# 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 ...
• 398k
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 ...
• 250k
Accepted

### 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 ...
• 334k
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 ...
• 250k

### 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 ...
• 44.2k
Accepted

• 79.4k

### "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 ...
• 11.9k
Accepted

### Is this halting time a non-standard integer?

Yes, this is basically right, although I would quibble a bit with this: In models of this theory, $M$ halts. I think it would be more accurate to say that models of this theory contain a nonstandard ...
• 437k
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• 334k
Accepted

### (Request for) simple constructive proof of existence of nonstandard model of PA

Ultrapowers are certainly important and worth understanding, but in my opinion - especially if we're looking at nonstandard models of arithmetic in particular - they are not optimally constructive. ...
• 250k
Accepted

### "Real-closed" vs "transfer principle"

1. Your "extensibility" condition is strictly stronger than being a real-closed field. You already know that every structure which satisfies your extensibility condition is a real-closed ...
• 11.9k