Questions tagged [peano-axioms]

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

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Set-up for the Paris-Harrington Theorem

In his book "Models of Peano Arithmetic" Kaye proves the Paris-Harrington Theorem. He starts off by introducing a "simplification" then proves the short Lemma 14.11 about it (see ...
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Motivation of indicator construction in Kaye

Kaye says the following in his book about models of $\textbf{PA}$ on p. 198: I have no clue what motivates the definition of $f_n(x, y)$. The reference made to Propositions 14.1, 14.2 don't help me ...
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Successor function and peano's axioms for a natural number system? [closed]

If set N=N* with initial element 0=0, what is a successor function such that: -The initial element does not have a predecessor -No element has more than one predecessor but: -There exists a counting ...
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prove $x+1 = 1+x$ using Peano axioms

I wanna prove that $x+1 = 1+x$ (without considering "$x+0=x$",and Im using the old definition of Peano axioms) This is my try: Using this basis: $(1):1+x = x^+$ $(2):x^+ +y=(x+y)^+$ ...
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Single sxiomatizability of r.e. extensions.

Beklemishev proves following lemma in [1] Lemma 2.21. Let $U$ be a consistent r.e. extension of $T$ of comlexity $\Sigma_n$. Then there is single $\Sigma_n$-sentence $\varphi$ such that $T+\varphi$ ...
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Construction of the addition function

I am reading a book called Analysis I by Herbert Amann and Joachim Escher. I am currently stuck on page 33 where they construct the addition operator using functions. One property the addition ...
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Which is the axiom: well ordering principle, principle of induction, both, or none?

From analysis 1 by Terence Tao, I learn that the principle of induction is a peano axiom. In many other analysis books, like analysis by Bartle and Sherbert, the well ordering principle is used to ...
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Proving $PAE ⊢ (Pr_S(\#(X → Y)) → (Pr_S(\#(X)) → Pr_S(\#(Y))))$, where $Pr_S(n)$ holds iff $n$ is the Gödel number of a formula provable from $S$

I'm trying to solve the following question set by my professor: Show that if $S$ is a definable set of sentences, and $Pr_S$ is an associated proof predicate, and $X$ and $Y$ are any formulae, then $...
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Proving a result similar to the diagonal lemma/fixed-point theorem

As the title explains, I'm trying to solve the following exercise that was left for the reader in my lecture notes: Show that for any two formulae $F(v_1)$ and $G(v_1)$ in $L_E$ (language of ...
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Prove that ever $\Sigma_0$ sentence is provable from PA (Peano Arithmetic, with exponentiation)

I am trying to study the intuition and steps leading up to proving Godel's Incompleteness theorems. In a text I am studying it asserts that every true $\Sigma_0$ sentence is provable from PAE (where ...
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$\text{Sat}_{\Delta_0}(x, y)$ is $\Delta_1(\textbf{PA})$ (Kaye's book)

In Kaye's "Models of Peano Arithmetic" in chapter 9 on satisfaction a lot of effort is expanded to prove that $\text{Sat}_{\Delta_0}(x, y)$, representing truth of $\Delta_0$-sentences, is a $...
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Set of natural numbers and Peano axioms

Under standard Peano axioms (below, from Wikipedia), what implies how the set of natural numbers actually looks like, e.g. that 1 = S(0), 2 = S(1), 3 = S(2), etc.? Why not for example 2 = S(0), 4 = S(...
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Peano axioms proof attempt

Given that $K$ is an ordered field satisfying the least upper bound property and $1$ as the multiplicative identity, the set of natural numbers $\mathbb{N}_K$ in $K$ is defined as: $1 \in \mathbb{N}_K$...
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Peano axioms proof

Given that $K$ is an ordered field satisfying the least upper bound property and $1$ as the multiplicative identity, the set of natural numbers $\mathbb{N}_K$ in $K$ is defined as: $1 \in \mathbb{N}_K$...
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$s(n)=n+2$ and Peano axioms

Define $s:\mathbb{N}\rightarrow \mathbb{N} $ given by $s(n)=n+2$, with $n\in\mathbb{N}$. Prove that $\mathbb{N}$ and $s$ satisfy every $n\in\mathbb{N}$ has only one sucessor and $s$ is one-to-one. ...
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$\pi$ is isomorphism from one Peano system $(N, S, e)$ to another $(N', S', e')$, then $\pi^{-1}$ is isomorphism from $(N', S', e')$ to $(N, S, e)$

This is an exercise from Cunningham's book "Set Theory: A First Course". Theorem: Let $(N, S, e)$ and $(N', S', e')$ be Peano systems. Let $\pi$ be an isomorphism from $(N, S, e)$ onto $(N', ...
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Peano Arithmetic - Decidability

I need to show that if Peano Arithmetic does not decide a sentence $\varphi$ then the standard model of Peano Arithmetic satisfies the negation of $\varphi$. I know this partly has to do with Godel's ...
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Isn't saying ${\mathbb{N}=\{0,1,2,3,…\}}$ with Peano axioms a little circular?

My title probably doesn't explain my worry / concern too well, but it's the best title I could think of. I am researching construction of real numbers for a college project, and as a consequence I am ...
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Proof critique of least number principle, please!

I am independently working through Elliot Mendelson's "Number Systems and the Foundations of Analysis," which I find very well-written and rigorous, along with providing a healthy set of ...
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Understanding nonstandard Peano arithmetic

I've had the idea of nonstandard Peano arithmetic introduced to me in the comments of this question. The concept that we could write down the axioms which produce the natural numbers and also produce ...
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Using Peano Axioms Prove That Greater Than Holds For Successor Functions

In first order logic, $\ge$ can be defined as $\exists x(u = x + w)$. I am trying to formally deduce a proof for: $\exists x(u = x + w) \vdash \exists x(s(u) = x + s(w))$ I can use peano axioms and ...
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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 $. ...
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provability and theorem

I am studying first order logic and I have a hard time understanding the link between provable formulas and theorem. In the book by Shoenfield, the predicate $ Pr_{T}(a,b)$ of is defined as the set of ...
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Logic check in Peano arithmetic problem

I am trying to prove this theorem I am using induction Working assumptions 6.10 m+n$^{+}=$(m+n)$^{+}$ 6.22 Theorem Let m $\leqslant$ n denote the fact that m ∈ n or m = n. Then the relation is an ...
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Need help with checking this proof.

Prove that if $m \leqslant n$ then $\exists!\ p\in\omega$. such that $m+p=n$ Correction:induction on $n$ thanks to @Brian M. Scott Working assumptions: 6.1 Definition By the set of the natural numbers ...
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Complete proof of 1+1=2

I'm searching for the proof that Started from Peano's axioms Using modern symbols Detailed as possible Could you give me a link to it? Thanks
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Exercise in Peano arithmetic and order relation

Show that if m<n. and k$\ne 0$ that km<kn Working assumptions: 6.22 Theorem Let m $\leqslant$ n denote the fact that m ∈ n or m = n. Then the relation is an order relation in ω. 6.19 Theorem (...
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Need help with Peano inequality theorem

The following deals with Peano arithmetic I have trouble dealing with inequalities Working assumptions : 6.22 Theorem Let m $\leqslant$ n denote the fact that m ∈ n or m = n. Then the relation is an ...
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Proof of Associativity and Commutativity For Multiplication and Addition of Real Numbers.

This fundamental proof is really bothering me for a long time. I have seen the proofs on proofwiki and other sites but it uses too much mathematical jargon. I would like a nice, intuitive proof using ...
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A basic question about $\Rightarrow$

I am a beginner in mathematical logic. Sometimes I find expressions like the following $$\textrm{PA}\vdash A\implies \textrm{PA}\vdash B\tag{$\star$}$$ For example, in Hilbert-Bernays-Lob derivability ...
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Prove: if $m \in n$ then $m^+ \subseteq n$.

The following is exercise 5(d), section 6.2, from A book of set theory, by Charles Pinter (pg. 122). 5. Prove the following, where $m, n, p \in \omega$. d) If $m \in n$, then $m^+ \subseteq n$. ...
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why peano defines 1 as natural?

as long as i have researched i have found here in this presumed book from Peano https://archive.org/details/arithmeticespri00peangoog/page/n6/mode/2up that actually peano has defined the 1 as the ...
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Is Peano Arithmetic the most-commonly used arithmetic system

I am learning mathematical logic. I want to ask whether the current arithmetic system that we are talking about is Peano Arithmetic. For instance, when we say we want to prove some conjectures, such ...
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Does the well ordering principle really implies mathematical induction?

Under the Peano Axioms, I want to prove that if the Axiom of Induction is substituted with the well-ordering principle (every non-empty subset of $ \mathbb N$ has a minimum element), everything will ...
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How can induction work on non-standard natural numbers?

When we consider the Peano axioms minus the induction scheme, we can have strange, but still quite understandable models in which there are "parallel strands" of numbers, as I imagine in the ...
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Can ZFC decide more values of the Busy Beaver function than PA?

This is related to a previous question. In that question, I asked whether ZFC can define the Busy Beaver function. I was told even Peano Arithmetic(PA) can define it, and also that PA can't decide ...
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Why are the Peano axioms axioms for natural numbers if they have non-standard models?

Why are the Peano axioms axioms for natural numbers if they have non-standard models? Shouldn't axioms determine an object up to isomorphism?
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Proof of commutativity of addition using Peano axioms

I'm studying the proof of commutativity of addition using only Peano axioms (with the distinguished element being 0 rather than 1), the definition of addition, and x+0=0+x=x. The main idea is ...
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Formalizing the number $\#\{n \leq \ell : \varphi(n) \}$ in PA

How do we formalize in $\mathbf{PA}$ that for some arithmetical formula $\varphi(x)$ there exists an $m$ that expresses the number of $n \leq \ell$ such that $\varphi(n)$, and from this obtain $\...
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Can theories with axioms beyond arithmetic make false promises about integer existence?

In this paper, author Nik Weaver warns that there could be questions of $\Sigma_1$-validity of ${\mathrm{ZFC}}$ set theory. As I understand it, he suggests that the axioms of a set theory might be ...
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Addition of natural numbers in Edmund Landau's Foundation of Analysis

I am reading the proof of addition of numbers. In the proof author first shows uniqueness of $x+y$ and then the existence of plus operation with the above listed properties. The second proof is as ...
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Formalization in PA in the Kritchman-Raz proof

In their paper Kritchman and Raz present a proof of Gödel's second theorem using Kolmogorov complexity. To make it work, they operate in some (weak) formal theory $T$ that incorporates some arithmetic,...
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Is there a least standard model of Peano Arithmetics?

Here's the informal part: I'm trying to get a better intuition about the differences between the standard model and the non-standard ones. Some people seem to be really good at imagining what non-...
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How to universalize $\text{Prov}(\ulcorner y < K(x)\urcorner) \to y < K(x)$ in a paper of Kikuchi

In Kikuchi's paper Kolmogorov complexity and the second incompleteness theorem he defines for $\Sigma_1$ binary predicates $R(x, y)$ the condition $$ \Gamma_{1}(R) \Leftrightarrow \forall x\forall y(R(...
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What subsystem of second-order arithmetic proves the weak Godel’s Theorem?

Godel’s Incompleteness Theorem states that any omega-consistent recursive theory in the language of first-order arithmetic is incomplete. But there is a weaker version of Godel’s Theorem that is ...
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The arithmetized completeness theorem

In Kikuchi's paper Kolmogorov complexity and the second incompleteness theorem he states the "arithmetized completeness theorem" as follows: Let $T$ be a recursively axiomatizable theory in ...
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How do you understand $a\times1=a$ in Peano axioms?

How do you understand $a\times1=a$ in Peano axioms? This should be understood as replacing $1$ with $a$? For example, when we multiply $3$ by $2$, that means: replace each $1$ in the number three with ...
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Provably total functions in $\mathsf{Q}$

I was interested in the relations between induction and recursion, and so a natural question (to my mind, anyway), was how much we can prove without appealing to induction, i.e. which functions are ...
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Show that $(P,S,1)$ is a Peano system if and only if it satisfies the Iteration Theorem

You can skip the first section(hints and useful things) if you want, I will keep editing this along I got more findings related to the question. Hints and useful things: I find out that the book from ...
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Prove trichotomy law of addition in $\mathbb{N}$ (Peano Axioms).

I need help in my proof trichotomy law of addition in $\mathbb{N}$ (Peano Axioms). I have already proved that addition is associative and commutative. Also I proved the cancellation law and some ...

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