Is forcing forded upon us?

Sat Aug 22 15:54:26 EDT 2020

If one regards a model of set theory as an elementary topos, then the forcing rules constitute the internal (Kripke-Joyal) semantics for toposes of sheaves over the given base topos. In this respect forcing is entirely natural. Kripke-Joyal semantics is (like the internal logic of  a topos) intuitionistic. This fact explains why Cohen’s original forcing rules had an intuitionistic flavour. This disappeared in Shoenfield‘s  notion of forcing and in Scott- Solovay’s Boolean-valued model approach.

— John Bell

Professor John L. Bell, FRSC
Department of Philosophy
Western University
London, Ontario
Canada
http://publish.uwo.ca/~jbell/

On Aug 22, 2020, at 12:01 PM, "fom-request at cs.nyu.edu" <fom-request at cs.nyu.edu> wrote:

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Today's Topics:

  1. Is forcing forced on us? (Timothy Y. Chow)
  2. Fwd: [PMA 37] Midwest PhilMath Workshop (Martin Davis)
  3. Is forcing force on us (John Baldwin)
  4. Re: Is forcing force on us (Timothy Y. Chow)

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Message: 1
Date: Fri, 21 Aug 2020 14:14:17 -0400 (EDT)
From: "Timothy Y. Chow" <tchow at math.princeton.edu>
To: fom at cs.nyu.edu
Subject: Is forcing forced on us?
Message-ID: <alpine.LRH.2.21.2008211401320.4722 at math.princeton.edu>
Content-Type: text/plain; format=flowed; charset=US-ASCII

I have recently been thinking (again) about giving a motivated exposition
of forcing.  In particular I asked a question yesterday on MathOverflow:

https://mathoverflow.net/q/369710

In the course of that discussion, I was reminded of one of the dreams in
Shelah's essay "Logical Dreams": Show that forcing is the unique method in
some non-trivial sense.  (Non-triviality is of course needed since forcing
isn't literally the unique method, but I think it's fairly clear what
Shelah is trying to say.)

To a mathematician who is accustomed to ordinary mathematical structures
but has no prior familiarity with axiomatic set theory, the forcing
machinery seems like an inordinately complicated way to extend a structure
(a countable transitive model M) to a slightly larger structure (M[G]).
I am trying to get a better sense for why the axioms of ZF seemingly force
forcing on us.

It occurred to me that if we throw out enough axioms of ZF then eventually
we will reach a point where can can build new models more easily.  So my
question is this: is there some weakening of ZF that is

1. not so weak as to be uninteresting, but
2. weak enough that some new (i.e., not forcing) method emerges to
construct nontrivial models?

Tim

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Message: 2
Date: Fri, 21 Aug 2020 11:52:53 -0700
From: Martin Davis <martin.david.davis at gmail.com>
To: fom at cs.nyu.edu
Subject: Fwd: [PMA 37] Midwest PhilMath Workshop
Message-ID:
   <CA+cpue+bbEFt8ucMekbgDuaUDnMNus6b-fyfqDm3yZ8Hh8dknQ at mail.gmail.com>
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---------- Forwarded message ---------
From: patricia.blanchette.1 <Patricia.Blanchette.1 at nd.edu>
Date: Fri, Aug 21, 2020 at 10:42 AM
Subject: [PMA 37] Midwest PhilMath Workshop
To: Philosophy of Mathematics Association <philosophyofmath at googlegroups.com

The annual Midwest PhilMath Workshop at the University of Notre Dame will
take place via Zoom this year. The downside: no good meals. The upside:
ease of participation!

Dates: Wednesdays in October, 2020.  Details TBA.  See this site for
updated info:
https://philosophy.nd.edu/news/events/mwpmw-2020/

To submit a paper (suitable for a 40-minute presentation), please send a
paper or extended abstract by September 1 to the organizers:

Paddy Blanchette (blanchette.1 at nd.edu
<https://mail.google.com/mail/?view=cm&fs=1&tf=1&to=blanchette.1@nd.edu>)

Tim Bays (bays.5 at nd.edu
<https://mail.google.com/mail/?view=cm&fs=1&tf=1&to=bays.5@nd.edu>)

Curtis Franks (cfranks at nd.edu
<https://mail.google.com/mail/?view=cm&fs=1&tf=1&to=cfranks@nd.edu>)

Jc Beall (jbeall at nd.edu
<https://mail.google.com/mail/?view=cm&fs=1&tf=1&to=jbeall@nd.edu>)

To be added to the list of participants, please send a note to Paddy
Blanchette (blanchette.1 at nd.edu
<https://mail.google.com/mail/?view=cm&fs=1&tf=1&to=blanchette.1@nd.edu>).

Thanks very much. We hope to see many of you there.

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Message: 3
Date: Fri, 21 Aug 2020 17:26:46 -0500
From: John Baldwin <jbaldwin at uic.edu>
To: Foundations of Mathematics <fom at cs.nyu.edu>, tchow at alum.mit.edu
Subject: Is forcing force on us
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   <CA+6jMnivZZ8682Q103Rg-OGXOAAqK34ckZfW273g7XmmF3CV2w at mail.gmail.com>
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Tim Chow wrote:
So my
question is this: is there some weakening of ZF that is

1. not so weak as to be uninteresting, but
2. weak enough that some new (i.e., not forcing) method emerges to
construct nontrivial models?

I reply:
The difficulty in finding a notion of `forcing for arithmetic'  suggests
this is difficult. But one might reverse the question. What is the
distinction between arithmetic and set theory that makes forcing possible?

John T. Baldwin
Professor Emeritus
Department of Mathematics, Statistics,
and Computer Science M/C 249
jbaldwin at uic.edu
851 S. Morgan
Chicago IL
60607
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Message: 4
Date: Fri, 21 Aug 2020 19:40:47 -0400 (EDT)
From: "Timothy Y. Chow" <tchow at math.princeton.edu>
To: Foundations of Mathematics <fom at cs.nyu.edu>
Subject: Re: Is forcing force on us
Message-ID: <alpine.LRH.2.21.2008211934300.24337 at math.princeton.edu>
Content-Type: text/plain; charset=US-ASCII; format=flowed

On Fri, 21 Aug 2020, John Baldwin wrote:
What is the distinction between arithmetic and set theory that makes
forcing possible?

I should clarify that I had in mind potentially "exotic" weakenings of ZF
that may not have been studied much.  In other words, I'm imagining
tinkering with the ZF axioms specifically to see what is forcing us to use
forcing (as opposed to tinkering with the axioms to create systems of
intrinsic interest).

But as for your question here, I tried asking something similar on
MathOverflow a while ago.

https://mathoverflow.net/q/100792

Tim

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