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.

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What is the strength of allowing multiple predecessor numbers?

If we have a theory of numbers, pairs of numbers, and sets of those, and axiomatize that the relation $<$ on numbers is both extensional and well founded, then this theory would prove all PA axioms ...
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Reflection schema

Peano Arithmetic consists of axioms $P_1, P_2, \ldots P_7$ plus first order classical logic. Let us call this theory $T$. This theory has its unprovable Gödel’s sentence $G$ such that $$ G\...
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Which sentences are "irreducibly" self-referential over $\mathsf{PA}$?

Previously asked at MSE. Below, all sentences/formulas are in the language of arithmetic, and for simplicity we conflate numbers with numerals and sentences with Godel numbers. Say that a sentence $\...
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6 votes
1 answer
240 views

Proving short consistency: can we do better than brute force search?

This is a minor variation of a question originally asked on MSE by user779130 and bountied by me, without success. Throughout, "length" refers to the number of symbols, not lines, in a proof....
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11 votes
2 answers
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How much induction does a p-adic valuation need?

Recently I learned a nice constructive proof of the irrationality of $\sqrt{2}$, which uses the 2-adic valuation of an integer: the count of how many times a number is divisible by 2. The valuation ...
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2 votes
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Why is the proof of decidability of arithmetic (Theorem 2.1) in Hamkins & Lewis (2000) enough?

Recently, I was reading the paper "Infinite Time Turing Machines" by Hamkins & Lewis. And one of the first theorems (Theorem 2.1) is about decidability of arithmetic. The proof is quite ...
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Is there a non-standard model of PA computable with infinitary computation?

By the Tennenbaum's theorem, there are no non-standard countable models of Peano Arithmetic that are computable using Turing machines. What about models of infinitary computation like infinite time ...
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4 votes
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Is there a simple proof of consistency of EA?

Let $\mathsf{EA}+\mathsf{CE}$ be elementary arithmetic with cut elimination theorem. Is there a simple (1-)consistency proof of $\mathsf{EA}$ over $\mathsf{EA}+\mathsf{CE}$? I think that a naïve ...
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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 ...
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Which arithmetical sentences have no counterexamples in the sense of Kreisel?

It is a well-known fact that given a first-order sentence $\psi$ in prenex normal form $\forall x_1 \exists y_1 \forall x_2 \exists y_2 \dots \forall x_n \exists y_n \theta(x_1,\dots,x_n,y_1,\dots,y_n)...
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Are there signatures escaping from Tennenbaum's Theorem?

By Tennenbaum's Theorem all recursive models of $\mathsf{PA}$ are isomorphic to the standard model. And by a result of Wilmer this holds even for models of the theory $\mathsf{IE}_1\subseteq \mathsf{I}...
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2 votes
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Representation of the equality relation between hereditarily finite sets in weak set theories

Consider General Set Theory ($ \mathsf { GST } $) axiomatized by the following. Axiom of Extensionality: The sets $ x $ and $ y $ are the same set if they have the same members: $$ \forall x \forall ...
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3 votes
1 answer
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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 ...
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7 votes
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Generic behavior of "polynomialish" models of $\mathsf{Q}$

(This question was originally asked and bountied at MSE - with different notation, some more explicit arguments, and topology in place of forcing.) Suppose $\mathcal{R}=(R_i)_{i\in\mathbb{N}}$ is a ...
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$\Sigma_n$-complete sets in the Levy hierarchy

Recall that a set $A \subseteq \mathbb N$ is (many-one, Turing) $\Sigma_n$-complete if it's $\Sigma_n$ and any other $\Sigma_n$ set (many-one, Turing) reduces to it. This definition actually makes ...
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4 votes
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Computably saturated Skolem hulls of Morley sequences in $\mathsf{PA}$

Recall that a model $M$ of a first-order theory $T$ (in a computable language $\mathcal{L}$) is computably saturated if for every finite tuple $\bar{a} \in M$ and every computable partial type $\Sigma(...
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What is the smallest countable limit ordinal in which 'lost melodies' occur

The question is in the title. This question is in response to the following paragraph found at the end of Prof. Hamkins' answer to my MathOverflow question, Are ITTM's necessary to compute Turing's &...
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6 votes
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Can this weakish system of arithmetic express multiplication for second-sort numbers?

Consider a 2-sorted first-order logic with equality (for first-sort entities). The first sort consists of numbers, the second sort (which will be capitalized) of unary functions. There is one constant,...
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How similar can a model of $I\Delta_0$ be to the intersection of all of its definable cuts?

Let $M$ be a model of $I\Delta_0$. Recall that a definable cut is a definable (possibly with parameters) subset $I$ of $M$ that is non-empty, downwards closed, and closed under successor. If we ...
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Is anything known about $\Delta_n$ bounding?

For a class $\Gamma \in \{ \Sigma_n, \Pi_n, \Delta_n \}$ in the arithmetical hierarchy, we can consider the induction, bounding, and least number principles for $\Gamma$: $\mathsf{I}\Gamma$ is $\big[ ...
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1 vote
1 answer
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Can $\mathsf{RCA}_0$ prove that every nonempty c.e. set $A \subseteq \mathbb{N}$ has a least element?

In other words, can $\mathsf{RCA}_0$ prove that for every function $f\colon \mathbb{N} \to \mathbb{N}$, there is $b \in \mathbb{N}$ such that $$ \exists k \in \mathbb{N},\ f(k) = b\quad \land\quad \...
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3 votes
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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 ...
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Proof of Tennenbaum's Theorem by McCarty

Tennenbaum's Theorem in its usual form states that for any countable non-standard model $M$ of PA there is no way to code the elements of $M$ as natural numbers such that either the addition or ...
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What is the proof-theoretic ordinal of bare $\mathsf{NFU}$?

On the Stanford Encyclopedia of Philosophy article on alternative axiomatic set theories, it is stated without reference that bare $\mathsf{NFU}$ (i.e., $\mathsf{NFU}$ without the axiom of infinity) ...
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3 votes
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Consistency and consistency strength of certain special cuts in $I\Delta_0$

Recall that $I\Delta_0$ is the theory in the language of arithmetic that consists of the axioms of $\mathsf{PA}$ with induction restricted to $\Delta_0$ formulas (i.e., formulas where all quantifiers ...
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Why doesn't $\mathsf{B}\Sigma_2$ hold in $\mathsf{RCA}_0$?

For a formula $\varphi(i,u)$ of arithmetic, the bounding principle for $\varphi$ is the statement $$\forall m \, \Big( \big( \forall i<m\ \exists u\ \varphi(i,u) \big) \to \big( \exists v\ \forall ...
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10 votes
1 answer
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What is the Turing degree of the monadic theory of the real line?

The monadic theory of the real line is the set of all sentences in the monadic second-order language of order which are true in $\mathbb{R}$. In this 1982 paper, Gurevich and Shelah show that true ...
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14 votes
2 answers
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How special is first-order $\mathsf{PA}$?

This is a modified version of a question which was asked and bountied at MSE without success. Below, "$\mathsf{PA}$" refers to first-order Peano arithmetic. There are various "...
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5 votes
1 answer
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Can a nonstandard model of $\mathsf{PA}$ be "$\Delta^1_1$-well-ordered?"

This was asked and bountied at MSE with no response: My question is the following: Is there a nonstandard model $\mathcal{M}\models\mathsf{PA}$ such that $\mathcal{M}$ has no $\Delta^1_1$-with-...
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8 votes
1 answer
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Paris-Harrington principles parametrized by functions $f:\mathbb N \to \mathbb N$

Recall that the Paris-Harrington Principle, $\mathsf{PH}$, is the statement that for each $e, r, k < \omega$ there is an $N < \omega$ so that given any coloring $c:[N]^e \to r$ there is an $H \...
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9 votes
1 answer
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What can $I\Delta_0$ prove?

What combinatorial and number-theoretic propositions can $I\Delta_0$ prove? Obviously there are an infinitude of them, but what are some well known theorems that can be proved in $I\Delta_0$, if any?
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8 votes
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Reverse mathematics of Noetherian rings over $\mathbb{Q}$

Take the Hilbert Basis Theorem over the rational numbers in this form in the language of Second Order Arithmetic:  For every $n\in N$ every ideal of the polynomial ring $\mathbb{Q}[x_1,\dots,x_n]$ is ...
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13 votes
3 answers
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Did Edward Nelson accept the incompleteness theorems?

Edward Nelson advocated weak versions of arithmetic (called predicative arithmetic) that couldn't prove the totality of exponentiation. Since his theory extends Robinson arithmetic, the incompleteness ...
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4 votes
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Can Robinson arithmetic prove any interesting theorems?

The motivation for my question is I'm curious whether studying Robinson arithmetic can be fruitful in the same sense as studying group theory. Robinson arithmetic is so weak that there are many ...
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9 votes
1 answer
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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[(\...
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3 votes
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Set theories that are complete modulo finite-order arithmetic

In a previous question, I asked whether there can be effectively axiomatizable set theories (at least as strong as, say, ZF) that are complete modulo first-order arithmetic, to which the answer is no; ...
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4 votes
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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 ...
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3 votes
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Interpretability of primitive recursive functions in Peano Arithmetic

Let $R$ be a set of defining equations for primitive recursive functions successively built up from $s, +, \cdot$. Is PA + $R$ interpretable in PA? (Interpretability understood in the sense of Tarski, ...
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6 votes
0 answers
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Nelson's contradiction in finitism

I have read up, in Shoenfield and elsewhere, on a lot of the details involved in Nelson's failed proof of the inconsistency of arithmetic. I understand the Kritchman-Raz proof; the proof of the ...
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1 vote
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Formalization in PA in the Kritchman-Raz proof

In their paper Kritchman and Raz present a proof of Gödel's second theorem using Kolmogorov complexity. To make it work, they operate in some (weak) formal theory $T$ that incorporates some arithmetic,...
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Peano axioms— mathematical induction and other axioms

The Peano axioms of $\Bbb N$ are: $1 \in \Bbb N$, i.e. $\Bbb N$ is not empty and contains an element denoted by $1$. Every natural number has a successor, i.e. $\forall n\in\Bbb N, \exists!s(n)\in\...
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2 votes
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Representing iteration of a function in PA

Let $\mathscr{L}$ be a (recursive) FOL language, with numeral symbols $\underline{0},\underline{1},\ldots$. Let $T$ be a recursive, consistent theory, containing PA (or even just Robinson arithmetic)....
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7 votes
1 answer
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Iterated Gentzen: or, a Sith objection to the proof of consistency of PA

$\DeclareMathOperator\PRA{PRA}\DeclareMathOperator\WF{WF}\DeclareMathOperator\Con{Con}\DeclareMathOperator\PA{PA}$Preamble: In the year … in a galaxy far far away, a nasty Sith named Darth Dubious (...
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9 votes
1 answer
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An internalized version of Tennenbaum's Theorem

Tennenbaum's celebrated 1959 theorem (see here for a reference) is certainly one of the key theorems in mathematical logic. Not so much for its proof, but because it helps "isolating" $N$ ...
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1 vote
1 answer
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Complete and consistent first-order theories that contain interesting phenomena

Gödel has shown that a consistent recursively axiomatizable first-order theory that can interpret Robinson arithmetic is incomplete. I think there is some sentimental value in working with a theory ...
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10 votes
1 answer
284 views

Concrete examples of statements not provable in PRA + $\epsilon_0$-induction that are provable in PA?

It is well-known that $\mathbf{PRA}$ plus $\epsilon_0$-induction on bounded formulas cannot prove all $\mathbf{PA}$ theorems (essentially because $I\Sigma_1$ plus $\epsilon_0$-induction on bounded ...
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5 votes
3 answers
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Are there first-order statements that second order PA proves that first order PA does not?

Are there first-order statements that second order PA proves that first order PA does not? Is this known one way or the other? Could you share an example? (edit: to clarify, by 'second order PA' I don'...
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4 votes
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The Return of Graham Arithmetics: adding induction up to $g_{64}$

In my previous question The inconsistency of Graham Arithmetics plus $ \forall n, n < g_{64}$, I introduced an extension of Robinson Arithmetics with the recursive definition of Tetraction, a small ...
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9 votes
0 answers
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Is there a definable model of PA whose domain is a proper class and whose complete theory is not definable?

Assume ZFC. Is there a formula of $\mathcal{L}_\in$ (without parameters) defining a model $\mathcal{M}$ of PA whose domain is a proper class but the complete theory of that model is not definable by ...
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12 votes
2 answers
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The inconsistency of Graham Arithmetics plus $ \forall n, n < g_{64}$

As you all know, Ronald Graham just passed away. He is famous for many fabulous contributions to finite combinatorics, and much much more, but perhaps none of them is as popular as the infamous ...
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