[FOM] Woodin's pair of articles on CH

Ali Enayat ali.enayat at gmail.com
Mon Jan 18 17:43:09 EST 2010

This note is in response to the recent discussion about Woodin's
reference to the well-known fact that the standard model for second
order number theory is "essentially the same" as the model
(M,epsilon), where M is the set of hereditarily countable sets.

As suggested out by Bill Tait, the above two structures are intimately
related at the interpretability level, i.e., they are
*bi-interpretable* . Note that this is stronger than saying that they
are mutually interpretable, e.g. the two theories ZF and ZFC are
mutually interpretable, but they are not bi-interpretable since, by an
old result of Cohen, ZF has a model with an automorphism of order 2,
but as noted by Harvey Friedman, ZFC cannot have such a model

[By the way, I highly recommend Friedman's paper INTERPRETATIONS,
ACCORDING TO TARSKI for a quick but deep introductionto the subject].

The aforementioned bi-interpretability of these two canonical
structures remains to be true even at the level of the first order
theories involved, i.e., the following two theories T_1 and T_2 are

T_1 =  Z_2 plus the Sigma^1_k choice scheme for all k (i.e., second
order number theory with the choice scheme) [See VII.6.1 of SOSA,
Definition VII.6.1, where SOSA is Steve Simpson's canonical text
Subsystems of Second Order Arithmetic].

T_2 =  ZF\{Power Set} + "every set is either finite or countable".

The bi-interpretability of the *standard models* of T_1 and T_2 was
first explicitly noted by Mostowski in the context of "beta-models" of
T_1.  I do not know who was the first to notice that using the same
idea, bi-interpretability continues to hold at the level of first
order theories, but Simpson has refined the bi-interpretability of
T_1 and T_2 by identifying set-theoretic equivalents of various
subsystems of T_1 that contain ATR_0 [see SOSA, Ch. VII.3, especially
the historical notes].

Best regards,

Ali Enayat 

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