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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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What is the Peano definition of subtraction?

I came up with this: (S.1) $a - a = 0$ (S.2) $a - b = S(a - S(b))$ This seems to work. At least for $a$$\ge$$b$. Is this the correct or most efficient formulation? Also, does there happen to be one for division? Of course I would imagine that that…
David Bandel
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Is there a formal definition of "Greater Than"

Intuitively, one can say that $S(n) > n$. But how do we prove it using the Peano Axioms. It seems like I need a formal statement as to what $>$ means.
Chris Cudmore
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There are no loops in a Peano system. I've attempted a proof. Is my proof correct?

Definition (Peano systems). Suppose $P$ is a set, $1 \in P$, and $S: P \to P$ is a function. The triple $(P, S, 1)$ is a Peano system if the following conditions hold. (P1) $\forall x (1 \neq S(x))$, (P2) $\forall x \forall y (x \neq y \implies…
Mostafizur Rahman
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Role of Induction among Peano Axioms

I am working with Peano Axioms as follows: $0 \in \mathbb{N}$. $n \in \mathbb{N} \implies S(n) \in \mathbb{N}$. $(\forall n \in \mathbb{N})(S(n) \neq 0)$. $n \neq m \implies S(n) \neq S(m)$. Let $A$ be a subset of $\mathbb{N}$. If $0 \in A$ and $n…
Promethèus
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Multiplication cancellation property by Peano axioms

I am trying to prove cancellation property of multiplication of natural numbers, $xy=xz$ implies $y=z$, with Peano axioms and arithmetic but not using or defining order on natural numbers. It can be done for addition. But for proving multiplication…
jnyan
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Why define addition with successor?

I'm reading Russell's Introduction to Mathematical Philosophy Russell defines the sum of two numbers in terms of successors. I don't understand why: Suppose we wish to define the sum of two numbers. Taking any number $m$, we define $m+0$ as $m$,…
zeynel
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Peano axioms without explicit reference to zero

Consider $(N, S)$ with S injective and not surjective and suppose the induction principle holds, where the zero in inductive principle is an element which is not in $S(N)$. Can I prove that $0$ is unique? In other words can we delete explicit…
asv
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Am I allowed to substitute terms when reasoning in Peano Arithmetic?

I feel a little silly asking this question, but here goes. I'm on Chapter 1 of "Forcing for Mathematicians" by Nik Weaver and doing some of the exercises. When doing stuff in Peano Arithmetic, can I substitute in terms? For example, axioms 3 and 4…
roundsquare
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How do Peano Axioms imply "nextness" with the successor?

Going with this explanation of Peano's Axioms, I cannot understand how/where the successor function is definitively stated to be the very next number in the case of natural numbers. In this treatment, it says The successor of $x$ is sometimes…
147pm
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Tao's definition of addition

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 can add $n++$ to $m$ by defining $(n++) + m \equiv…
Chris
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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 notion of successors (the successor of $n$ is taken…
user170039
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Name this binary operation on $\Bbb N$

An operation $\ominus$ for the natural numbers is defined as follows: $$a\ominus 0 = 0\ominus a = a\\S(a)\ominus S(b) = a\ominus b$$ Here $S$ is the successor function. The operation $a\ominus b$ is equivalent in value to $|a-b|$, except negative…
Regret
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Are elements not "reachable" from 0 prevented in Peano Arithmetic?

I am having a discussion about Peano Arithmetic and I was told that there is actually nothing in Peano Arithmetic to prevent one from defining an element a with S^k(a)=a. I am only able to derive that a is not S^j(0) for any j, but I don't see how…
Thomas Bartscher
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Is a set $T$ containing all natural numbers identical to $\mathbb{N}$?

In this comment, I see that Mauro Allegranza and I were saying the same thing as shown in the table below: Allegranza Zeynel $0 \in T$ $0 \in T$ $n \in T$ $n \in T$ $S(n) \in T$ $S(n) \in T$ $ T = \mathbb{N}$ $T$ contains all natural…
zeynel
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An alternative axiom to the peano axiom of induction?

$\def\N{\mathbf{N}}$ The peano axioms: $0\in\N$ $n\in\N \implies S(n)\in\N$ $\forall n\in\N, S(n)\neq0$ $\forall n,m\in\N, n=m \iff S(n)=S(m)$ $[(0\in X) \wedge(\forall n\in\N, n\in X \implies S(n)\in X)]\implies X\supseteq\mathbf{N}$ I…
user1020500
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