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Questions tagged [peano-axioms]

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

In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are a set of axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearly unchanged in a number of metamathematical investigations, including research into fundamental questions of consistency and completeness of number theory.

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Find all functions that compose with the successor function

In Mac Lane/Birkhoff's Algebra, they spend some time discussing the natural numbers and give the Peano Axioms, roughly (from memory) $\sigma$ is injective 0 is not the successor of any element the principle of induction, which yields the natural…
Burnsba
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(How) can we derive "primary school rules of arithmetic" from the peano axioms?

The Peano axioms are intended to be able to prove very general statements about arithmetic, such as "all natural numbers can be written as the sum of two primes". However, how can we use the peano axioms to mathematically derive all the rules that…
user56834
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Closed under equality

In the Wikipedia article on "Peano axioms" I read this (source): For all $a$ and $b$, if $a$ is a natural number and $a = b$, then $b$ is also a natural number. That is, the natural numbers are closed under equality. Seems legit (to a computer…
pancake
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Peano Equivalent for Rationals

First let us establish that There exists a bijection between rationals, $\mathbb{Q}$ and natural numbers $\mathbb{N}$. Using the Peano Axiom's (and considering Gödel's incompleteness theorem), we can fundamentally describe the nature of…
Eli Sadoff
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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 application of $S$ to $0$, denoted as $S^n(0)$."…
Johny Diala
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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 $0$ is a natural number. Axiom 2.2 If $n$ is a natural number then…
user170039
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Prove $n

The problem is, Prove that $n
user170039
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Is Russell's proof of addition with Peano's 5. Axiom valid?

This is a follow up question to my previous question: Why define addition with successor? In this one I'd like to ask about Russell's use of Peano's 5. Axiom to prove his definition of addition: Suppose we wish to define the sum of two numbers.…
zeynel
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Peano axioms and meaning of successor map in Jacobson's Basic Algebra I

From Jacobson's Basic Algebra I on P. 16, the Peano axioms are stated as: $0 \neq a^+$ for any $a$ (that is, $0$ is not in the image of $\mathbb{N}$ under $a \to a^+$). $a\to a^{+}$ is injective. (Axiom of induction). Any subset of $\mathbb{N}$…
Richard K Yu
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Strategy to proof the existence of a number/condition

The following are the Peano Postulates: There exists a set $\mathbb{N}$ with an element $1 \in \mathbb{N}$ and a function $s:\mathbb{N} \to \mathbb{N}$ that satisfy the following three properties. a. There is no $n \in \mathbb{N}$ such that…
mauna
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Can we replace these two Peano axioms with this single axiom?

Peano Axioms from Mathworld: Zero is a number. If $a$ is a number, the successor of $a$ is a number. zero is not the successor of a number. Two numbers of which the successors are equal are themselves equal. (induction axiom.) If a…
user986614
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Prove property involving successor function

Let $a \in \mathbb{N} $ and $s: \mathbb{N} \longrightarrow \mathbb{N} $ be the successor function defined in Peano postulates. Then, prove the following $$ 1 + a = s(a) = a + 1 $$ Now, there is a theorem, which says that there is a unique binary…
user9026
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Which combination of Peano axioms shows that $0\neq 1?$

Please vote to close this question. It's really dumb as when I was reading the Peano axioms, axiom 8 didn't register. Don't waste your time reading this question.... I also cannot delete it (I have tried but it won't let me). $1$ is defined as…
Adam Rubinson
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Gentzen Consistency Proof and Peano's 9th Axiom. Was PA consistent as originally stated or consistent only with a weaker 9th Axiom?

I have done meta-proofs of the consistency of FOL (Studied about 40 years ago), but have not done any for PA and have not looked at (and maybe now could not follow) Gentzen or the other proofs of PA. In looking at some characterizations of his…
Lee Hester
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In what sense can you prove something in Peano Arithmetic

On Wikipedia it says: "... the Gödel sentence of Peano arithmetic, is not provable nor disprovable in Peano arithmetic." When talking about proving something in Peano Arithmetic, are we talking about just using the axioms of Peano Arithmetic (or…
johannesCmayer
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