**452 peano-axioms questions.**

It is well-known that validity in Peano Arithmetic is undecidable. It is less well-known that validity is already undecidable in True Arithmetic (the theory of the standard model of Peano Arithmetic). ...

Presburger Arithmetic is decidable theory but weaker than Peano Arithmetic. Are there systems in some sense that are:
stronger than Presburger but weaker than Peano and remain decidable?
weaker than ...

How do you prove commutativity of multiplication using peano's axioms.I know we have to use induction and I have already proved n*1=1*n.But I cant think of how to prove the inductive step.

The Goldbach Conjecture asserts:
It is possible to write every even number greater that 2 as the sum of
two primes.
Assume I can prove that the Goldbach Conjecture is unprovable from the Peano ...

Suppose you have a candidate structure, and you want to prove it satisfies the Peano axioms. For example, let 1 serve as the first element, and let $s(x)=x/(1+x)$. It's easy to see the non-induction ...

Assume we have a countable, non-standard model of PA where 0=1 is provable. We construct a TM that recursively enumerates every proof in PA. This TM halts if it finds a proof of $0=1$.
Will this TM ...

I have a couple of questions concerning the reflection schema over PA,
Suppose that we want to consider a deviant reflection schema over PA formalized by $Prov_{Ros-PA}(\varphi)\rightarrow \varphi$, ...

Let there be a set $\Bbb{N}$ defined by these 3 axioms:
There exists a set $\Bbb{N}$ such that $1\in \Bbb{N}$ and a function $s:\Bbb{N}\rightarrow\Bbb{N}$ satisfying these properties:
$$\not\exists ...

This question is about the $\operatorname{seq}$ "notation" (for lack of a better word) defined in
https://math.stackexchange.com/a/312915/13675
Can someone give some concrete examples illustrating ...

I have taken no formal mathematics logic course yet, I'm sorry for unclear parts of this question.
I've learned about Presburger Arithmetics few days ago, it seemed really interesting. But since then,...

In the first order theory of $\mathbb N$ with $+$ and $\cdot$ is the set of formulas with bounded quantifiers (universal and existential) decidable, i.e. can we decide for a given such sentence if it ...

Consider the elementary theory of the category of sets (ETCS). Inside this framework, we have that $(\mathbb N, 0, s)$ is a natural number objet and that :
it is a model of the Peano axioms,
it is ...

It seems to me that if one is to "believe" in PA, then one must "believe" in Con(PA) (this is, in some sense, what it means to believe in the theory!). Similarly, I think such a wishful thinker might ...

Consider the set of statements of arithmetic, such that:
the statement contains no existential quantifiers, only universal quantifiers;
the statement contains only logical ...

I suspect that it will be hard to correctly convey this question, but here goes:
How its normally done:
The way I've been taught, and what is normally done in mathematical logic, is as follows:
We ...

Gödel's first incompleteness theorem excludes the possibility of formulating a consistent and decidable set of first order sentences which are true in standard arithmetic from which the truth/...

I want to prove $a < b$ iff $S(a) \leq b$ but I can't figure it out for the life of me, how to even begin.
$S(a)$ is the successor of $a$ and $a < b$ is defined as $a \leq b$ with $a \neq b$. ...

In Terence Tao's Analyis Volume One. He presents that following Lemma. I would like to know if the argument presented below is correct? For your ease i have presented the relevant Peano's axiom as ...

I didn't know how to formulate a more clear title for this question:
Take arithmetic to be the structure $\mathcal N= (\mathbb N, \sigma, +,•, 0,1)$ with its standard interpretation. When I use the ...

In another proof of mine I had written:
Since $d \neq 0$ we can write $S(k) = d$ for some $k$ without violating Peano's 3rd axiom.
Apparently this isn't a valid step in a proof so I want to more ...

Proof 1.1: $a < b \iff b = a + d$ for some positive natural number $d$
First we prove that $a < b \implies b = a + d$ for positive $d$. We will prove it by contradiction. Suppose that $d$ is ...

Please check if my proof has any error! I'm very happy to receive any suggestion to improve my proof. Many thanks for your help.
Definition:
$m \text{ divides } n \Leftrightarrow \exists p, n=pm$.
...

In the usual natural number definition of addition $0+a=a$ is taken as true by definition. This feels like it should be something we derive from $0+0=0$, instead? As in, let's define addition this way:...

PA$^{-}\vdash I\Sigma_n \leftrightarrow I\Pi_n$.
Here $I\Sigma_n$ refers to the induction principle restricted to $\Sigma_n$ formulas. PA$^{-}$ is just PA without induction.
I was reading the paper ...

I would like to prove this theorem with only basic properties of Peano's axioms, addition, and multiplication. Please have a check of my below proof. Many thanks for your help!
In my definition:
$...

I am reading about non-standard models of peano arithmetic, and came across a theorem by Friedman that states the following.
Every non-standard countable model of (peano) arithmetic is isomorphic ...

Is my proof correct?
If we define multiplication for natural numbers as
$a \times S(b) = (a \times b) + a$
$a \times 0 = 0$
And addition as
$a + 0 = a$
$a + S(b) = S(a+b)$
Where $S(n)$ is the ...

I am totally confused when Tao gets into recursive definitions (page 26).
Paraphrasing, the axioms of natural numbers let us define sequences recursively. Suppose we want to build a sequence $a_0, ...

Is my proof correct?
The trichotomy of order for natural numbers states: Let $a,b$ be natural numbers. Then exactly one of the following statements is true:
I. $a < b$
II. $a = b$
III. $a > ...

After defining things like natural numbers and addition, I'd like to prove some things about the operator $\leq$ and ask if they are correct.
Definition 1: Let $a$ and $b$ be natural numbers. We say ...

In Tao's Analysis vol 1 we have various proofs from properties and operations on natural numbers, as well as axioms.
For example additive identity $a+0=a$.
But then in some proofs we apply these ...

Are my proofs correct?
Additive Identity: $a + 0 = a$
Definition of Addition: $a + S(b) = S(a + b)$
where $S(a)$ is the successor of $a$.
Claim: $0 + a = a$.
Base Case: When $a=0$, we have $0 + ...

Every natural number, with the exception of $0$, has a predecessor: $\mathbb{N}^{+} = \mathbb{N} \backslash \{0\}$
I know what predecessor means but can't understand this equation.

The Peano axioms are intended to be able to prove very general statements about arithmetic, such as "all natural numbers can be written as the sum of two primes".
However, how can we use the peano ...

In textbook A Course in Mathematical Analysis by prof D. J. H. Garling, the author proves the cancellation law in multiplication before he moves on to prove the trichotomy of order. I have tried to ...

We can extend Robinson arithmetic (Q) to include exponentiation, using the axioms
1) $x^0=1$ and 2) $x^{S(n)} = x^n*x$. Wiki page
How much can this extension actually say about exponentiation? ...

I often see two variations in how the principle of strong induction is stated:
First Variation: $\Big(B\!\subseteq\!\mathbb{N}\wedge1\!\in\!B\wedge\big(\forall x[x\!\leq\!k\rightarrow x\!\in\!B]\...

In the Peano axioms, the concept of addition is described:
$$a+0=a$$
$$a+S(b)=S(a+b)$$
where $S(n)$ is the successor function. As far as I can tell $a+0=a$ is (even in books such as Tao's Analysis ...

There are two versions I know for mathematical induction, as well as structural induction. One says for all subset $S$ of $\Bbb N$, $1\in S\wedge (\forall n,~n\in S\rightarrow s(n)\in S)\rightarrow S=\...

From Enderton's mathematical logic book sec 1.1, there is a thing called construction sequence for wffs. For example, $P\wedge Q\to R$ can be thought as
$\langle P,~Q,~P\wedge Q,~R,~P\wedge Q\to R\...

I read somewhere that the Peano's axioms can be derived out of ZFC. But if that is the case ZFC would be incomplete right( by Godel's incompleteness theorem)? But since ZFC is in first order logic , ...

Is $K_{n-1}$ a $\Sigma_{n-1}$ elementary substruture of $K_n$?
Let $M$ be any non-standard model of PA.
$K_n$ is define to be the set of $\Sigma_n$-definable elements of $M$.
I have a feeling the ...

Hello my question is related to Why is it impossible to define multiplication in Presburger arithmetic? and to How is exponentiation defined in Peano arithmetic?. I would have preferred to add it as a ...

The original Peano axioms were based on a single unary operator $\operatorname{succ}$ and one second-order induction axiom: $\lbrace \operatorname{succ} \rbrace + \operatorname{IND}_2$
Peano ...

Before write this question, I looked around enough in this forum for a possible answer and although there are many similar questions, I couldn't find one answer which understand or satisfies me. I did ...

One thing that has bothered me so far while learning about the Peano axioms is that the use of parentheses just comes out of nowhere and we automatically assume they are true in how they work.
For ...

Context: I keep running into circular reasoning in attempting to derive strong induction (more generally "induction" whether it be weak or strong) from the well-ordering principle. Assume:
Peano ...

As an exercise I wanted to prove that addition was commutative using the Peano axioms.
To quickly restate the definition of addition (where $S(n)$ is the successor function of $n$, which we can show ...

I wanted to try to prove the commutative property of addition before reading too much about it and "spoiling" things for myself. So I am curious how close I got.
First, some axioms (statements/...

So I am learning about the successor function $S(n)$ where we have $S(n) = n+1$ basically. So $S(0) = 1, S(1) = 2, S(2) = 3, S(3) = 4$, etc.
But are we explicitly mapping $0$ through $9$ "by hand" ...

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