[FOM] The necessity of forcing
joeshipman at aol.com
Fri May 11 10:30:26 EDT 2007
If my claim (when reasonably reinterpreted to avoid Rosser's sentence
and similar contrived examples) is true, then we can say that all
sentences shown relatively consistent with ZFC without forcing are
consistent with each other and assume them all as axioms. But does
this amount to anything more than assuming V=L? (Yes, because V=M is
stronger still; but can we go further than this?)
The question is not whether forcing is the unique method, it's whether
it is the unique method for showing the relative consistency of results
that aren't consequences of V=M.
If so, this is a very serious concern, because neither of those two
methods works for arithmetical sentences. Is there ANY non-contrived
arithmetical sentence A such that ZFC |- "both A and ~A are relatively
consistent with ZFC" ?
If the answer to the last question is "No", then the only method for
showing arithmetical sentences independent is to show that they imply
Con(ZFC). But the existence of Rosser's sentence is then very
disturbing. It may be that there is a whole realm of "sideways
independence results", for which finding a mathematically natural
example is very hard.
My personal opinion is that mathematically natural examples will
eventually be found, but this won't happen until after P is shown to be
different from NP, and after mathematically natural "Turing degrees"
other than 0, 0', 0'', ... , are found. All these problems have a
family resemblance. and a resemblance to CH, because they hinge on the
difficulty of finding something "in between" S and O(S), where S is a
set and O is an operator that represents some kind of step up to a
higher logical level.
From: tchow at alum.mit.edu
To: joeshipman at aol.com
Sent: Fri, 11 May 2007 9:47 AM
Subject: Re: [FOM] The necessity of forcing
> CLAIM: There is no important result (provable in ZFC) of the form
> If ZFC is consistent, then neither A nor ~A is a theorem of ZFC
> which has been proven without using forcing in at least one of the
> halves of the result.
Problem 2.4 of Saharon Shelah's "Logical Dreams" is "Show that forcing
the unique method in a non-trivial sense." Unfortunately he doesn't
really elaborate on this problem, but it sounds consistent with your
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