[FOM] Mac lane set theory
Jeremy Bem
jeremy1 at gmail.com
Mon Dec 21 19:20:03 EST 2009
For me, the appeal of weaker set theories is the possibility of
incorporating them into a straightforward worldview such as the
following:
"Alice is a Platonist. She believes in the objective reality of sets,
and that first-order statements in the language of set theory have
definite truth values. She believes that the axioms of ZC (Zermelo
set theory with choice) are true, and so of course she believes that
ZC is consistent. She also believes that by putting E := the empty
set, S := E U Pow(E) U Pow(Pow(E)) U …, M := S U Pow(S) U
Pow(Pow(S)) U …, she obtains a set which is a model of ZC. This
confirms her belief that ZC is consistent; in fact, it is for her an
(informal) proof of the consistency of ZC. She believes (and can
prove formally) the fundamental theorem of algebra, Goodstein's
theorem, and many other standard, beautiful results. She doesn't know
the truth value of Borel determinacy, projective determinacy, or the
continuum hypothesis."
Is there an objection to this worldview, and is there a similarly
straightforward worldview based on ZFC? In particular, it doesn't
seem that one could get the same reassurance that Alice gets from M,
from the definition of the Von Neumann universe. I've never been at
peace with that definition, but in particular, V isn't a set, is it?
So it isn't a model in the usual sense of first-order logic. This
seems like a real sacrifice, not ad hoc, and not worth making just for
Borel determinacy.
As far as Mac Lane set theory is concerned, perhaps Alice should stick
to that when doing actual formalization in algebra or similar. I
believe ZC can prove the consistency of Mac which can itself prove
many beautiful results? So Alice would then have *formal* reassurance
about her formal proofs. And she could also use the very practical
HOL theorem proving systems which you yourself proved equivalent to
Mac :)
-Jeremy
On Sun, Dec 20, 2009 at 11:47 AM, Thomas Forster
<T.Forster at dpmms.cam.ac.uk> wrote:
>
> There is a body of opinion out there (i suspect not represented on FOM
> but...) that holds that Ordinary Mathematics can be captured by Mac lane
> set theory (the fragment of Zermelo set theory with Delta-0 separation
> instead of full separation).
>
> I take it that the proponents of this view have a reasoned organic view of
> ordinary mathematics which is secure enough for them to form a view about
> the claims of relevance of replacement, large cardinals etc. They also
> have a story about how the view is adequately covered by Mac lane set
> theory.
>
> I have never understood this point of view, but i would like to. (These
> people who think that Borel Determinacy is not part of ordinary
> mathematics - what do they mean and why do they mean it?) In particular i
> would like to understand how these people get a non ad-hoc concept out of
> what seems - to me - the incredibly ad-hoc (because not time-invariant)
> concept of ``ordinary mathematics''. Having grasped that concept i would
> then be in the market for arguments that it is all captured by Mac lane
> set theory. I suspect the thought is merely that mathematics is to be
> identified with the nth order theory of the reals/complexes for n
> arbitrarily large, but it might be more interesting than that
>
> If any list-members can point me at any literature where these ideas are
> *explained* (as opposed to merely baldy stated) I would be grateful to
> them.
>
> Thomas
>
>
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