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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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Peano axioms. Exercise

If $ mk = nk $ and $ k \neq 0 $, then $ m = n $. I try to do it by induction. Let $ X = \{k \in \omega \ \wedge \ k \neq 0 \mid mk = nk \Rightarrow n = k \} $ Clearly $ 1 \in X $. Suppose $ k \in X $. To show that $ k ^ + \in X $, I assume that $ mk…
KaizeO
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How to show $m+n = n+m$ in the set of Naturals with zero.

I understand the axioms defined on the set of natural numbers that $m+0:=m$ and that $m+n’:=(m+n)’$. Where does the commutativity come into this?
Partey5
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Proof involving Peano axiom

Prove that $n \ne n+1 $ for all $ n \in \mathbb N.$ How do I prove this statement using only Peano axioms?
Twilight
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About the proposition2.1.11 of Analysis I by Terrence Tao

I'm confused about why this proposition is true if this property is not true at all. $Proposition$ 2.1.11. A certain property $P(n)$ is true for every natural number n. Proof. We use induction. We first verify the base case $n = 0$, i.e., we prove…
廖倪徵
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Why natural numbers are generalized in $\mathbb{Z}$ to ensure the group structure respect to the sum?

The natural numbers $\mathbb{N}$ are defined through the Peano axioms and then generalized in $\mathbb{Z}$ to ensure the group structure with respect to the sum. Why do we need to ensure such group structure? The most complex structures are…
Ken
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Is Peano arithmetic complete for quantifier free formulas and strict arithmetical formulas?

By a quantifier free arithmetical sentence I mean a fully quantified sentence in the language of Peano arithemtic $``PA"$, in prenex normal form having no existential quantifier. By a strict arithmetical sentence I mean a quantifier free…
Zuhair
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Peano's Axiom: Is it implied that successor of a number is not the number itself?

Using the Peano's Axioms from MathWorld as the basis, I'm wondering if it is implied that the successor of a number is not the number itself, or is it deducible?
KGhatak
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Peano Induction confusion

In a beginning Analysis book, I found this basic representation of induction: Let $A$ be any set or collection of natural numbers $\mathbb{N}$ with the properties (1) $1 \in A$ and (2) $k + 1 \in A$ wherever $k \in A$, then $A =…
147pm
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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 there exists an unique isomorphism that moves first…
José Osorio
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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 $S(A)\subseteq A \Rightarrow A=P$ But i've got a…
José Osorio
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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 prove it doesn't meet the other requirements?
José Osorio
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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: $1 \in \mathbb{N}$. For each $n \in \mathbb{N},\:$…
147pm
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PA subsets consistency

IF i have T in piano arithmetic and suppose I add a new symbol c suppose it is T1 T1=T U {c>1,c>1+1,c>1+1+1.......} Is it true that every finite subset of T1 is consistent and can we show that a structure M |= T1 I have read in a notes that it is…
Maaya
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Natural numbers proof via Peano's axioms (not trichotomy)

Prove that for each $x,y$ an element of the natural numbers ($\mathbb{N}$), $xy$. So at least one is true. I have the definition of order to work with and the basic algebra of the natural numbers to work with (i.e. commutativity…
Steve D.
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A (quantifier-free) is true in standard model of PA ==> PA |- A ??

Is the following statement correct ? A is a formula in PA without a quantifier and A is true for the standard model of arithmetic, i.e. the model |N = (N,+,×,0,1,<) This means: |N |= A ==> A is proofable in PA. This means: PA |- A Example: |N |=…
user508589
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