# What is Peano: Definition and 17 Discussions

Giuseppe Peano (; Italian: [dʒuˈzɛppe peˈaːno]; 27 August 1858 – 20 April 1932) was an Italian mathematician and glottologist. The author of over 200 books and papers, he was a founder of mathematical logic and set theory, to which he contributed much notation. The standard axiomatization of the natural numbers is named the Peano axioms in his honor. As part of this effort, he made key contributions to the modern rigorous and systematic treatment of the method of mathematical induction. He spent most of his career teaching mathematics at the University of Turin.
He also wrote an international auxiliary language, Latino sine flexione ("Latin without inflections"), which is a simplified version of Classical Latin. Most of his books and papers are in Latino sine flexione, others are in Italian.

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1. ### Prove ##a + b = b + a## using Peano postulates

Following is a set of Peano postulates I am using as defined in the book "Th real numbers and real analysis" by Ethan Bloch. There exists a set ##\mathbb{N}## with an element ##1 \in \mathbb{N}## and a function ##s: \mathbb{N} \rightarrow \mathbb{N} ## that satisfy the following three...
2. ### Prove ##(a+b) + c = a + (b+c)## using Peano postulates

I have to prove the associative law for addition ##(a+b) + c = a + (b+c)## using Peano postulates, given that ##a, b, c \in \mathbb{N}##. Now define the set $$G = \{ z \in \mathbb{N} |\forall\; x, y \in \mathbb{N} \quad (x + y) + z = x + (y + z) \}$$ Obviously, ## G \subseteq \mathbb{N} ##...
3. ### I Proof that 1 is an odd number using Peano Axioms of naturals

So I was just writing a proof that every natural number is either even or odd. I went in two directions and both require that 1 is odd, in fact I think that 1 must always be odd for every such proof as the nature of naturals is inductive from 1. I am using the version where 1 is the smallest...
4. ### I Hilbert's omega rule, induction, omega-consistency

As I understand Hilbert's omega rule for a first-order proposition P over the natural numbers, P(0) &P(1) &P(2) &... ⇒ ∀n∈ℕ P(n) :star: which seems to be the same as ω-consistency. Is there a difference? Further, the axiom schema of induction has each axiom for a proposition P over the...
5. ### Prove a statement using Peano's Axioms

Homework Statement Let, m, n be natural numbers and S(n) the succesor of n. If S(n)*m = nm + m Prove that m*S(n) = nm + m Homework Equations The Attempt at a Solution
6. ### I Interval of existence and uniqueness of a separable 1st ODE

Problem: y'=((x-1)/(x^2))*(y^2) , y(1)=1 . Find solutions satisfying the initial condition, and determine the intervals where they exist and where they are unique. Attempt at solution: Let f(x,y)=((x-1)/(x^2))*(y^2), which is continuous near any (x0,y0) provided x0≠0 so a solution with y(x0)=y0...
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### I Cancellation law multiplication natural numbers

Hello everyone. I wanted to prove the following theorem, using the axioms of Peano. Let ##a,b,c \in \mathbb{N}##. If ##ac = bc##, then ##a = b##. I thought, this was a pretty straightforward proof, but I think I might be doing something wrong. Proof: Let ##G := \{c \in \mathbb{N}|## if ##a,b...
8. ### Learning Peano Arithmetic

It seems that learning PA is necessary if you want to understand the relationship between logic and math. Should I track down this book at the library, a chore which will take up an hour of my precious time The principles of arithmetic, presented by a new method" in Jean van Heijenoort...
9. ### Can Sets be Defined from Peano Axioms Alone?

Is it possible to define sets from just the peano axioms? Usually when people use the peano axioms as the basis of their math they just assume the existence of sets but without axioms regarding sets we technically shouldn't just say they exist. Oh, also there are two versions of the...
10. ### Was Giuseppe Peano Influenced by Euclid in Developing Axiomatic Systems?

Hello there, First off: I realize this is not a history of math sub-forum, but I could not find any such thing. Now, secondly: I'm writing an essay on the history axiomatic systems, and I found one or two sources that indicate that Giuseppe Peano was in fact influence by Euclid.I wonder...
11. ### Peano axioms for natural numbers - prove 0.5 ∉ N

i am studying real analysis from terence tao lecture notes for analysis I. http://www.math.ucla.edu/~tao/resource/general/131ah.1.03w/ from what i understand , property is just like any other statement. for example P(0.5) is P(0) with the 0s replaced with 0.5 . so the notes says (assumes ?)...
12. ### [LOGIC] Proof by Induction in Peano Arithmetic

I have to do the following using these axioms PA1-7, the others below it are previously proved results I can use too. [Sa] means the successor of a. Base Case: y = S0 x.S0 = S0 → x.0 + x = S0 → 0 + x = S0 → x = S0 & y=S0 Now the induction step is usually y=a to y=Sa, however this does...
13. ### How do primes come out of Peano arithmetic?

Let (N, s(n), 0) be a Peano space. That is, N=\{1,2,3,\dots \} is a set in which http://en.wikipedia.org/wiki/Peano_arithmetic" can be used. We can then define: 0=\varnothing, 1=\{0\}, 2=\{0,1\},\dots \implies n=\{0,1,2,\dots ,n-2,n-1\} s(a)=a\cup \{a\}\implies s(a)=a+1 From here we...
14. ### Peano Existence Theorem confusion

I'm working on a proof of the Peano Existence Theorem. It says that: For a continuous ordinary differential equation dx/dt = f(x, t), where x is in R^n and f is continuous on |t-t_0|<=a, ||x-x_0||<=b when you have an initial value, x(t_0) = x_0, then there is a solution on |t-t_0|<= c...
15. ### Validate Peano Arithmetic Approach

Homework Statement Show that the natural numbers satisfy commutativity of multiplication and distributivity of multiplication over addition. Homework Equations The Attempt at a Solution I'm wondering if there is any potential circularity in this reasoning. I proved distributivity...
16. ### Exploring Peano Curves: Links and Books for Study

Can anyone point me towards an online link or some books where I can study these bad boys? I am supposed to write like a 5-10 page paper on Peano curves in which I prove a few interesting things regarding them.
17. ### Anyone familiar with Peano' (or Grassman) algebra?

I'm an engineer, not a mathematician... the professor has assigned several proof questions, and I'm having difficulty answering them... (This may need to be moved to homework help, but the topic is unusual, so I thought I'd get better response here) Terminology: v is a join operation...