ZFC

ZFC

Post by Amar » Fri, 03 Jun 2005 06:20:32


Hello,

I have seen some posters use the term ZFC when explaining concepts of
undecidability.
Can someone give an intuitive explanation of the concept?

Thanks
Amar
 
 
 

ZFC

Post by tcho » Fri, 03 Jun 2005 08:52:40

In article < XXXX@XXXXX.COM >,


http://www.yqcomputer.com/
http://www.yqcomputer.com/

ZFC is a set of axioms for set theory. It is widely accepted that almost
all of mathematics can be formalized in ZFC, so ZFC is considered a
"foundation for mathematics."
--
Tim Chow tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences

 
 
 

ZFC

Post by Jesse Alam » Fri, 03 Jun 2005 14:46:10


XXXX@XXXXX.COM writes:


Your statement makes me wonder what kinds of mathematics could *not*
be formalized in ZFC. Are there any fairly well agreed-upon examples?

Jesse

--
Jesse Alama ( XXXX@XXXXX.COM )
 
 
 

ZFC

Post by Jesse Alam » Fri, 03 Jun 2005 14:55:22


XXXX@XXXXX.COM writes:


It's clear that a very wide range of mathematical objects could be
regarded as sets: natural numbers, real numbers, topological spaces,
graphs, *** spheres, vector spaces... What strikes me as a more
interesting claim is that the kinds of *reasoning* we perform to make
judgments about these objects and their relations can also be treated
within ZFC. Does that claim "ZFC is a foundation for mathematics"
concerns merely the scope of the mathematical objects that can be
treated within ZFC?

Jesse

--
Jesse Alama ( XXXX@XXXXX.COM )
 
 
 

ZFC

Post by tcho » Fri, 03 Jun 2005 23:21:12

In article < XXXX@XXXXX.COM >,


No, the reasoning is also included in the scope.

Roughly speaking, the completeness theorem of first-order logic is the
basis for thinking that our reasoning is adequately captured by the
usual formal rules. That is, the completeness theorem tells us that if
some statement S about the objects in question cannot be proved from
the given axioms using the known rules, then we can exhibit something
that satisfies the given axioms and not S; therefore, there is no valid
principle of reasoning whose addition to the known ones would let us
prove any new consequences of the axioms.
--
Tim Chow tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences
 
 
 

ZFC

Post by tcho » Fri, 03 Jun 2005 23:41:32

In article < XXXX@XXXXX.COM >,


Set theorists routinely make use of axioms that go beyond the axioms of
ZFC---large cardinal axioms, determinacy axioms, etc. However, you might
argue that all the math in question can be phrased in the form, "Yadda yadda
follows from ZFC + [extra axiom]." These if-then statements can be easily
formalized in ZFC (in fact, in much weaker systems). A similar argument
might be made about investigations into alternatives to ZFC, like Quine's
NF. That is, one might argue that the "real" mathematical content of these
investigations is adequately captured by statements of the form "X follows
from NF."

For a stronger example: As soon as you fix a particular logical language,
logicians will be able to show you how to construct statements not
expressible in that language. Here's a nice example.

http://www.yqcomputer.com/

These examples might be considered artificial, though. There aren't many
"naturally occurring" candidates for things not formalizable in ZFC.
There are some fancy constructions in algebraic geometry that might fill
the bill; see

http://www.yqcomputer.com/

But even these examples can be handled, at worst, by very mild extensions
of ZFC.
--
Tim Chow tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences
 
 
 

ZFC

Post by haber » Mon, 06 Jun 2005 22:33:20

In article <429e4a48$0$564$ XXXX@XXXXX.COM >,




Most of it, like the stuff discussed in this group, I gather. If you
though go to math researchers in category theory, homological algebra and
stuff, it is not so clear what to use. The interested one can check out
the group sci.math.research, where the question pops up every now and
then.


This might be formally true if, in addition, the question of the
consistency could be resolved. From the Wiki above:
<quote>
While most metamathematicians believe that these [ZF] axioms are
consistent (in the sense that no contradiction can be derived from them),
this has not been proved. In fact, since they are the basis of ordinary
mathematics, their consistency (if true) cannot be proved by ordinary
mathematics; this is a consequence of Gel's second incompleteness
theorem. On the other hand, the consistency of ZFC can be proved by
assuming the existence of an inaccessible cardinal.
<end of quote>

One does not really know how to attack this problem it seems. (See though
the consistency proof at the end of Mendlson's book "Introduction to
mathematical logic", in a system much stronger than the one used by
Goedel.) But any math constructed from below in a reasonable fashion is
not expected to depend on its outcome.

Also see:
http://www.yqcomputer.com/
http://www.yqcomputer.com/ %F6del_set_theory

--
Hans Aberg
 
 
 

ZFC

Post by examachin » Tue, 07 Jun 2005 01:10:02


I would think that the logic used to formalize a given set theory
is the foundation, rather than the set theory itself. After all,
when we look beneath set theory we find a logical language, but
when we try to look beneath logic, there is nothing.

Regards,
 
 
 

ZFC

Post by tcho » Tue, 07 Jun 2005 02:49:52

In article < XXXX@XXXXX.COM >,


Even in these areas, there are standard dodges that allow most things to be
formalized in ZFC. Though it's true that there are some unclear issues at
the outer limits.


ZFC is still considered a foundation for mathematics regardless.

It's true that if you insist to clinging to the intuition behind Hilbert's
program, maintaining that (1) a mathematical issue cannot be "resolved"
unless it is given a classical mathematical proof, and (2) the consistency
of a proposed foundation for mathematics must be "resolved" before that
foundation can be accepted, then you're still going to be left *** ,
probably forever. But most people have learned to let go of one or the
other of these intuitions and accept ZFC as a foundation for mathematics.
--
Tim Chow tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences
 
 
 

ZFC

Post by tcho » Tue, 07 Jun 2005 02:58:49

In article < XXXX@XXXXX.COM >,


Usually, when people say that ZFC is a foundation for mathematics, they are
implicitly thinking of both the logic and the set theory together.

But, the matter is controversial, even among those who are knowledgeable
about logic and set theory. See for example:

http://www.yqcomputer.com/
--
Tim Chow tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences
 
 
 

ZFC

Post by Barb Kno » Tue, 07 Jun 2005 08:02:50

In article < XXXX@XXXXX.COM >,





Not necessarily. The foundational proofs of the consistency and
completeness of FOL itself use models, those models being represented in
some set theory. So in this sense the logic itself stands on set theory
(which is of course expressed in a logic...) It's turtles all the way down.


--
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| BBB a a r bbb | Quidquid latine dictum sit,
| B B a a r b b | altum viditur.
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ZFC

Post by haber » Tue, 07 Jun 2005 08:24:24

In article <42a33b40$0$576$ XXXX@XXXXX.COM >,



The foundations of mathematics are rather unclear. Though, if one does not
use math that touches into this foundational area too much, one needs not
to worry too much about it, according to customary belief. The belief is
that all "essential math", results that sufficiently constructively are
proving something about common mathematical objects, such as integers and
other common sets built up from there (real numbers, manifolds, functional
analysis, number theory, etc.), can be proved within something much
smaller than ZFC. But it is a belief, not firm knowledge. The best way to
find out more is to check out the sci.math.research group.

--
Hans Aberg
 
 
 

ZFC

Post by haber » Tue, 07 Jun 2005 08:31:58

In article <42a33d59$0$576$ XXXX@XXXXX.COM >,



These axiomatic set theories are clearly intended to be the foundation of
mathematics, as well as other theories, like type theories, higher order
theories, even topos theory and other stuff, of not exactly clear
relation. But if one says to people hard at work on those foundations that
the foundation has already been found, one will probably upset them a
great deal. :-)

--
Hans Aberg
 
 
 

ZFC

Post by tcho » Tue, 07 Jun 2005 10:26:34

In article < XXXX@XXXXX.COM >,


Sci.math.research, which I have followed since its inception in 1991, is
not in fact the best way to find out more about this subject. The best
first step is to study Stephen Simpson's book "Subsystems of Second-Order
Arithmetic," which I have recently mentioned in another thread.

http://www.yqcomputer.com/

This book alone provides a giant step towards converting your "belief"
into "firm knowledge," and it also provides many pointers to the large
literature on the topic.
--
Tim Chow tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences
 
 
 

ZFC

Post by tcho » Tue, 07 Jun 2005 10:31:23

In article < XXXX@XXXXX.COM >,


Who exactly do you have in mind? Certainly there are philosophers of
mathematics who will regard the foundations of mathematics as "not having
been found," but most of them are not "hard at work on those foundations,"
but rather hard at work writing papers in philosophy. People like Harvey
Friedman and his collaborators are certainly hard at work on the foundations
of mathematics, but they don't regard the foundations of mathematics as
"unclear" and will certainly not be upset if you say that ZFC is a
foundation for mathematics.
--
Tim Chow tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences