As the title says, why is the smallest example of an admissible set hereditarily finite set?
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3What does "admissible" mean for you here? Wikipedia gives one definition of which it says that the smallest admissible set at all is the set of all hereditary finite sets, which is not itself finite. – hmakholm left over Monica Feb 16 '13 at 15:40
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@hwe: is it possible to add further clarification to your question and make it clearer or more specific? Regards – Amzoti Feb 16 '13 at 17:12
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@hwe Could you please clarify your question? – Potato Feb 16 '13 at 19:39
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Whether you include this in the standard formulation of Kripke-Platek set theory $\mathsf{KP}$, or consider it part of logic, "there is a set" holds (see here), so no admissible set (that is, no model of $\mathsf{KP}$) is empty.
But then, by $\Sigma_0$-separation, there is an empty set.
Repeated application of pairing gives us now that any admissible set has as elements $$ \emptyset,\{\emptyset\},\{\{\emptyset\}\},\{\{\{\emptyset\}\}\},\dots $$ and arguing with extensionality shows that all these sets are different.
It follows that no model of $\mathsf{KP}$ can be finite, much less hereditarily finite.
If what you meant to ask is why the set $V_\omega=HF$ of all hereditarily finite sets is a model of $\mathsf{KP}$ without infinity, perhaps asking it as a different question is more appropriate. When the axiom of infinity is considered as well, the smallest admissible set is $L_{\omega_1^{CK}}$, where $\omega_1^{CK}$, the Church-Kleene ordinal, is the smallest non-recursive ordinal.
Andrés E. Caicedo
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