# peano-axioms's questions - Chinese 1answer

452 peano-axioms questions.

### 3 Decidability of quantifier-free formulae in Peano- and True Arithmetic

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). ...

### 4 Systems of arithmetic models [on hold]

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 ...

### 2 proof of commutativity of multiplication for natural numbers using Peano's axiom

2 answers, 1.824 views natural-numbers peano-axioms
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.

### 10 Is it a paradox if I prove something as unprovable?

1 answers, 1.360 views logic proof-theory peano-axioms
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 ...

### 1 How to prove Peano induction axiom for a candidate model

0 answers, 30 views logic induction peano-axioms
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 ...

### 1 How Long is the Shortest Proof of 0=1?

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 ...

### 3 Reflection schema over PA for Rosser provability predicate

1 answers, 66 views logic peano-axioms
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 $Prov_{Ros-PA}(\varphi)\rightarrow \varphi$, ...

### 1 Mathematical induction - the set version v.s. predicate version

0 answers, 54 views logic induction set-theory peano-axioms

### 3 About ZFC, peano's axioms, first order logic and completeness?

2 answers, 271 views first-order-logic peano-axioms
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 $K_n$ [closed]

1 answers, 42 views logic model-theory peano-axioms
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 ...

### 2 Why is it impossible to define multiplication in Presburger arithmetic yet possible to define exponentiation in Peano Arithmethic?

1 answers, 129 views peano-axioms presburger-arithmetic
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 ...

### Different arithmetics

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 ...

### 2 An infinite set of axioms in ZF? What does that mean?

1 answers, 101 views logic set-theory axioms peano-axioms
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 ...

### 17 Do we have to prove how parentheses work in the Peano axioms?

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 ...

### 2 Proof of Strong Induction Using Well-Ordering Principle

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 ...

### How do I know what I need to show when using induction? (Peano axioms)

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 ...

### 4 Prove addition is commutative using axioms, definitions, and induction

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/...

### 1 Some questions about the successor function

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" ...