Articles of peano axioms

Why induction is defined as an implication instead of an “if and only if” statement?

The formal definition of induction, taken from wikipedia, is written as $$\forall P.\,[[P(0)\land \forall (k\in \mathbb {N} ).\,[P(k)\implies P(k+1)]]\implies \forall (n\in \mathbb {N} ).\,P(n)]$$ But then, it is this version true? $$\forall P.\,[[P(0)\land \forall (k\in \mathbb {N} ).\,[P(k)\implies P(k+1)]]\iff \forall (n\in \mathbb {N} ).\,P(n)]\tag{1}$$ If it is not true can you show me why? (I […]

Prove if $x<y$ then $x+z<y+z$

This question already has an answer here: For each $x,y,z∈\mathbb N$, if $x<y$ then $x+z<y+z$ 2 answers Prove that if $x\leq y$ then $x+z\leq y+z$ 1 answer

proof of commutativity of multiplication for natural numbers using Peano's axiom

How do you prove commutativity of multiplication using peano’s axioms.I know we have to use induction and I have already proved n*1=1*n.But I cant think of how to prove the inductive step.

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 comment to one of the above discussions but I don’t have commenting powers yet 🙁 Anyway, when I look at the […]

How to prove that $+$ is commutative on the natural numbers?

Let $N$ be a non empty set. Let $s:N\to N$ a function satisfying: there is only one element in $N-s(N)$ (denoted by $1$); $s$ is injective; for any subset $X\subset N$, if $1\in X$ and $(n\in N \Rightarrow s(n)\in N)$ then $X=N$. We define a binary operation ‘$+$’ on $N$ by $$m+n=s^n(m)$$ where $s^n$ is […]

How can the Gödel sentence be Pi_1

The Gödel sentence must be provable or unprovable by itself – you have to resolve all definitions until it only uses the elementary symbols of Peano arithmetic. What is the correct way to resolve definitions of primitive recursive functions so that the Gödel sentence stays Π1? According to http://www.staff.science.uu.nl/~ooste110/syllabi/peanomoeder.pdf page 10, the formula that corresponds […]

Are there natural numbers that are not the descendant of 0?

Based on the Peano Axioms (wich are a way to correctly absolutely define the set of natural numbers – correct me if i’m wrong) it is possible to construct a set of symbols that doesn’t quite look the way i imagine the natural numbers: If there is a circle of other symbols next to the […]

Are there meaningful statements about the numbers which cannot be proved with Peano+(independence from Peano)

There are meaningful statements about the natural numbers not provable in Peano. Can all such statements be proven by Peano+(Their independence from Peano). For example can Strengthened Finite Ramsey Theorem https://en.wikipedia.org/wiki/Paris%E2%80%93Harrington_theorem#The_strengthened_finite_Ramsey_theorem be proven using only Peano+(the independence of Strengthened Finite Ramsey Theorem from Peano), and if so, are there any counterexamples?

Not Skolem's Paradox – Part 2

This is a followup to a previous question: Not Skolem’s Paradox. Assume we have a countable, non-standard model of Peano Arithmetic in ZFC. Let $N^*$ be the universe of this model and let $p \in N^*$ be a non-standard natural number larger than any standard natural number. In ZFC we can define the set $X […]

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.