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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Lunar arithmetic to help the set $\left\{ \mathbf{+}, \mathbf{\cdot} \right \}$ functionally complete?

If the goal of lunar arithmetic (=lunatic) to develop a minimal set of operations that can perform all the arithmetic: From the typical equation in part 9, is this possible to combining members of the ...
8 votes
3 answers
951 views

Dedekind-Peano axioms, but numbers have at most one successor

One can consider a variant of the Dedekind-Peano axioms in which one replaces the assumption that every number has exactly one successor by the assumption that every number has at most one successor, ...
5 votes
1 answer
119 views

Does visible nonstandardness imply visible ill-foundedness?

For $X\subseteq\mathfrak{M}\models \mathsf{TA}$, say that $X$ is $\mathfrak{M}$-disruptive iff there is some formula $\varphi$ in the language of arithmetic + a new unary predicate symbol $U$ such ...
2 votes
0 answers
98 views

Can we extend the projectively extended real line with a single number that stands for division of zero by zero?

If we work within $\hat{\mathbb R} = \mathbb R \cup \{\infty\}$, i.e. one point compactification of the real line. We extend $<$ relation on $\mathbb R$ to $\hat <$ defined as: $ x \ \hat{<} \...
5 votes
1 answer
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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 $Z_2$ ...
3 votes
1 answer
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Can we always know if an algebraic rule over the reals is preserved over the extended reals or not?

Recall a prior posting titled Is there an effective way to generalize this approach of affinely extending the number line?, and especially the accepted answer given to it. So we are working in $\sf ...
3 votes
0 answers
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Self-referential Quinean proof of Löb's Theorem

Given its relevance for Open-source game theory, Dr. Andrew Critch asks the following about provability logic: We conjecture that Löb’s Theorem can be proven without the use of the modal fixed point $...
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1 vote
1 answer
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Is there an effective way to generalize this approach of affinely extending the number line?

The general approach here is a follow up of the approach outlined in a prior posting on extending the projectively extended real line. In particular arithmetic operators break down to ternery ...
-2 votes
1 answer
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Is this extension of the projectively extended real line, consistent?

This posting has been Edited. The edited material shall be noted. The projectively extended real line $\hat {\mathbb R}= \mathbb R \cup \{\infty\}$ is one system which allows division by zero! Yet it ...
7 votes
1 answer
186 views

What is known about first order logic of $\mathbb{N}$ with + and a unary predicate?

In "Weak Second-Order Arithmetic and Finite Automata", Büchi claims that the first order theory of $\mathbb{N}$ with + and a predicate for recognizing powers of 2 ($Pw_2$) is expressively ...
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30 votes
2 answers
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Do we expect that sufficiently large computable ordinals settle every question of arithmetic?

I came across a post by Ron Maimon on physics.SE that makes what seems to me to be a very interesting conjecture I've never seen before about what it would take to settle every question of arithmetic. ...
9 votes
5 answers
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What is the canonical way to extend Peano's axioms to the set of all integers?

My first idea on how to do this would be: $0\in\mathbf Z$ $\forall x\in\mathbf Z,Sx\in\mathbf Z\land Px\in\mathbf Z$ $P$ and $S$ are injective $\forall x\in\mathbf Z,PSx=SPx=x$ some induction axiom ...
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32 votes
2 answers
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What is the logical status of the sentence combining the ideas of Löb and Rosser, "this sentence is provable before any proof of its negation"?

Logicians are familiar with the variety of self-referential sentences expressible in the language of arithmetic: The Gödel sentence, "this sentence is not provable", which indeed is not ...
1 vote
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Can this type theory interpret second order arithmetic?

Language: multi-sorted first order logic with equality and membership, where for each natural $t$ there is a set $x^t$ of sort $t$. Equality "$=$" only occurs between variables of the same ...
6 votes
1 answer
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A "negative" standard system

For $\mathcal{M}$ a (countable) nonstandard model of $\mathsf{PA}$, let $\mathsf{SS}(\mathcal{M})$ be the set of sets of natural numbers coded by elements of $\mathcal{M}$. There are various ways to ...
27 votes
2 answers
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Can we interpret arithmetic in set theory, with exactly PA as the ZFC provable consequences?

There are many interpretations of arithmetic in set theory. The Zermelo interpretation, for example, begins with the empty set and applies the singleton operator as successor: $$0=\{\ \}$$ $$1=\{0\}$$ ...
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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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1 answer
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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\...
11 votes
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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 $\...
6 votes
1 answer
271 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....
11 votes
2 answers
891 views

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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3 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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1 answer
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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 ...
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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13 votes
1 answer
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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}...
2 votes
1 answer
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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 ...
4 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 ...
4 votes
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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 ...
4 votes
0 answers
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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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0 votes
1 answer
247 views

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 &...
6 votes
0 answers
204 views

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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7 votes
0 answers
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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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3 votes
0 answers
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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[ ...
2 votes
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 \...
3 votes
0 answers
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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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6 votes
0 answers
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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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12 votes
1 answer
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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
1 answer
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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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1 vote
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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 ...
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 ...
14 votes
2 answers
781 views

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 "...
5 votes
1 answer
220 views

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-...
8 votes
1 answer
241 views

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 \...
9 votes
1 answer
349 views

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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9 votes
0 answers
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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 ...
16 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
0 answers
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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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