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Questions tagged [peano-axioms]

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

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Peano axioms and first-order logic with $\exists^{\infty}$

All Peano axioms except the induction axiom are statements in first-order logic. The induction axiom is written as $\forall X(0 \in X \land \forall n(n \in \mathbb{N} \rightarrow (n \in X \land n' \in ...
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Peano axiom of induction- why cannot it be replaced by something simpler?

The Peano axiom of induction for natural numbers says that For any property $P(n)$ , if $P(0)$ holds, and that whenever $P(n)$ holds, $P(n++)$ holds, then $P(n)$ holds for all natural numbers. Can ...
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Why we care only about $\Pi_1$ parameters in axiom schemes?

There is exist tradition to formulate axiom schemes such as induction, comprehension or replacment in a way like "for each formula $\phi(y,\overline{x})$ with free variables $y$ and $\overline{x}=x_1,....
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Must heredity be explicilty stated in addtion to Peano's axioms when defining natural numbers?

My question is stated in bold text. This question pertains to the following formulation of Peano's axioms used to formalize an introduction of the natural numbers (beginning with 1) consisting of ...
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How can I prove transitivity of < in Peano Arithmetic

I wish to prove $$(\forall x)(\forall y)(\forall z)((x < y \land y < z) \rightarrow x < z)$$ only using the rules of Peano Arithmetic (PA) including the induction rule. I have seen other ...
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Basis of Mathematics

I was reading up on Goedel's Theorems and was wondering what exactly is the basis of Mathematics today and is the discussion around an axiomatic system settled? I know that there are several attempts ...
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How do we know PA is incomparable with PRA + $\epsilon_0$?

Gödel 2 says that no subtheory of PA can prove Con$_{PA}$, and even though most natural theories $T$ extending PA can prove Con$_{PA}$, this is relatively uninteresting since anyone doubting the ...
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Is there a computable and complete “probabilistic” theory of arithmetic?

Let $\mathbb T$ be a probability distribution over complete and consistent theories of first order arithmetic that contain $PA$. Additionally, we will require that for any sentence $\phi$ in the ...
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How do Peano's arithmetical axioms guarantee that we can construct the natural number set?

I'm probably not understanding but I don’t see how the axioms can guarantee total construction of the natural number set. The successor function as well as the axiom of induction guarantee that the ...
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Unprovable sentence, but provably it is provable.

Work in Peano Arithmetic, PA. Let Prov(n) be a standard proof predicate, so that $PA \vdash Prov(\ulcorner \phi \urcorner) \text{ iff } PA \vdash \phi$ . By Löb's theorem, we know that if $PA \vdash ...
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How did mathematicans prove things before ZFC or Peano

I do not get it. How was a+b=b+a or a (b +c) = ab + ac proved by ancient mathematicians before set theory or Peano axioms? If you cannot logically prove the laws of arithmetic without Peano or set ...
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Axiomatizing a “bounded” companion to PA

There's nothing special about PA here, I'm just focusing on it since it's strong enough to ignore lots of minor technical issues around foundations. If switching to some other theory would yield a ...
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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 ...
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Is it necessary to prove uniqueness of Peano addition?

If we define addition as follows: Define $a+0=a$. For all $a,b\in\mathbb N$ such that $a+b$ is defined, define $a+S(b)=S(a+b)$. It's easy to show through induction that this defines addition for all ...
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Consistency of PA from a Formalist Perspective

In this lengthy thread there's people bickering back and forth about the consistency of PA (Peano Arithmetic) and misunderstandings abound. In reading it I came to an understanding I found useful, ...
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How do Peano Axioms imply “nextness” with the successor?

Going with this explanation of Peano's Axioms, I cannot understand how/where the successor function is definitively stated to be the very next number in the case of natural numbers. In this treatment, ...
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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 ...
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How is induction (without hypothesis) to be used in this proof that $a\ne{b}\implies{a<b\lor{b<a}}$ for $\mathbb{N}$?

The following is from Fundamentals of Mathematics, Volume 1 Foundations of Mathematics: The Real Number System and Algebra, Edited by H. Behnke, F. Bachmann, K. Fladt, W. Süss and H. Kunle. It's part ...
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Is this sufficient to prove the associativity of natural number addition?

This question is regarding the same proposition as is discussed in How should this proof of the associativity of natural number addition be understood? Here I wish to ask if a much simpler proof than ...
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How should this proof of the associativity of natural number addition be understood?

The context of this question is the proofs of the basic laws of arithmetic beginning with Peano's axioms which are stated starting with the number 1, rather than 0. The modified principle of induction ...
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A property of representable functions?

Recall that a function $f:\mathbb{N}^k\rightarrow\mathbb{N}$ is representable in Peano arithmetic if there exists a formula $\varphi(x_1,\dots ,x_k,y)$ such that for every $n_1,\dots,n_k,m\in\mathbb{N}...
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Peano Axiom Proofs: Proving $a < b$, if and only if $a + + \leq b$

As for where I am getting my Peano Axioms, its from Terrance Tao's Analysis I text (math.unm.edu/~crisp/courses/math401/tao.pdf). I am unsure whether my proof is correct for proving the forward ...
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$a < b$ and $c<d$ imply $a+c < b+d$

$a < b$ and $c<d$ imply $a+c < b+d$ when $a,b,c,d$ are arbitrary nonnegative integers. I know that (assuming we include zero) $$\begin{align*} a<b \Leftrightarrow (\exists x\in \mathbb ...
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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 ...
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Show that first order Peano's axioms capture the natural numbers regarding satisfiablity

Denote be $\mathcal P_{MO}$ the set of the monadic second order axioms of Peano. Then as shown by Dedekind any two model are isomorphic to $\mathbb N$. Hence for a monadic second order sentence we ...
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Induction with two variables in PA

This probably has been asked before, but apologies, I don't know how to locate it. I want to prove $\forall x,y: P(x, y)$. My premises are: $$P(0, 0) \wedge \\ [\forall x: P(x, 0)] \wedge \\ [\...
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proof of formula with Peano-axioms

For all natural numbers n define $\Delta_n$ as: $\Delta_0$ is the constant $0$ and $\Delta_{n+1}$ is $S(\Delta_n)$. Here is S the function for the follower, i.e. $\forall x: S(x) = x+1$. 1)I want ...
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Difference between first and second order induction?

Can anyone explain the difference between induction as it's stated in first order logic and that from second order logic? I don't understand the difference as it pertains to things like Peano axioms.
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Algebraically Constructing the Natural Numbers Using a Binary Operation Satisfying Some Properties

Here is the proposed theory: Definition: Let $M$ be a nonempty set with a binary operation $+$ satisfying the following properties: P-0: The operation $+: M \times M \to M$ is both associative and ...
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Why do we need the following set to have an infinite set?

By Dedekind's definition: Definition: We say that some set $S$ is infinite if there exists an injection $f:S\rightarrow S$ such that $\operatorname{Im}(f)\neq S$ (or $f(S)\neq S)$. Now, the book ...
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Definition of nonstandard models without enumerations

To define nonstandard models of Peano arithmetic or set theory, many articles use enumerations like $x>0, x>1, \dots$ where $x$ is said to be a nonstandard natural number, ie a number that is ...
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What is the best book to learn Peano Arithmetic

My goal is to develop an automated proof program for Peano arithmetic. I want to know as much proofs as possible in this system for both my mathematical culture and to enrich the database of my future ...
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Can we prove that the peano axioms are true for $(\mathbb N, \sigma)$ in type theory?

In mathematical logic that I'm used to (i.e. we have first-order formulas and a sequent calculus to derive formula's from axioms), we never prove that the peano axioms are true for the natural numbers,...
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Using the Definition of Dedekind-Infinite to Replace the Axiom of Infinity

The axiom of infinity in ZFC (see this wikipedia link) allows us to construct the natural numbers. But I think we can drop that axiom and replace it with this one: Axiom of Infinity: There exists a ...
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Well-orders on non-standard models of Peano arithmetic

The standard model of Peano's arithmetic, $\mathbb{N}$, has the useful property that the order $\leq$ is a well-order. However, being a well-order cannot be expressed in the language of first-order ...
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Can every true arithmetical sentence be proved from the first-order Peano arithmetic system?

Consider the first-order Peano Arithmetic axioms (which consist of the standard succesor, addition and multiplication axioms, along with first-order induction axioms, as detailed in Wikipedia). This ...
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Bijection between two Peano triples.

Let $(P_1, \sigma_1, s_1)$ and $(P_2, \sigma_2, s_2)$ be two Peano triples. Can we show that there exists a bijection $g:P_1 \to P_2$ such that $g(\sigma_1) = \sigma_2$ and $g \circ s_1 = s_2 \circ g$...
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Does Gödel's incompleteness theorem invoke a Law of Excluded Middle contradiction? [closed]

Does Gödel's incompleteness theorem cause the Law of Non Contradiction to contradict its self? If so, would this be a considered a conjecture?
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Why Presburger arithmetic cannot prove pigeonhole principle

The pigeonhole principle is the scheme $\forall \bar{a},s \{\forall x\!<\!s\!+\!1 \exists y\!<\!s\psi(x,y,\bar{a}) \to \exists x_1,x_2\!<\!s\!+\!1\exists y\!<\!s[x_1\neq x_2 \wedge \psi(...
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Existence of Natural Numbers as an Axiom

By natural numbers $\mathbb{N}$ I understand any set satisfying Peano axioms: $0 \in \mathbb{N}$ $\sigma : \mathbb{N} \to \mathbb{N}$ $\forall n \in \mathbb{N} \; . \; \sigma(n) \neq 0$ $\forall n,m \...
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What is Complete Induction without Hypothesis?: Ordering of $\mathbb{N}$ from Peano's Axioms

The following is from Fundamentals of Mathematics, Volume 1 Foundations of Mathematics: The Real Number System and Algebra; Edited by H. Behnke, F. Bachmann, K. Fladt, W. Suess and H. Kunle I've used ...
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Do induction axioms of Peano Arithmetic have other simple equivalent forms?

For the example of Presburger Arithmetic, which consists of basic axioms for successor, addition, as well as an infinite set of induction axioms. However, it is well known that Presburger Arithmetic ...
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Proving well ordering principle from PMI (PCI) from peano axioms

Axiomatically, set $\mathbb{N}$ is constructed via injective function $s:\mathbb{N}\rightarrow \mathbb{N}$ and an element $1\in\mathbb{N}$ and we have that $\forall n\in\mathbb{N}:s(n)\neq1$. Together ...
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Proving the addition law for equality, eg. $+$ is compatible with $=$ on $\mathbb{N}$

Using following definition of addition (Peano arithmetic): Definition:$+:\Bbb{N}\times\Bbb{N}\rightarrow \Bbb{N}$ $$\forall a\in\mathbb{N}:a+1=s(a)\tag{i}$$ $$\forall a,b\in\mathbb{N}:a+s(b)=s(...
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Proving the commutative law for 1-based $\Bbb{N}$

We will work with $1$-based natural numbers, that is $0\notin\Bbb{N}$. Let $s:\Bbb{N}\rightarrow\Bbb{N}$ be the function axiomatically given in peano axioms, we define addition and multiplication as ...
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Inductive proof using Fitch software

I am trying to prove the integer square root theorem $\forall x: \mathbb{N}, \exists y : \mathbb{N}((y^2 \leq x) \land (x < (y+1)^2))$ for $\lfloor \sqrt{x} \rfloor$. In words: for any natural ...
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George Boolos and Gödel's Second Incompleteness Theorem

In Mind, Vol. 103, January 1994, pp. 1-3, George Boolos wrote: And so on. In fact, if a claim can be proved, then it can be proved that the claim can be proved. And that too can be proved. ...
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Is Fermat's last theorem provable in Peano arithmetic?

The sentence $S$ which Gödel in his proof of the incompleteness theorem proves to be be unprovable in the system of Peano arithmetic can be proved (as a true theorem of PA) outside PA (and necessarily ...
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What's the relationship between ZFC and Peano axioms? Are they overlapping, complementary or separate?

What's the relationship between ZFC and Peano axioms? Are they overlapping, complementary or separate? I'm particularly wondering whether natural number stuff like order can expressed in ZFC at all. ...
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Proof of induction from higher starting point

I'd like to request to verify this proof that for a arbitrary natural $n_0$ this holds: $[C(n_0+1) $ and $\forall n>n_0 : C (n) \implies C (n+1)] \implies \forall n> n_0 : C (n) $ ($C (n)$ means ...