I was trying to define "natural numbers" consider my reasoning and maybe you can complete my construction.
Steps:
1)Let S be the set of all FINITE sets (note that S itself is infinite)
2)Consider the following Axiom (Axiom of Disjointness): For any finite set K there exists another set K' with the same cardinality as K having the property that the interestion of K and K' is {}. IMPORTANT- this axiom probably cannot be proven because we do not have a definition for a set.
3)Define a natural number as the cardinality of any element of set S.
4)Define addition of two natural numbers as the union of two disjoint sets.
5)Demonstrate that addition is well-defined by using the Axiom of Disjointness.
6)Define "zero" as the cardinality of the empty set.
This is were I will stop. Perhaps you can complete this construction. Maybe use Zorn's Lemma to show there is a minimal element, prove the Well-Ordering Theorem for this set and define partial ordering.
Yes zero is a natural number.
Well that thread is from three years ago.
Now I know how natural numbers are defined.
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For any set define .
Let , , , , ...
That is basically how we define natural numbers. We just need to explain what " ... " means. Once we do that we have the natural numbers.