# [FOM] A question about \omega-consistent theories

Dan Isaacson daniel.isaacson at philosophy.ox.ac.uk
Mon Apr 27 14:54:20 EDT 2009

Every omega-consistent Sigma0-complete theory is Sigma2-sound, so a fortiori
Sigma0-sound.  To construct an omega-consistent theory which proves a false
sentence, let S be any sound theory in which syntax can be arithmetized.
Let K be a sentence that 'says' "This sentence, when added to S results in
an omega-inconsistent system" (such exists by the diagonal lemma).  Then S +
K must be omega-consistent.  If not then K is true, in which case S + K is
sound and hence omega-consistent, which implies K is false.  Hence K is
false and S + K is an omega-consistent theory which proves a false Sigma3
sentence, namely K.  This result is due to Kreisel, in a review of a paper
of Henkin, "A generalization of the concept of omega-consistency", Math
Reviews Vol 16 No 2 (Feb 1955), p. 103.

Daniel Isaacson

Faculty of Philosophy
Oxford University
10 Merton Street
Oxford OX1 4JJ, U.K.
daniel.isaacson at philosophy.ox.ac.uk

-----Original Message-----
From: fom-bounces at cs.nyu.edu [mailto:fom-bounces at cs.nyu.edu] On Behalf Of
Arnon Avron
Sent: 27 April 2009 17:06
To: fom at cs.nyu.edu
Subject: [FOM] A question about \omega-consistent theories

Can anybody give me an example of a (not necessarily recursive
or r.e.)  \omega-consistent theory in the language of PA which
proves all true quantifiers-free sentences in this language,
but also some sentence which is  not true (in the standard model of PA)?

Thanks

Arnon Avron

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