Questions tagged [peano-axioms]
For questions on Peano axioms, a set of axioms for the natural numbers.
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Has this logic of relations been done before?
First Order Logic of Relations "FOLR":
Language: first order logic with Equality "$=$"(and its axioms), and Membership "$ \in $", nLinks "$\overrightarrow {x_1,..,...
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Peano's axioms: why is n++ $\neq$ n? [duplicate]
Why is the successor of n, n++, unequal to n?
I've been reading about Peano's axioms in Analysis 1 by Terence Tao recently (which I am thoroughly enjoying :)).
In one of the exercises I found myself ...
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Gentzen Consistency Proof and Peano's 9th Axiom. Was PA consistent as originally stated or consistent only with a weaker 9th Axiom?
I have done meta-proofs of the consistency of FOL (Studied about 40 years ago), but have not done any for PA and have not looked at (and maybe now could not follow) Gentzen or the other proofs of PA. ...
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How to justify the necessity of the Axioms?
I am studying the logical fundaments of mathematics, but very often I have trouble to understand Peano's and ZFC/ZFC axioms.
In Tao's book Analysis I, I found very helpful when he points out what ...
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What are the inference rules of Peano Arithmetic? [duplicate]
There are many examples in the literature (for example, in this question)
where the author says that something "is provable in Peano arithmetic" or "is not provable in Peano arithmetic&...
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Can Peano arithmetic prove the consistency of "baby arithmetic"?
I am reading Peter Smith's An Introduction to Gödel's Theorems. In chapter 10, he defines "baby arithmetic" $\mathsf{BA}$ to be the zeroth-order version of Peano arithmetic ($\mathsf{PA}$) ...
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Why does this nonstandard model of Robinson Arithmetic fail to be a model of PA?
As talked about in this question, there is a nonstandard model of Robinson arithmetic of the form:
$z_0 + z_1\omega + z_2\omega^2 + z_3\omega^3 + ... + z_n\omega^n$
in which $\omega$ is a formal ...
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Why was it important for Peano arithmetic to prove ITS OWN consistency?
A paraphrase of Gödel's Second Incompleteness Theorem into non-technical language states that, if a formal system is powerful enough to express Peano arithmetic, that system is unable to prove its own ...
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Are positive integer structures axiomatizable?
Recall the first-order Peano axioms for structures of the form $(N;0,S,+,*)$:
$\neg \exists x Sx = 0$
$\forall x \forall y (Sx = Sy \rightarrow x=y)$
The axiom schema of induction (which are really ...
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Can the fact that NBG without choice is a conservative extension of ZF be proved in Peano Arithmetic?
Consider NBG without choice. Denote this theory NBG'. It is fairly easy once one knows a bit of model theory to demonstrate that this theory is a conservative extension of ZF.
An outline of the ...
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Successor arithmetic is not finitely axiomatizable
Consider the theory $\text{Th}(\mathbb{N}, 0, S)$, with $S$ being the successor function. In his book A Course in Model Theory, Poizat claims (p. 109) that this theory is not finitely axiomatizable. ...
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Models of successor arithmetic
Consider the theory $\text{Th}(\mathbb{N}, S, 0)$, which we know to be axiomatized by the following axioms:
$\forall x (S(x) \neq 0)$
$\forall x \forall y (S(x) = S(y) \rightarrow x=y)$
$\forall x (x ...
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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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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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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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Formal induction over two variables
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 $$\...
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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 ...
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(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. ...
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Gentzen's consistency proof of PA
What would be some good introcutory references to start reading about Gentzen's consistency proof of Peano arithmetic?
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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 ...
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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 ...
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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 ...
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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 &...
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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.
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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 ...
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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 ...
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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.
...
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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 ...
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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 ...
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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 ...
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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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Finiteness, finite sets and representing its elements.
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 ...
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Can the natural numbers contain an element that is not representable by a number?
I read the following document: https://www.math.wustl.edu/~freiwald/310peanof.pdf . In this document, the author wants to formalize that natural numbers, that are informally thought of as a collection ...
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Proof that the elements are distinct with Peano's axioms.
Consider the function successor function $s: \mathbb{N} \to \mathbb{N}$ and the Peano's axioms:
P1) $s: \mathbb{N} \to \mathbb{N}$ is injective.
P1) $\mathbb{N} \setminus s(\mathbb{N})$ has only one ...
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Peano axioms set theory
Arithmetic is the core of the study of numbers; from natural numbers you can construct the other numbers. Now, I have heard that set theory is the most fundamental core of mathematics; for this reason ...
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Are the Inductive Axiom and the Well ordering Principle really equivalent?
I am new to formal math, so apologies if this is naive.
In class, we stated 4 of Peano's axioms. For the fifth, my professor claimed that we may either write the Well Ordering Principle or the ...
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In what sense can you prove something in Peano Arithmetic
On Wikipedia it says:
"... the Gödel sentence of Peano arithmetic, is not provable nor disprovable in Peano arithmetic."
When talking about proving something in Peano Arithmetic, are we ...
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Are the Peano Axioms necessary to prove $\forall{x}\exists{y}\ x<y$ (LPL 16.41)
[This is problem 16.41 in Barwise & Etchemendy's "Language, Proof, and Logic".]
The only premise given is $\forall{x}\forall{y}\ (x<y \iff \exists z (x+s(z)=y))$
From this, it asks ...
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Peano Axioms and modeling sets.
I am assuming the standard Peano-Axioms, which can be found here https://en.wikipedia.org/wiki/Peano_axioms under "Formulation". Does one assume that a set of objects called $\mathbf{N}$ ...
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Understanding the natural numbers and Peano's axioms
I am reading the book Analysis $I$ by Terence Tao, where the following is written: [...] $\mathbf{N}$ should consist of $0$ and everything which can be obtained from incrementing: $\mathbf{N}$ should ...
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Existence of closures of sets
I am reading Stillwell's Elements of Algebra. And in Chapter 1, he introduces the real quadratic closure of $\mathbb Q$ as
the set of the numbers obtainable form $\mathbb Q$ by square roots of ...
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Zero divisors and the Peano Axioms
Using the Peano axioms, how can it be proven that the only zero divisor of the natural numbers is $0$ (or there is no zero divisor depending on your definition).
Can it even be done? Do new axioms ...
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Incompleteness theorem: Peano arithmetic vs. standard model of arithmetic
Context (from
https://en.wikipedia.org/wiki/Non-standard_model_of_arithmetic#From_the_incompleteness_theorems):
The incompleteness theorems show that a particular sentence G, the
Gödel sentence of ...
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Does Peano Arithmetic prove all identities involving addition, multiplication, and exponentiation?
Consider the structure $(\mathbb{N};+,*,\uparrow,0,1)$, where $+$ denotes addition, $*$ denotes multiplication, and $\uparrow$ denotes exponentiation. Does Peano Arithmetic, augmented with the axioms $...
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How does second-order arithmetic rule out non-standard numbers?
According to 1 there are axioms of second-order arithmetic which categorically characterize the natural numbers up to isomorphism. The axioms are:
$$\forall x\,\neg(x+1=0)$$
$$\forall x\,\forall y(x+1=...
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