For questions on Peano axioms, a set of axioms for the natural numbers.

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Are all models of Peano arithmetic elementary equivalent?

By Löwenheim-Skolem we know there are models of (first order) PA that are not isomorphic to the standard model, but are elementary equivalent to it, i.e. they satisfy the same set of first-order ...
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61 views

How is a formal system including only a first-order axiomatization of induction stronger than a system without?

Stumbled upon another aspect of Peano arithmetic that I find confusing... I understand that what I write in the title is in fact the case, e.g. certain statements provable in PA not being provable in ...
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1answer
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Peano and consistency, how to understand it rightly.

I'm struggling with the notion of consistency, and a few cases : I'm writing in the following $Con(T)$ to denote the arithmetic formula which expresses the consistency of $T$, for $T$ a consistent ...
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1answer
123 views

Why a system becomes incomplete once it's capable of doing arithmetic?

For a Formal axiomatic system to obey Godel's incompleteness theorems, It has to be powerful enough to incorporate Peano Axioms. Why It does not apply to say, Presburger arithmetic or the axioms of ...
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Expressing quantifier free $\mathcal{L}_{PA}$-formulae $\varphi(y,\vec{x})$ with polynomials

I'm stuck at the following exercise: I want to show that for every quantifier-free formula in the language of $\mathsf{PA}$ there are polynomials $P(y,\vec{x}),Q(y,\vec{x})$ such that for all ...
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1answer
50 views

How does PA prove all $\Delta_0$-formulas which are true in the standard model?

Let $\varphi(x_1,\dots,x_n)$ be a $\Delta_0$-formula, i.e. a formula in which every quantifier is bounded. I want to prove that $$ \text{PA}\vdash\varphi(\overline{n_1},\dots,\overline{n_k}) \iff ...
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The necessity of the axiom of induction

$\underline{First\ question}$ Let $P(n)$ be a proposition about $n$. In standard mathematical induction, we require: (1)$P(0)$ holds. (2)If $P(n)$ holds, $P(n+1)$holds. Here we use "the axiom of ...
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Regarding Peano's Axioms

According to the Wikipedia entry on the Peano axioms: "the number 1 can be defined as $S(0)$, 2 as $S(S(0))$ (which is also $S(1)$), and, in general, any natural number n as the result of n-fold ...
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2answers
60 views

Variations in the successor fuction from Peano's axioms

Concerning the successor function in Peano's axioms, what prevents me from defining it in the following way: 0 to 2, 2 to 1, 1 to 4, 4 to 3, 3 to 6, 6 to 5, 5 to 8, 8 to 7 ... and so on. It seems ...
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1answer
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Which one of the following logical propositions is to be preferred?

I'm trying to update the symbolism of Giuseppe Peano's "Arithmetices Principia", to make the translation freely available. Might I ask you, which of the following might be a correct mathematical ...
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Transfinite Induction in Peano Arithmetic

I have heard that Peano Arithmetic (PA) cannot perform transfinite induction up to $\varepsilon_0$. This seems to imply that it can induct up to smaller ordinals, like $\omega$ or $\omega^\omega$ or ...
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Set of axioms for finite subset of Natural Numbers

I would like to get a set of Peano like first-order axioms for a finite subset of natural numbers $N'$ such that $0 \leq N' \leq Max$, with $Max$ denoting the upper-bound. (So my signature might be ...
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3answers
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Prove that the greatest common factor of $m+n$ and $m^2+n^2$ is 1 or 2 if $m$ and $n$ are relatively prime.

Prove that the greatest common factor of $m+n$ and $m^2+n^2$ is $1$ or $2$ if $m$ and $n$ are relatively prime natural numbers. Can anyone give a step-by-step answer for this?
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Why doesn't inequality hold as a property in natural number induction?

It is said that all natural numbers follow the rule of induction: if a said property holds for one number and for its successor, it holds for all natural numbers. But, let us define the following ...
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1answer
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Modifying the Peano Axioms to allow multiple successors

If you took the familiar Peano Axioms and replaced the axiom $x \in \mathbb{N} \implies \exists y\in \mathbb{N}(y =S(x))$ with $x \in \mathbb{M} \implies (\exists y_1\in \mathbb{M})(\exists ...
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2answers
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embdedding standard models of PA into nonstandard models

Maybe it's well known to experts, but is there always an embedding $f$ of the standard model of Peano arithmetic into a nonstandard model? By Peano arithmetic I mean its first-order version, with the ...
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1answer
64 views

Failures of categoricity not related to size?

The paradigmatic cases I know of theories failing to be categorical are cases that seem tied to the language's inability to distinguish between different infinite cardinalities of the theories domain. ...
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4answers
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Peano's Axioms: Mathematical Philosophy

In Peano Axioms, why is it necessary to define number and successor. Does not using them imply that we know what they mean? Or could they have just as easily been any two arbitrary terms which are not ...
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2answers
68 views

Prove that the set of all integers $>0$ is the smallest inductive set

An inductive set is a set $I$ such that $1 \in I$ and if $x \in I$ then $x+1 \in I.$ Some authors define the set of all integers $>0$ as the smallest inductive set, say Apostol's Analysis. But I ...
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2answers
142 views

How is first-order logic derived from the natural numbers?

I've heard that first-order logic comes from Peano Arithmetic, yet I can't see how. We don't need numbers to formulate quantifiers, variables, functions, constants, relations or even sets. Insted, it ...
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1answer
157 views

Construction of a model of Peano Arithmetic

I'm studying the axioms of Zermelo-Frankel Set Theory at the moment. I already know the following six axioms: The axiom of empty set The axiom of extensionality The axiom of pairing The axiom of ...
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48 views

Strong logical system without principle of explosion

Are there some logical systems strong enough to contain theorems of first/second order Peano Arithmetic but constructed in such way that principle of explosion does not hold for them?
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Is there a difference between induction in Peano Arithmetic and Presburger Arithmetic

Following this question I still do not get clearly the difference between defining exponentiation in PA but impossiblity of recursively define multiplication in Presburger Arithmetics I was thinking ...
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Peano's Axioms and Induction

I was reading Landau's Foundations of Analysis. He starts his construction of number systems by stating five axioms. My question is related to the fifth, the axiom of induction: Let there be given a ...
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4answers
127 views

Counting numbers vs Natural numbers; Peano Axioms

I can feel that my question is going to be a somewhat lengthy one, but I will try my best to deliver it in as short a form as I can manage. So to begin, I've always thought that the numbers such as ...
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2answers
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Check my proof of “overspill” in non-standard models of Peano arithmetic.

Proposition Let $\mathcal{M}$ be a nonstandard model of Peano arithmetic, $\phi(v,\bar{w})$ a formula in the language of arithmetic, and $\bar{a} \in \mathbb{M}$. Show that if $\mathcal{M} \models ...
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1answer
37 views

Proving the order relation in $\mathrm{PA}$ is total.

Let $\mathrm{PA}$ be the first order logic axioms of Peano Arithmetic. Define an order relation by: $$ x\leq y\; \text{ if }\; (\exists z)(x+z=y). $$ Can it be proved that this relation is total?
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Gödel's Second Incompleteness Theorem and Arithmetically Non-Definable Theories

My recursion theory knowledge has become a bit rusty, so I will appreciate any corrections for misstatements. Gödel's incompleteness theorem is often exploited by philosophical discussions which ...
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101 views

Incompleteness theorem

Correct me if I am wrong at any point! Godel's incompleteness theorem allows us to express "PA is consistent" in the language of Peano arithmetic, and shows that this is not provable in PA. Let's ...
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1answer
90 views

Prove true in natural numbers (Peano Arithmetic)

While reviewing old exercise sheets, I have found this question and am having difficulties understanding some of the logic: Let $\mathbb{N}$(natural numbers) be a model for Peano Arithmetic, that ...
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114 views

Prove $x' \neq x$ using Peano axioms

I am looking at Edmund Landau, Foundation of analysis and do not agree with is proof of Theorem 2 part 2. I put the pages here for easy reference (http://pbrd.co/1y89p7b and http://pbrd.co/1y89A2s). ...
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1answer
128 views

Definition of the sum of natural numbers

After define the natural numbers using the Peano axioms, I'm trying to understand the definition of sum between natural numbers, let $s$ be the successor function used in the Peano axioms. The most ...
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1answer
54 views

Why is it impossible to define multiplication in Presburger arithmetic yet possible to define exponentiation in Peano Arithmethic?

Hello my question is related to Why is it impossible to define multiplication in Presburger arithmetic? and to How is exponentiation defined in Peano arithmetic?. I would have preferred to add it as a ...
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3answers
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Possible not countable extension of the natural numbers?

This question comes from:Is $1234567891011121314151617181920212223......$ an integer? We define $\mathcal{A}$ as the set of infinite strings of digits $$ \bar a_i=a_0 a_1a_2a_3\cdots a_i \cdots ...
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2answers
207 views

Why don't we use Presburger's arithmetic instead of Peano's arithmetic?

I was reading about quantifier elimination and discovered the Presburger Arithmetic, the article mentions two points about it: It is decidable, complete and consistent. It omits multiplication ...
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4answers
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Another problem on Peano Axioms (indirectly) from Tao's Analysis book

In Terence Tao's book Analysis I, the definition of $1$ is given right after stating the first two axioms, namely the following axioms, Axiom 1. $0$ is a natural number. Then Tao elaborates the ...
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1answer
71 views

Why are all computable functions representable in PA?

I'm trying to understand the proof of the first incompleteness theorem, and more specifically, the diagonal lemma. Suppose $GN(x)$ is the Gödel Number of a formula $x$. The first step of the diagonal ...
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1answer
112 views

Reference request - Outline of Edward Nelson's Inconsistency Proof

Edward Nelson retracted his inconsistency proof before it was published. Unfortunately, the outline given by Nelson has been removed. Is there a copy of it on the web? I am interested in how the ...
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Why isn't it necessary to postulate the existence of $1$?

These are the Peano axioms, I'll focus on the second one now: If $a$ is a number, the successor of $a$ is a number. Basically, here is defined the successor function $S(n)=n+1$. My question is, ...
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For which subsystems T of 2nd order arithmetic is there a model of T + $\neg$Con(T)?

A theory T might have the following property: there is a model of T + $\neg$Con(T) 1st order PA has this property, but full 2nd order PA doesn't. Among subsystems of 2nd order arithmetic, which ones ...
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149 views

Defining addition in second order logic

(before saying it's duplicate, read whole question) I was told by someone that we can define addition and multiplication purely in terms of successor function, provided that we work in second order ...
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2answers
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How can I prove this proposition from Peano Axioms?

The problem is, Prove that for each positive natural number $a$ there exists a natural number $b$ such that $b{++}=a$. Using only the followings, Peano Axioms. Axiom 2.1 ...
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1answer
127 views

Prove $n<m{++}\leq n{++}\iff m=n$ from Peano Axioms

The problem is, Prove that $n<m{++}\leq n{++}\iff m=n$. Using only the followings, Peano Axioms (see the axioms here). Definition of Addition: Let $m$ be a natural number. We ...
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A question on Terence Tao's representation of Peano Axioms

While reading Terence Tao's book on Analysis I had some questions regarding the implication of the Peano Axioms. After writing the following four axioms (which I will write without changing their ...
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Existence of nonstandard elementary extensions of $PA$?

My question follows from the 1958 result of MacDowell–Specker (located originally in Modelle der Arithmetik, J. Symbolic Logic Volume 38, Issue 4 (1973), 651-652) of the proof of the following ...
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1answer
114 views

Non-standard models for Peano Axioms

This might be an easy question, but I still struggle to comprehend non-standard models for Peano axioms. I understand that Godel Theorem tells us that the theory defined by Peano axioms is not ...
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1answer
86 views

Model of Robinson Arithmetic but not Peano Arithmetic

I am curious how can structure of with linear successor function that do not admin induction looks like. In other words I would like to see structure of Peano Arithmetic without induction. I believe ...
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2answers
73 views

Are these two definitions of the natural numbers equivalent?

If we consider two definitions of the natural numbers: Definition 1 $N$ is the set that satisfies all of: There is an element $0$ in $N$. For each element $n$ in $N$, there is the successor of ...
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1answer
81 views

Proving that successor of a number is not zero

I am stuck proving the simple claim that $Sx \neq 0$ in say Peano Arithmetic (the first order theory of arithmetic) or Robinson Arithmetic or Presburger Arithmetic. I see that this is sometimes added ...
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118 views

Is $\neg Con(PA)$ true in a model of $PA+\neg Con(PA)$?

It is known that if Peano arithmetic (PA) is consistent then $PA+\neg Con(PA)$ is also consistent. One way we can express $Con(PA)$ is $\forall x(\neg P(x))$ where $P(x)$ expresses that $x$ encodes a ...