3.8 Natural numbers in ZF[246]
In this section we will build a model of the natural numbers inside the ZF set theory; this model satisfies Peano’s axioms [1XD] and has an order relation that satisfies [26H], so this model enjoys all properties described in Sec. [1X9]; for this reason in this section we will mostly discuss properties that are specific of this model.
Successor
Natural numbers in ZF
Using the axiom of infinity [243] we can prove the existence of the set of natural numbers.
We can also prove directly the induction principle.
To prove the above theorem, the exercises in the following section can be used.
The ordered set \(ℕ,⊆\) then enjoys these properties.
More details are in the course notes (Chap. 1 Sec. 7 in [ 3 ] ); or [ 14 ] , [ 13 ] .
Transitive sets
The previous exercises prove Theorem [24D], then by results of Sec. [1X9] we obtain that \((ℕ,≤)\) is well ordered.
Here following are other interesting exercises.
Ordinals
Perusing the above results we can give some elements of the theory of ordinals.
[ (da sistemare) ]