories, $T^+$ and $\sf PA$.
For purposes of self inclusiveness I'll re-iterate $T$ and its extensions.
$\textbf{Logic:}$ ...
- 11.3k
-4
votes
0
answers
135
views
Which arithmetic\set theory is synonymous with this theory?
$\textbf{Logic:}$ Mono-sorted first order logic with equality.
$\textbf{Extralogical Primitives: } <, \in$
Define: $x > y \iff y < x$
Define: $x \leq y \iff x < y \lor x=y$
$ \textbf{...
- 11.3k
-4
votes
1
answer
173
views
To which arithmetic\set theory this theory is bi-interpretable?
$\textbf{Logic:}$ Mono-sorted first order logic with equality.
$\textbf{Extralogical Primitives: } <, \in$
$ \textbf{Axioms:}$
$ \textbf{Order:} \ x < y < z \to x < z $
$ \textbf{...
- 11.3k
-2
votes
1
answer
211
views
Can this theory interpret Peano arithmetic?
Logic: Bi-sorted first order logic with equality, first sort written in lower case range over natural numbers, the second sort written in upper case range over sets of naturals, "$=$" has no ...
- 11.3k
6
votes
1
answer
197
views
Is there a 1-generic degree g such that Th(D(< g)) is more complicated than true arithmetic?
I am currently reading an article titled "Embedding and Coding Below a 1-Generic Degree" by Greenberg and Montalbán(link to a free source:https://pi.math.cornell.edu/~erlkonig/Papers/...
- 893
2
votes
0
answers
100
views
Realizing arithmetic hierarchy in algebraic number theory
Is it possible to realize arithmetic hierarchy in algebraic number theory?
For example, consider a $\Pi^0_4$ statement of the form $\forall x \exists y \forall z \exists w \phi(x,y,z,w)=0$ where $\phi$...
- 593
2
votes
1
answer
154
views
The Dirichlet principle and arithmetical induction
Let us consider the Dirichlet principle as follows: for all natural numbers $n > k > 0$, there is no injection from $\{0, \dots, n-1\}$ into $\{0, \dots, k-1\}$.
Is it true that in some non-...
1
vote
1
answer
180
views
Natural functions outside $\sf PA$?
Can theories stronger than $\sf PA$ manage to define functions from the naturals to the naturals, that $\sf PA$ cannot? If so, what are examples of those functions?
- 11.3k
1
vote
1
answer
146
views
Can PA define functions related to higher theories?
Working in $\sf ZFC$ we can define a function $F$ from naturals to naturals such that $F(0) = \, ^\circ\ulcorner r({\sf Z}) \urcorner$, where $r({\sf Z})$ is the Rosser sentence of Zermelo set theory $...
- 11.3k
1
vote
2
answers
340
views
Heuristic interpretations of the PA-unprovability of Goodstein's Theorem
I've relatively recently learned about Goodstein's Theorem and its unprovability in Peano arithmetic (the Kirby-Paris Theorem). I do not have any real knowledge of formal logic; but I think I've seen ...
- 708
6
votes
1
answer
743
views
Which part(s) of this proof of Goodstein's Theorem are not expressible in Peano arithmetic?
EDIT: Noah Schweber helpfully points out that $\mathsf{ACA}_0$ is a conservative extension of Peano arithmetic in which certain aspects of my proof not expressible in Peano arithmetic are expressible. ...
- 708
5
votes
0
answers
109
views
Computational complexity of arithmetic sentences over classical theories
Below, I use the term "tracker" rather than "realizer" since I'm not requiring the relevant objects to be computable.
Define the relation "$f$ tracks $\varphi$" for $f:\...
- 20.7k
4
votes
1
answer
269
views
Does Peano's axioms prove $\alpha$-induction for primitive recursive sequences for every concrete $\alpha < \varepsilon_0$?
It is well-known that Peano's axioms (PA) cannot prove $\varepsilon_0$-induction for primitive recursive sequences (PRWO($\varepsilon_0$)), because PA + PRWO($\varepsilon_0$) proves the consistency of ...
23
votes
1
answer
3k
views
What is known about the theory of natural numbers with only 0, successor and max?
Consider the first-order theory whose intended/standard model is the natural numbers $\mathbb{N}$, with constant $0\in \mathbb{N}$, with an injective successor operation $s$ such that $0$ is not a ...
- 35.5k
4
votes
1
answer
193
views
Further research on relevant realizability etc
I just read Dunn's 1979 paper Relevant Robinson's Arithmetic, and the end especially caught my interest. Following the surprising role of constant functions in collapsing "relevant Q with zero&...
- 20.7k
1
vote
0
answers
123
views
Is possibile to define transfinite sum and product recursively? [closed]
On mathstackexchange a few days ago I published the following question where I asked about "transfinite" sum and products but actually nobody answered or gave an opinion with a comment: thus ...
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
4
votes
1
answer
155
views
Does this hierarchy of fragments of $I \Sigma_1$ collapse?
Does anyone know whether the following hierarchy of fragments of
$\mathrm{I} \Sigma_1$ (or rather
$\mathrm{I} \Pi_1$) collapses or not?
Let $\Sigma^b_n$ denote formulas in the language of arithmetic ...
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
4
votes
1
answer
515
views
Truth Values of Statements in non-standard models
Excuse me, if the question sounds too naive.
Non-standard models of PA will have statements of non-standard lengths, basically infinite. And it is also true that every statement of a theory will have ...
- 1
14
votes
1
answer
526
views
Is there a theory between HA and PA that doesn't have Markov's rule?
A theory $T$ admits Markov's rule when
For every formula $\phi(n)$, if $$T \vdash \forall n \in \mathbb N. \phi(n) \lor \lnot \phi(n)$$ and $$T \vdash \lnot \lnot \exists n \in \mathbb N. \phi(n)$$ ...
- 6,403
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
4
votes
1
answer
258
views
What is the theory of statements with a provably *bounded* realizer (according to PA)?
$\let\T\mathrm\def\kr{\mathrel{\mathbf r}}$This is a follow up to Kleene realizability in Peano arithmetic.
We can summarize the results from Emil Jeřábek's answer as follows:
\begin{gather*}
T_1 = \{ ...
- 6,403
7
votes
1
answer
334
views
Why include $0$ and $1$ in the signature of Presburger arithmetic?
I recently became interested in some problems concerning decidability of extensions of Presburger arithmetic. However, being a number-theorist rather than a logician, I am confused by some notational ...
- 1,914
-2
votes
1
answer
211
views
Would this alteration safeguard the resulting theory from inconsistency?
If we replace "Emergence" axiom in the theory $T$ presented at posting "What is the set theory synonymous with this order-set theory?" with the following axiom, call the resulting ...
- 11.3k
3
votes
1
answer
169
views
Would this alteration of $T$ affect its synonymy with PA?
If we replace the single axiom "Emergence" in the axiomatic system $T$ presented at posting [What is the set theory synonymous with this order-set theory], by the following schema. Would the ...
- 11.3k
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
1
vote
1
answer
313
views
What is the set theory synonymous with this order-set theory?
Let $ T$ be a theory written in Mono-sorted first order logic with equality, with extralogical primitives: $<, \in$.
Define: $x \leq y \iff x < y \lor x=y$
Axioms:
$\textbf{Well ordering: }\\\...
- 11.3k
12
votes
4
answers
1k
views
Is this theory synonymous with PA?
Language: Mono-sorted first order logic with equality.
Extralogical Primitives: $<, \in$
Define: $x \leq y \iff x < y \lor x=y$
$\textbf{Well ordering: }\\\textit{Transitive:} \ x < y \land ...
- 11.3k
1
vote
1
answer
221
views
Seeking clarification of ultrapower nonstandard model of arithmetic
I've read that one nonstandard model of arithmetic is:
take $\mathbb{N}^\mathbb{N}$, the set of countably infinite sequences of natural numbers
take a quotent that gives the ultrapower: identify ...
- 1,288
-2
votes
1
answer
416
views
Defining the set of natural numbers in the first order Peano arithmetic [closed]
The question seems simple, but I'm not sure:
let's consider a first order Peano Arithmetic and its standard model $N = \{ 0,1,2,3,... \}$ of natural numbers.
A question: how can we define the whole ...
- 19
3
votes
1
answer
123
views
Kleene normal form theorem for r.e. relations proven in arithmetical theories
After reading the relevant chapters of Classical Recursion Theory (freely available from here), I have the following questions concerning Theorem II.1.10 (Normal form theorem) and Theorem IV.1.9 (...
- 31
8
votes
1
answer
588
views
Con(PA) via non-well-foundedness?
Lumsdaine made the following interesting
comment:
if Con(PA) fails in a non-standard model, it means it contains a
“proof of non-standard length” of a contradiction from PA. With a
little work, one ...
- 16.6k
15
votes
5
answers
2k
views
In what sense does the sentence $\operatorname{con}(\mathsf{PA})$ "say" that $\mathsf{PA}$ is consistent?
It seems common amongst logicians to think of "truth" as being relative to a particular structure. Consider, for instance, the first-order theory of groups. The sentence $\forall x\forall y(...
- 961
14
votes
1
answer
646
views
Extensions of $PA+\neg Con(PA)$ with large consistency strength
There is a large hierarchy of theories strengthening $PA$ eg $PA+Con(PA)$, $PA+Con(PA+Con(PA))$,..., second-order arithmetic and $ZFC$, ordered by consistency strength.
Is there an extension of $PA+\...
- 345
3
votes
0
answers
206
views
Independence and truth in PA
By $\textbf{PA}$ I will mean the usual first-order Peano Arithmetic. I will denote an element of $\mathbb{N}$ by $n$, and by $[n]$ I will denote the corresponding term in the language of $\textbf{PA}$:...
- 3,318
6
votes
0
answers
407
views
Can Set Theory be turned into Infinite Arithmetic?
The following system I'd label as "Infinite Arithmetic" is simply an endeavor to extend second order arithmetic to the infinite ordinal world, and extending with it the representation of ...
- 11.3k
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
1
vote
0
answers
129
views
Is set theory interpretable in infinite primitive recursive arithmetic?
In A Formalization of the Theory of Ordinal Numbers, Takeuti interprets $\sf ZFC$ in a first order theory extending first order arithmetic to the infinite ordinal realm, while at the same time ...
- 11.3k
2
votes
1
answer
329
views
Can set theory be interpreted in infinite arithmetic?
Is the following system of infinite arithmetic consistent?
If so, can it interpret $\sf ZFC$?
Language: first order logic
Primitives: $\operatorname{Card}, <, + , \times,\text{^}$
where $\...
- 11.3k
10
votes
2
answers
436
views
The additive structure of clusters of nonstandard models of arithmetic
Given $\frak M$ a countable nonstandard model of $\sf PA$ and let $a\in M$ be a nonstandard element. A "cluster around $a$" is the set of successors and predecessors of $a$, a cluster is a ...
- 1,676
0
votes
0
answers
73
views
Least number principle for IOpen fragment of Peano Arithmetic
Is it possible to prove the least number principle in IOpen fragment of Peano Arithmetic, i.e. having induction only for Open formulas?
- 19
-3
votes
1
answer
638
views
Analysis I, simpler proof of Tao's construction of the integers [closed]
In chapter 4 of Analysis I by Terence Tao, we have the following note about the set theoretic construction of the integers:
In the language of set theory, what we are doing here is starting with the ...
- 23
6
votes
0
answers
192
views
How to show that $\omega^\omega$ is well-founded in PA?
By induction on $n$ variables I can show that for any meta-natural number $n$, PA proves well-foundedness of $\omega^n$. However it is well known that PA proves well-foundedness of $\omega^\omega$ ...
- 101
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
2
votes
0
answers
75
views
Can all the strongly provable theorems of $\sf PA+\neg Con(PA)$ be captured in an effective manner through alternative kind of provability?
If we extend $\sf PA$ with the following axiom asserting its own inconsistency:
Inconsistency: $\exists x: \operatorname {Proof}_{\sf PA} (x, \ulcorner 0=1 \urcorner)$
For short denote this axiom by $\...
- 11.3k
2
votes
2
answers
294
views
Can we use remote provability to prove the first incompleteness theorem sans $\omega$-consistency?
Let $\mathcal g_1$ denote the usual Godel sentence defined as: $$ \mathcal g_1 \iff \neg\exists x:\operatorname {Proof}_T(x, \ulcorner \mathcal g_1 \urcorner)$$
Lets suppose that $\sf T$ is ...
- 11.3k