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

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An infinite set of axioms in ZF? What does that mean?

Before write this question, I lookeded around enough in this forum for a possible answer and although there are many similar questions, I couldn't find one answer which understand or satisfies me. I ...
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1answer
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How to show that Peano axioms prove that if $\varphi$ defines a non-empty set, then it has a least element? [on hold]

Show the following statement in PA $\forall v_1\dotso\forall v_k\,(\exists v_0\,\varphi\to\exists v_0(\varphi\wedge\forall v_{k+1}<v_0\,\neg\varphi^{v_0}_{v_{k+1}}))$ With $v_0, v_1, \...
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1answer
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Prove that addition preserves order. (for natural numbers)

Prove that addition preserves order. $a ≥ b$ if and only if $a+c ≥ b+c$. (using peano axioms) I try to do it by induction on $c$. Can I use $(a+c)++ ≥ (b+c)++$. I am not sure because first we will ...
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Prove that order is antisymmetric. (for natural numbers)

Prove that order is antisymmetric.(for natural numbers)i.e. If $ a \leq b$ and $b\leq a$ then $a=b$. I do not want a proof based on set theory. I am following the book Analysis 1 by Tao. It should be ...
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1answer
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Peano Arithmetic, proof

Show in PA: $\forall v_0\forall v_1 (v_0<v_1\rightarrow \exists v_2\quad v_0+v_2=v_1)$ Hello, I have a question to this task, because I do not know, how to proof this. I give the definition ...
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1answer
23 views

example of nonstandard model of PA that is not recursively saturated

I know that every nonstandard model of PA realizes any recursive type of given quantifier complexity (say $\Sigma_n$, for some $n$). I suppose there must be recursive types that are not always ...
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What definition of isomorphism is used here?

I've found many diferent definitions of isomorphisms depending on the theory you are working on, sadly my book doesn't give an expicit definition. I'm trying to prove that given two Peano's systems ...
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What if I can't find a set that meet the requirements for -a requirement-?

For a system to be Peano's you requite 3 things: 1) First element isn't a succesor of any other element 2) "Successor" function is injective. 3) If $A \subseteq P$, First element is in $A$, and $S(A)\...
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This proof of peano's systems seemed so easy, perhaps i'm wrong?

Let $P=\{1,2,3,4\}$, where the succesor of a number is $S(n)=n+1$ and $S(4)=1.$ This last one means that 1 is the succesor of another number, hence this can't be a Peano's system. Do I have to ...
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What is an example of a non standard model of Peano Arithmetic?

According to here, there is the "standard" model of Peano Arithmetic. This is defined as $0,1,2,...$ in the usual sense. What would be an example of a nonstandard model of Peano Arithmetic? What would ...
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Decidability of quantifier-free formulae in Peano- and True Arithmetic

It is well-known that validity in Peano Arithmetic is undecidable. It is less well-known that validity is already undecidable in True Arithmetic (the theory of the standard model of Peano Arithmetic). ...
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What does this theorem say in english?

Let $(P,Sc,1)$ a Peano's system, $G:P\times P\rightarrow P, H:P\rightarrow P$ are functions. Then $\exists ! F:P\times P\rightarrow P$ such that i)$F(x,1)=H(x)\forall x\in P$ ii)$F(x,Sc(y))=G(x,F(x,...
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1answer
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I'd like some clarification in this theorem proof.

Let $(P,Sc,1)$ a Peano's system, then $P=\{1\}\cup Sc\{P\}$ They use the third Peano's axiom, in which if $A\subseteq P, 1\in A$ and $Sc(a)\subseteq A\Rightarrow A=P$. But their proof says in the ...
2
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1answer
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Prove that the system $(P, S, 0)$ satisfy Peano Axioms.

Peano Axioms. Let $\mathbb N \neq \emptyset$ be a set and $S:\mathbb N \to \mathbb N$ a function. The elements of $\mathbb N$ are the natural numbers. If $n \in \mathbb N$ then $S(n)$ is the ...
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1answer
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Connection between number theory and the Von Neumann construction of naturals

There are many unsolved conjectures and hypothesis in number theory. For example, the twin primes conjecture, Goldbach's conjecture, the Riemann hypothesis, infinitude of Mersenne's primes, and many ...
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Proof from axioms of $\mathsf{PA}$: every natural number has remainder $0$ or $1$ or $2$ when divided by $3$

Using only the axioms of $\mathsf{PA}$, I want to prove this fact. It came up in a previous year's exam paper, and seems more difficult than I had anticipated... The question was to sketch the idea, ...
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1answer
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Exercise in Peano Arithmetic

I'm doing some excercises from the book "The Incompleteness Phenomenom" from Goldstern and Judah. I'm doing exercise about Peano Arithmetic and I have to show that: $\vdash(\forall z\leq x(z\geq 1 \...
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1answer
25 views

Prove that it is a theorem

I'm doing some excercises from the book "The Incompleteness Phenomenom" from Goldstern and Judah. I'm doing exercise about Peano Arithmetic and I have to show that: $\vdash p|x \wedge r·s=p \to r|x$...
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Give a derivation in Peano Arithmetic

I'm doing some excercises from the book "The Incompleteness Phenomenom" from Goldstern and Judah. Show that: $\vdash \exists y[y>1 \wedge \forall z(z\leq 0\wedge z>1\to z|y)]$ Hint: Show that $...
3
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1answer
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Does no non-standard model of Peano Arithmetic make the integers a principal ideal domain?

Though I do not find a reference now, I have heard no non-standard model of Peano Arithmetic has a principal ideal domain as its ring of integers. Is that right? Is it trivial? Or is there a good ...
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Is the axiom of induction constructively verifiable for a non-standard model of Peano arithmetic?

There exist models of the natural numbers which include infinite numbers. Such models are called non-standard models of arithmetic. (Proof: by the compactness theorem, there exist models of Peano ...
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Peano and induction according to Schaum's Outline

As some of you may have noticed, I've been coming at Peano from a few different angles. This time I'm stuck on what the Schaum's Outline Abstract Algebra version might mean. Here's Schaum's Peano: $...
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Peano axioms with only sets and mapping

I've got Serge Lang's Undergraduate Algebra (2nd edition). In the Appendix is a treatment of the Peano Axioms, but, as he says: " The rules of the game from now on allow us to use only sets and ...
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1answer
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Could relational operators be used to construct formal theory of natural numbers which is “stronger” than Peano Axioms?

This is a beginner's question about foundational construction of (alternative?) number theory. The notion of mathematical equality is closely related to logico-philosophical notion of 'Law of Identity'...
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Metamathematic: Cover the case if X=Y

I want to formalize: "If X is less than Y, Then U is equal to Y ", and have been told that $$ \bf [\forall V \sim X=(Y+V)]U=Y $$ does not cover the case X=Y. Therefore I have rewritten it as $$ \bf [\...
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Peano Induction Axiom

This is a typical rendition the Peano Axiom of Induction: If subset $S \subseteq \mathbb{N}$ contains $1$ and is closed under the successor function (i.e., $n \in S$ implies $\sigma\text{n} \in S$ ...
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2answers
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About ZFC, peano's axioms, first oder logic and completeness?

I read somewhere that the peano's axioms can be derived out of ZFC. But if that is the case ZFC would be incomplete right( by Godel's incompleteness theorem)? But since ZFC is in first order logic , ...
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Does Tennenbaum's theorem apply to Modular Arithmetic?

I recently asked on math.stackexchange Are the algebraic numbers recursive? I had assumed the field of algebraic numbers is a model of a theory I call Modular Arithmetic. I also assumed Tennenbaum's ...
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Are the algebraic numbers recursive?

I am interested in a theory I call Modular Arithmetic (MA). MA has the same axioms as first order Peano arithmetic (PA) except $\forall x Sx \neq 0$ is replaced with $\exists x Sx=0$. MA has finite ...
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How can addition be non-recursive?

Tennenbaum's theorem says neither addition nor multiplication can be recursive in a non-standard model of arithmetic. I assume recursive means computable and computable means computable by a Turing ...
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Independence of First-Order Peano Axioms

In class was given these 7 (first-order) axioms of Peano arithmetic (the + denotes successor): enter image description here The task task is to prove that these axioms are independent. I have figured ...
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Is it possible to formalize all mathematics in terms of ordinals only?

Our experience shows that all finitary mathematical objects could be encoded using the natural numbers, and all operations on those objects could be expressed in terms of a few basic operations on ...
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What is the method to solve these kind of questions?

Here is the exercise : We define in $\textbf{PA}$ : "$x \mid y \equiv \exists z \ y=xz"$ "$\text{irr}(x) \equiv \forall z(z\mid x \rightarrow z=1 \vee z=x)$" "$\text{prime}(x)\equiv x>1 \wedge \...
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3answers
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Why might Dieudonne have been “begging the question” by appealing to second-order Peano Axioms?

Following a comment by Peter Smith, I've been reading A. R. D. Mathias's paper The Ignorance of Bourbaki. Parts of the paper are above my head, but I understand it well enough for my own amateurish ...
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2answers
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How to prove that doesnt exist a natural number such that is equal to it successor from Peano axioms?

Im getting a hard time trying to prove the general for any natural number $n$ such that $$\nexists n\in\Bbb N: S(n)= n$$ From the second Peano axiom we know that $$\nexists n\in\Bbb N: S(1)= n$$ ...
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Why is the Axiom of Infinity necessary?

I am having trouble seeing why the Axiom of Infinity is necessary to construct an infinite set. According to a professor of who's mine teaching a class on "infinity," the Peano axioms are only ...
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Understanding Demonstration There's no Bijection between a set and a proper set of it.

I'm in need of help to understand the demonstratrion of the theorem: Given $I_n=\{ p \in N / p\le n \}$ there's no bijection $f: A \to I_n$ between the set $I_n$ and the proper subset of it $A$. The ...
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1answer
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Defining Primes in Non-standard Models of Peano Arithmetic

I was recently reading a post basically discussing an "intuition" that Goldbach's Conjecture may be a statement which is undecidable (the post does not specify which axiomatic system the statement is ...
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3answers
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How to prove $\mathbb{N}\subseteq\mathbb{R}$ given basic field axiomatisation.

I understand that $\mathbb{R}$ can be defined axiomatically in a number of different ways, including as an extension from more basic number systems (e.g., $\mathbb{N}$, $\mathbb{Z}$, or $\mathbb{Q}$) ...
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2answers
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Can the Recursion Theorem be proved in Peano Arithmetic?

A recursive function on $\mathbb{N}$ can be defined as follows: Given an element $a \in \mathbb{N}$ and a function $f:\mathbb{N}\rightarrow\mathbb{N}$, we can define a function $g:\mathbb{N}\...
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1answer
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Is there an non provable sentence from Peano Arithmetic?

I'm trying to deduce the following sentence using only Peano Axioms: "There exist infinite prime numbers" Since PA is known to be incomplete, its possible there is no such proof supporting the ...
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On what sets other than $\mathbb{N}$ might we use proof by induction?

Suppose we have a set $S$ with $s_1\in S$ and $f: S\to S$ and $n\subset S$ such that $n=\{s_1, f(s_1), f(f(s_1)), \cdots \}$ ($n$ not necessarily infinite). To establish properties of $n$, can we ...
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Peano Arithmetic before Gödel

If I understand correctly, Gödel was the one to discover how to encode finite sequences of integers in Peano Arithmetic, with his Chinese Remainder Theorem trick (his "beta function"). As a ...
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Arithmetic theory

Assuming that Peano’s first-order arithmetic (PA) is indeed a consistent, effective, and arithmetic theory. Is there a way to extend PA to a theory $T$, such that $T$ is consistent, effective but ...
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1answer
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Proving the artihmetic property of a theory

When can we say a threorom is arithmetic? Is it by following the first 9 axioms of Peano arithmetics? Or should it only allow us to express idioms of natural numbers? For example, can we add a claim ...
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1answer
116 views

Tao's definition of addition [closed]

In Tao's Real Analysis he defines addition: Let $m$ be a natural number. To add zero to $m$, we define $0+m \equiv m$. Now suppose inductively that we have define how to add $n$ to $m$. Then we ...
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Peano Arithmetic: How would this formalized statement be correct?

Using Peano Axioms I have formalized the following: x is the square of an odd prime number For some odd prime number x' , x is its square IF x is some odd prime number, THEN x is the square of x' IF ...
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1answer
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How do we know $\mathbb{N}$ is a model of Peano Arithmetic?

The induction axiom in the theory of Peano Arithmetic (PA) is actually an axiom scheme such that for every formula $\phi(x,\bar{y})$ with free variables $x,\bar{y}$ ($\bar{y}$ being a string of ...
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1answer
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Stuck on GEB chapter 9 - is b a MU number? is b a TNT number?

I'm reading through Gödel, Escher, Bach, and I found myself stuck at chapter 9. I've been rereading through several times already, but I must be missing something. To clarify my background, I'm a ...
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A subtle point regarding the axiom of induction

The fifth of Peano's axioms states the following: If $S\subset \mathbb{N}$ such that $1 \in S$ and $n \in S \Rightarrow \sigma(n) \in S$, then $S = \mathbb{N}$ (where $\sigma(n)$ is the successor ...