4
votes
2answers
774 views

Axiom to exclude nonstandard natural numbers

In Peano Arithmetic, the induction axiom states that there is no proper subset of the natural numbers that contains 0 and is closed under the successor function. This is intended to rule out the ...
5
votes
2answers
849 views

Set Theory inside Arithmetics via the Ackermann Yoga

Among the basic results of logic which, simple as they are, never fail to intrigue me, is Ackermann's interpretation of $ZF$-Infinity in $PA$ (see for refs this MO question and here for an excellent ...
10
votes
2answers
597 views

Is Robinson Arithmetic biinterpretable with some theory in LST?

Let ZFC$^{\text{fin}}$ be ZFC minus the axiom of infinity plus the negation of the axiom of infinity. It is well-known that ZFC$^{\text{fin}}$ is biinterpretable with Peano Arithmetic. In this sense ...
7
votes
4answers
1k views

Incompleteness and nonstandard models of arithmetic

The following are a collection of doubts, some of which shall have concrete answers while others may have not. Any kind of help will be welcome. Reading Peter Smith's "Gödel Without (Too Many) ...
1
vote
3answers
2k views

“Interesting” properties of sets of natural numbers

On Wikipedia, there is a list of properties of sets of reals, which are in some sense "interesting": just have a look. I could not find a comparable list of properties of sets of natural numbers ...