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

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

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 )

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

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

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

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

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

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

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

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

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

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

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.

--

---------------------------

| BBB b \ Barbara at LivingHistory stop co stop uk

| B B aa rrr b |

| BBB a a r bbb | Quidquid latine dictum sit,

| B B a a r b b | altum viditur.

| BBB aa a r bbb |

-----------------------------

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.

--

---------------------------

| BBB b \ Barbara at LivingHistory stop co stop uk

| B B aa rrr b |

| BBB a a r bbb | Quidquid latine dictum sit,

| B B a a r b b | altum viditur.

| BBB aa a r bbb |

-----------------------------

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

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

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

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

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

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

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

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

1. Zfc ***Hot stuff - check this out !!! Zfc

3. ZFC ?

5. zfc and Orpheus. By now well OT.

6. Not zfc

7. ZFC chat

8. ZFC

9. argonet.co.uk/zfc (was: Clean installing a RISC PC/A7000)

10. OT: Music and language (was zfc and Orpheus)

11. You all need to report to ZFC

13. Talking of Jokes was zfc and Orpheus

15. OT : zfc and Orpheus... Chip & pin