[FOM] Certificates are fully practical
Harvey Friedman
hmflogic at gmail.com
Sun Sep 29 03:16:51 EDT 2013
On Sat, Sep 28, 2013 at 2:49 PM, Timothy Y. Chow <tchow at alum.mit.edu> wrote:
>
> What I believe your parable shows is the following: We know already that
> the continuum hypothesis is neither provable nor disprovable in ZFC. It
> would be interesting to strengthen this result by showing that there is no
> infinitary proof or disproof either, in some suitable sense of "infinitary
> proof."
>
> There is an obvious L_omega_1,omega form of ZFC, and using the standard
complete set of countably infinitary axioms and rules for L_omega_1,omega,
this extension of ZFC has a standard set of countably infinitary axioms and
rules.
It can be easily seen that this version of ZFC neither proves nor refutes
the continuum hypothesis, assuming ATR_0 + "there is an omega model of ZFC
containing any given subset of omega".
I leave it to others to extend this result for stronger logics, where one
does not have, or does not expect to have, a complete set of axioms and
rules for the logic.
Harvey Friedman
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