# Tag Info

Accepted

### What is an example of a non standard model of Peano Arithmetic?

Peano arithmetic is a first-order theory, and therefore if it has an infinite model---and it has---then it has models of every cardinality. Not only that, because it has a model which is pointwise ...
Accepted

### How does Gödel Completeness fail in second-order logic?

The property that "every consistent theory has a model" does not hold for second-order logic. Consider, for example the second-order Peano axioms, which are well known to have only $\mathbb N$ as ...
Accepted

### "Natural" non-standard models of Peano.

Certainly the good money is on the (current) nonexistence of such an example. But that's boring. Here are a couple positive observations, although - in my opinion - each falls short of an actual ...
Accepted

### How can addition be non-recursive?

Tennenbaum's theorem doesn't claim that your TM doesn't work inside the model (if its operation is suitably arithmetized in well-known ways). It says that there cannot exists a TM that works outside ...

### How does Gödel Completeness fail in second-order logic?

First of all, even in the first-order case that "proof" doesn't work: how do you get $M$? You need the assumption that, if $K'$ is consistent, then it has a model; but this is exactly what you are ...

### How does Gödel Completeness fail in second-order logic?

Take the language of arithmetic, augmented by a single constant symbol $c$. Now add to the Peano [second-order] axioms the following schema, $0<c$, $s(0)<c$ and so on. If this theory is ...

### Explicit countable elementary extension of $\mathbb{N}$

Unfortunately, Tennenbaum's theorem (https://en.wikipedia.org/wiki/Tennenbaum%27s_theorem) shows that any such extension is non-computable. So there isn't really an explicit example. That said, if ...
Accepted

### How does induction fail in computable nonstandard models?

Here's an even simpler one: "Every number is either even or odd." That is, $$\forall x\exists y(x=y+y \mbox{ or } x=y+y+1).$$ The polynomial $x$ is a counterexample. Ignoring the specific model and ...
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### The satisfaction relation is undefinable, but does it still "exist"?

Option 2 is exactly what is going on. For each meta-language natural number $n$, we can write down a formula in the language of set theory that formalizes $\vDash_n$. However, we cannot define a ...

### What is an example of a non standard model of Peano Arithmetic?

It should be mentioned that one of the most concrete nonstandard models of PA was developed by Skolem in the 1930s in ZF (without the axiom of choice, unlike the constructions mentioned in the other ...
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### Non-standard model of arithmetic - why is adding new constants to the model acceptable?

There's nothing mysterious or underhand going on here. Though there is an important point in the background, which is simply this: In a standard formal language (like the language of Peano ...

### Why do models of ZF which are not $\omega$-models have non-standard formulas whose length is "infinitely large natural numbers"?

For every natural $n$, $\phi_n$ is a sentence, where $\phi_0$ is $\forall x\,(x=x)$ and $\phi_{n+1}$ is $(\phi_n\land\phi_n)$. By recursion, there is a sentence in the theory that codes this claim and ...

### "Natural" non-standard models of Peano.

This answer is a remark on "all methods I've seen of constructing non-standard models never in any sense uniquely specify any non-standard model". It shows that you can uniquely specify a ...
Accepted

### Is there a consistent arithmetically definable extension of PA that proves its own consistency?

Here is the answer I had made over on MathOverflow: Surprisingly, the answer is yes! Well, let me say that the answer is yes for what I find to be a reasonable way to understand what you've asked. ...
Accepted