# Questions tagged [peano-axioms]

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

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### If I define addition in the following way, how can I prove that it's commutative?

$a+b=a$, if $b=0$ $a+b=S(a)+S^{-1}(b)$, if $b\not=0$ Here a and b are natural numbers defined according to the Peano axioms, while S represents the successor function. Basically, I am trying to prove ...
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### Why is a unary numeral system bijective?

In this treatment of unary numeral system, it is called "bijective base-1 numeral system." I understand bijective from functions, but why is it used here? Also, the article says the Peano as ...
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### Does the side of the successor function in the second postulate of the definition of addition matter?

In Peano arithmetic addition is usually defined with the following two postulates: $(1a):p + 0 = p$ $(2a):p + S(q) = S(p+q)$ Lets say I put the successor term of the second postulate on the left? ...
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### How much arithmetic can we find definably in the surreals?

Playing fast and loose with size issues for simplicity, let $\mathfrak{S}$ be the structure of the surreal numbers equipped with addition, multiplication, and the simplicity order. I'm curious how ...
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### Question regarding natural numbers in Tao’s Analysis 1.

This doubt might be a mistake on my part, but its confused me for a while now so I decided to ask here. In Terrence Tao’s “Analysis 1”, at the beginning of the chapter about Natural Numbers (where he ...
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### Which sentences are irreducibly self-referential?

Below, all sentences/formulas are in the language of arithmetic, and for simplicity we conflate numbers with numerals and sentences with Godel numbers. Now asked at MO. Say that a sentence $\varphi$ ...
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### Is there a nonstandard model of Peano's arithmetic where any element has only finitely many prime divisors?

By the compactness theorem, there are nonstandard models where don't have finite prime decomposition, and some elements are divisible by infinitely many distinct primes. But what if we want a nice ...
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### Are there any arithmetic statements where there does not exist a proof proving the statement?

Because of Godel's Incompleteness Theorems, there must exist certain statements about the numbers which have no proof via the Peano axioms. This includes The Strengthened Ramsey Theorem. However, this ...
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### Proving $H_a: a≠0$ and $a, b, c$ are natural numbers such that $ab = ac \rightarrow b=c$, proper use of induction

Learning on my own from the ground up, I'm developing a series of basic proofs using Peano's axioms and I suspect I'm misusing the proposition $ka=kb \rightarrow b=c$ in my inductive hypothesis $h_k$ ...
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### How can I proove only one of these three holds? $m>n, m=n, m<n$.

I'm going to finish constructing real numbers from the Peano axiom. I've just finished constructing natural numbers and integers and now I'm going to construct an inequality within the integer range, ...
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### Prove that intersection of a non-empty set of natural numbers is itself a natural number using ZF set theory and Peano axioms

Update I was going to use this question to solve another question using intersection approach. I could find an answer for that question using another approach and posted an answer there but I still ...
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### Halmos's set theory If $E$ is a non-empty set of natural numbers, exists some $k$ in $E$ such that $k \leqq m$ for all $m\in E$. Prove by intersection

Update I saw a similar question which provided a proof by using axiom of induction and I wrote an answer by adapting to one of the answers of that question. But I hope someone can give another proof ...
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Induction. According to the Peano Axioms in this article https://en.wikipedia.org/wiki/Peano_axioms, one axioms states that if $\varphi$ is a unary predicate such that $\varphi(0)$ is true and $$\... • 195 2 votes 1 answer 84 views ### How exactly are you allowed to use free set/predicate variables in first-order arithmetic? In first-order arithmetic, you can't quantify over sets of numbers. However, you can include sets as free variables. I don't think this is just a meta-linguistic thing, as I've read papers about Peano ... • 21 3 votes 2 answers 134 views ### (Request for) simple constructive proof of existence of nonstandard model of PA I know of two straightforward nonconstructive proofs of the existence of nonstandard models of arithmetic. By the existence of the standard model of PA, PA is satisfiable and has an infinite model. ... • 6,426 0 votes 0 answers 40 views ### Gentzen's consistency proof of PA What would be some good introcutory references to start reading about Gentzen's consistency proof of Peano arithmetic? • 1 1 vote 1 answer 51 views ### Peano axioms without identity and function symbols Can the first-order Peano axioms be reformulated without identity and without function symbols? I tried doing this by characterizing one of the following relations axiomatically, but end up with ... • 5,060 0 votes 0 answers 20 views ### Equivalence of two characterizations of initial segments of \mathbb N In proving the recursion theorem, Tao in his Analysis, asks to show the existence of the sets \{0, \ldots, n\} directly from the Peano axioms so that +, \ge, etc. are not yet defined (which he ... • 1,967 3 votes 1 answer 77 views ### Interesting examples of first-order, one-sorted proper extensions of PA Are there any interesting nontrivial examples of first-order, one-sorted theories of natural numbers (i.e. theories whose quantifiers range over natural numbers only and not, say, sets of natural ... • 5,060 3 votes 0 answers 48 views ### Soundness for first order While reading the book by Peter Smith I came across two different definitions of soundness, the general definition A theory T is sound iff its axioms are true (on the interpretation built into T’s ... • 1,971 2 votes 1 answer 62 views ### How to prove non-equality using Peano axioms? Using the five Peano axioms of first-order theory arithmetic, one can prove '1+1=2', '2+3=5' or '342+637=979' (don't try the last one!). However, my question is how to prove a non-equality such as &... • 79 0 votes 0 answers 40 views ### How can I encode a proof in PA as a godel number? It seems straightforward to encode a wff in PA as a number. I can't see how to encode a proof. Could someone please tell me how this is usally done? Thanks. 5 votes 1 answer 109 views ### Satisfaction of Peano postulates in topos with natural numbers object Let \mathcal{T} a topos with a natural numbers object, noted N. Also assume that \mathcal{T} is not degenerate, meaning that its initial and terminal objects \emptyset and * are not ... • 909 0 votes 1 answer 42 views ### Peano axioms and Induction to prove s(x) \cdot y = x \cdot y + y Using the following axioms A1: \forall x \in \mathbb{N}: x + 0 = x A2: \forall x,y \in \mathbb{N}: x + s(y) = s(x+y) M1: \forall x \in \mathbb{N}: x \cdot0 = 0 M2: \forall x,y \in \mathbb{N}: x ... 4 votes 1 answer 154 views ### What is the difference between "Peano arithmetic," "second-order arithmetic," and "second-order Peano arithmetic?" I think this needs to be clarified, so it would be helpful to see an answer to this somewhere. I've seen the following terms: Peano arithmetic. Second-order arithmetic. Second-order Peano arithmetic. ... • 1,924 1 vote 0 answers 66 views ### Are the naturals really a subset of the real numbers? [duplicate] Ok, so this question seems obvious, right? But what I mean is the numbers in the way they are logically / axiomaticaly defined in the foundations of Mathematics. As far as I know, the naturals are in ... • 21 0 votes 3 answers 56 views ### Landau Foundations of Analysis Axiom 4: Is it necessary? Landau gives 5 axioms as the foundations for deriving the theorems in the first chapter: Axiom 1: 1 is a natural number. Axiom 2: If x = y then x' = y'. Axiom 3: 1 is not a successor to any ... 0 votes 2 answers 59 views ### How to prove that there are n natural numbers that are less or equal than n and what properties are allowed to use in induction. Let n \in \mathbf{N}. I wondered how to prove that there are exactly n natural numbers that are smaller or equal than n. This seems somewhat circular which confuses me. I guess the way to do ... • 241 5 votes 1 answer 67 views ### a stronger fixed point theorem Given a unary predicate \phi, the Fixed Point Lemma of PA tells us that there is a sentence S such that:$$\mbox{PA} \vdash S \leftrightarrow \phi (\ulcorner S \urcorner) (Note that $\phi$ doesn'...
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A set $S$ is called finite if there exists a bijection from $S$ to $\{1,...,n\}$ for one $n \in \mathbf{N}$. It is then common to write its elements as $s_1,...,s_n$. I now wonder, why this is ...