# Tag Info

### Peano axiom of induction with "no junk"

The set of all the dominoes in the picture satisfies axioms 1-8. (The caption says this, if properly interpreted. The dark dominoes by themselves do not satisfy axiom 8. They satisfy axiom 1 only if ...

### How do Peano's axioms make it clear what the successor is equal to?

The general concept of an axiom system does not tell you what the objects are that it refers to, it tells you only properties of those objects and of the interactions between them. The axioms do not ...

### In the Peano Axioms, how do we know that the successor of a natural number $n$ is $n + 1$? (It's never explicitly stated that it should be "$+1$")

1 is defined as $S(0)$, where $S$ denotes the successor function. The relevant convention does not concern what is known about numbers, but only what we call them. Of course, our preconceptions about ...

### Peano axiom of induction with "no junk"

The idea is that the first $8$ axioms describe natural numbers together with some possible "junk". Those junk may have various forms. Axioms 1–8 are also satisfied by the set of all light ...
Accepted

### Can Goodstein's theorem be false in some model of Peano arithmetic? How?

I think it's helpful to consider $(i)$ a simpler statement than Goodstein's theorem and $(ii)$ a weaker theory than $\mathsf{PA}$. The reason for the former is that Goodstein's theorem has lots of ...
Accepted

### How do Peano's axioms make it clear what the successor is equal to?

`how do we know that the successor of $1$ is $2$ based on Peano's axioms' $2$ is defined to be the successor of $1$, so $2=1'$. Define addition by the following two rules: $x+1=x'$ $x+y'=(x+y)'$ So, ...
Accepted

### What is the mathematical definition of "standard arithmetic/standard natural numbers"?

Your use of the word "intuitive" means that we're entering philosophical waters. In ZFC, as you know, one can prove the formal version of the assertion "there is, up to isomorphism, ...

### First order theories stronger than PA

We have to distinguish between superficially different theories, and theories that are "essentially different". For example, you ask about adding exponentiation to PA. As noted in the ...
Accepted

### How do Robinson arithmetics axioms prevent this model of N?

This image is really only about how the successor function behaves, so is more suitable to a discussion of the Peano axioms than to a discussion of Peano Arithmetic. But more to the point: in order to ...

### If Goldbach's conjecture G is undecidable in PA, then can we prove $\mathbb N\models G$?

If we could prove (in any given system) that $G$ is undecidable in PA, then we could prove there that it is true. But it is quite possible that $G$ happens to be undecidable (and true) but there is ...
Accepted

### Are the natural numbers definable in ZFC-Inf

Yes. If $\mathfrak{M}\models\mathsf{ZF}$-$\mathsf{Inf}$, the "natural numbers of $\mathfrak{M}$" are the things which $\mathfrak{M}$ thinks satisfy the following formula: $\nu(x):\quad$ $x$ ...
We can use the definition of Dedekind infinite to avoid appealing to "foreverness" or the successor function. To do this we say a set $S$ is infinite if there exists some $S' \subsetneq S$ ...