[FOM] strong hypotheses and the theory of N

Ali Enayat ali.enayat at gmail.com
Mon Mar 22 14:21:16 EDT 2010

In my earlier posting (March 17), I specified a recursive set of
axioms, without using Craig's trick, of the arithmetical consequences
of ZF. In light of the recent postings of J. Shipman (March 20), and
A. Koskensilta (March 16),  I wish to add the following two comments:

1. The proof of the axiomatization I provided is not hard; it uses the
ZF-reflection theorem in one direction, and in the other direction,
relies on the so-called Sigma_1-completeness of PA; the latter
direction can also be established model-theoretically using
recursively saturated models.

2. The set of arithmetical consequences of ZF is not finitely
axiomatizable over PA, and indeed, not axiomatizable over PA by any
set of arithmetical axioms with a bounded number of quantifier
alternations.  So, Koskensilta's is correct in answering "NO" to
Shipman's question only if one takes the view that the naturality
condition on the extra axioms requires them to only use a bounded
number of quantifiers.


Ali Enayat

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