Re: Theory T.

On Jul 14, 7:24 am, Aatu Koskensilta <aatu.koskensi...@xxxxxx> wrote:
Chris Menzel <cmen...@xxxxxxxxxxxxxxxxxxxx> writes:
On Mon, 13 Jul 2009 11:52:21 -0700 (PDT), zuhair <zaljo...@xxxxxxxxx> said:
On Jul 13, 9:03 am, Aatu Koskensilta <aatu.koskensi...@xxxxxx> wrote:

How is ZF ad hoc? It is a very natural system, both from a mathematical
and conceptual point of view.

Yea especially Replacement ha.

Huh.  So you've got a set S.  And you can describe a way of correlating
each member x of S with some other object y.  And you find it ad hoc to
assert that those objects that can be correlated with the members of S
themselves constitute a set.  Weird.


Aatu, your writings deserve well-reasoned, informed, and contemplative
replies. That said here are some comments.

As it happens I'm frantically trying to make good on my promise to write
a few deeply thoughtful little essays with footnotes; one for Moeblee on
the irrelevance of finitism, and the idea that some mathematical
statements are, so to speak, contentual, their truth or falsity turning
on the mathematical nature of things, while others are more a matter of
stipulation, mathematical imaginings, and so on; and one for Marc,
Rupert and Daryl, on replacement, ("predicatively meaningful")
reflection, and the iterative conception of set. The latter bears the
pointlessly provocative and pompous title /Who cares about the iterative
conception of set? or: Reflections on reflection -- a replacement for
replacement?/. In it I put forth in particular the following claims and
observations:

 o That Zermelo-Fraenkel set theory is /mathematically natural/ in that
   it basically just stipulates the universe of sets contains enough
   sets for the construction of the basic furniture of the world of our
   usual mathematical experience, and that the world of sets is closed
   under operations mathematicians routinely subject sets and structures
   to in their mathematical work, whether they care about axiomatics or
   not (and usually they do not).


But, it's not closed under union, a typical operation on sets, because
there is no universe in ZF.

Of course you know I argue that there is already ZF's universe in ZF's
universe because ZF's universe is the Russell set in theories with
less restricted comprehension.

The notion of infinities in foundational logic and as well in applied
mathematics that are not regular (i.e. well-founded) infinities sees
examples in universal considerations (re paradoxes of Cantor, Russell,
Burali-Forti) and compactified (for some implicitly so) number spaces.

 o That the iterative conception of set, on the other hand, justifies
   either a theory that is weaker than ZFC, viz. Zermelo set theory +
   "the powerset-operation can be iterated along any well-ordering"
   (this principle justifies reflection for a restricted class of
   formulas in the language of set theory[1]), or a theory that is stronger
   than ZFC, viz. ZFC + "various small large cardinals, up to somewhere
   below the first weakly compact".


Again some see in the transfer principle reason to have them go right
through the compact.

 o That, as a brute fact of mathematical life, the mathematical
   naturalness of ZFC is more significant than the philosophical delight
   one gets from pondering the iterative conception and various related
   finer philosophical points.


The cumulative hierarchy of ordinals, in their cumulative limit
hierarchy, is mathematically consequent to the formulation of the
axiom of infinity in ZF, which might not be the "natural" inclusive
truism about those things.

For finite combinatorics ZF is complete.

 o That the equivalence of first-order reflection and replacement is
   just a technical artefact, of no conceptual significance; and that
   because of this the sometimes made suggestion that the iterative
   conception + reflection is basically captured in ZFC is muddle-headed
   nonsense. (I forget the title and author of the book where much was
   made of this supposed revelation...)


Two axioms might be independent yet separable into component
statements that aren't.

 o A fifth insightful observation about all this which alas eludes me at
   the moment.

There's loads of erratic rambling and pointless maundering on various
philosophical and historical matters, and a few technical appendices,
but for that you'll have to wait for the real thing.

Footnotes:
[1]  It's been a while since I worked this stuff out and I have quite
forgotten the details. I have a hazy recollection this class was Sigma-2
or Pi-2 (in the Levy hierarchy), which classes have certain absoluteness
properties...

--
Aatu Koskensilta (aatu.koskensi...@xxxxxx)

"Wovon mann nicht sprechen kann, darüber muss man schweigen"
 - Ludwig Wittgenstein, Tractatus Logico-Philosophicus

Thanks,

Ross
.

Relevant Pages