Questions tagged [theories-of-arithmetic]
Theories of arithmetic in first-order logic, such as Peano arithmetic, second-order arithmetic, Heyting arithmetic, and their subsystems and extensions. Models of Peano arithmetic.
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Are there different levels of homomorphisms/isomorphisms between formal theories?
I will start with an analogy. We know that $R^2$ and $R$ are isomorphic as sets, but not isomorphic as vector spaces or topological spaces. So we have different notions of isomorphisms, each notion ...
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Optimal Subsystem of $\mathsf{PA}_{2}$ for Proving Existence of Set of Gödel Codes of True Arithmetic
Let $\mathcal{N}$ denote the standard model of first-order Peano arithmetic, $\mathsf{PA}$, and let true arithmetic, $\mathsf{Th}(\mathcal{N})$, be the set of sentences in $\mathsf{L}_{\mathsf{PA}}$ ...
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Axioms of Soft Logic
Someone here bumped into the papers by Moshe Klein & Oded Maimon on Soft Logic? I try to understand whether their axioms actually exclude the zero-product property.
Here are the axioms from one of ...
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Is there a second incompleteness theorem for Rosser-provability?
In question What are the adequacy conditions for Rosser Provability? I asked for the adequacy conditions for Rosser-provability ($\rho$) as compared with the Hilbert-Bernays-Löb derivability ...
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Quantifier elimination for natural numbers with order and successor
It is well-known that the first-order theory of (N, <, suc, 0), the natural numbers with order and successor function, admits quantifier elimination. But I could not find any reference. Does anyone ...
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Can custom functions be defined in Peano Arithmetic or the language of First-order Arithmetic?
Can we specify custom recursively-defined functions in the language of First-order Arithmetic?
I know that we can define functions in Second-order Arithmetic ($Z_2$). For example, we could define ...
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Is the set of theorems of a PA + “PA is inconsistent” equivalent to the halting set?
In what follows, we let $T$ be a consistent, recursively axiomatizable theory that includes $\mathsf{PA}$ (Peano arithmetic).
Definition: Let us say that the theory $T$ is creative when the set of ...
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Why does collapsing $\mathit{Ord}$ preserve $\mathsf{Z}_2$?
In Cheng/Schindler's Harrington's principle in higher order arithmetic, the following claim is made in the proof of Theorem 3.1:
Let $G$ be $\mathit{Col}(\omega,<\mathit{Ord})$-generic over $L$. ...
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Can Witnessing theorems lead to code extraction from proofs? (bounded arithmetics)
In bounded arithmetic (or bounded reverse mathematics) we study formal systems so weak that the structures of their proofs correspond to some known complexity classes such as PTIME, LOGSPACE or AC0.
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If we work in first-order intuitionistic logic, does the Well-ordering Principle schema imply the induction schema over $PA^-$?
Work over $PA^-$, i.e. Peano Arithmetic without the induction schema, which is the theory of a discrete ordered commutative unital semiring. By induction for an arithmetic formula $\varphi(x)$, we ...
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Simple true $\Pi^0_1$ statements independent of weak arithmetics
I originally asked this question on Math StackExchange here, but I have copied it here as I now feel it is more appropriate for this site.
There is an explicitly known 549-state Turing machine where, ...
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Cut-elimination in a Hilbert-type system?
I have a question regarding the paper "Provably computable functions and the fast growing hierarchy" by Buchholz and Wainer (1987).
The authors perform their analysis on a Gentzen-type ...
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Proof-theoretic ordinals of $\mathsf{ACA}_0$ plus $\Pi^1_n$-induction
The proof-theoretic ordinal of $\mathsf{ACA}_0$ is $\varepsilon_0$ (Simpson, Subsystems of Second Order Arithmetic, 2009, theorem IX.5.7), and the proof-theoretic ordinal of $\mathsf{ACA}$ is $\...
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Do indicators offer something that other proofs of unprovability don't?
My question is about a technique in proving that a statement is unprovable in some theory of first-order arithmetic. To make this definition, work in a nonstandard model $M$ of Peano arithmetic, and ...
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What is the weakest theory that settles all interesting first-order arithmetic sentences?
Woodin's program of refuting CH, as summarized in 1, continues the following assertions (roughly as in Propositions 7, 13, and 20 of that paper):
In any model of $\text{ZFC}$, the theory of $(H(\omega)...
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Is this cardinality motivated theory of arithmetic synonymous with PA?
Logic: Strict Bi-sorted first order logic with equality,
Signature: $=; \in; ||$
, first sort written in lower case range over natural numbers, the second sort written in upper case range over sets of ...
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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{...
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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 ...
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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 ...
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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.
$\textbf{Logic:}$ ...
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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{...
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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 ...
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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/...
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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 $\phi$...
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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-...
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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?
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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 $...
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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 ...
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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. ...
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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:\...
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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 ...
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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 ...
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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&...
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$\Pi^0_1$ 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 ...
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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 ...
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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 ...
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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 ...
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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 $$T \vdash \lnot \lnot \exists n \in \mathbb N. \phi(n)$$ ...
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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 ...
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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 = \{ ...
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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 ...
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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 ...
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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 ...
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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-...
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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: }\\\...
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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 ...
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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 ...
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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 ...
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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 (...
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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 ...
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