How do We Know that ZF is the Axiomatization that Proves everything Provable?
Charlie-Boo
How do We Know that ZF is the Axiomatization that Proves everything
provable?
People say that ZF proves everything provable. I wonder:
1. How can we know that?
2. Doesn't Godel say that there are true statements that are not
provable in ZF but are provable in an extension of ZF?
3. Why ZF and not the other axiomatizations of Set Theory (ZFC, NBG,
KPU etc.)?
4. If ZF proves all of Mathematics that can be proven, then these
other axioms of Set Theory must be wrong. So why do we still use
them?
C-B
Jan Burse
Bye
Charlie-Boo
Then ZFC is said to prove everything provable? Ok, then still the
questions remain regarding ZFC.
> Bye- Hide quoted text -
>
> - Show quoted text -
LordBeotian
"Charlie-Boo" <shyma...@gmail.com> ha scritto
> How do We Know that ZF is the Axiomatization that Proves everything
> provable?
Probably you should have said "everything provable *by ordinary
mathematics*".
The answer would depend on what we mean by "ordinary mathematics".
elsiemelsi
The Australian philosopher colin leslie dean argues thatThe Skolem paradox
destroys the incompleteness of ZFC
http://gamahucherpress.yellowgum.com/book/philosophy/GODEL5.pdf
The Skolem pardox shows ZFC is inconsistent
Undecidability of ZFC is based on the assumption that it is consistent
therefore the presence of the Skolem paradox shows ZFC is not consistent
so all those proofs that show the incompleteness of ZFC are destroyed
undermined and complete rubbish
"At present we can do no more than note that we have one more reason here
to entertain reservations about set theory and that for the time being no
way of rehabilitating this theory is known." – (John von Neumann)
"Skolem's work implies 'no categorical axiomatisation of set theory (hence
geometry, arithmetic [and any other theory with a set-theoretic model]...)
seems to exist at all'." – (John von Neumann)
"Neither have the books yet been closed on the antinomy, nor has agreement
on its significance and possible solution yet been reached." – (Abraham
Fraenkel)
"I believed that it was so clear that axiomatization in terms of sets was
not a satisfactory ultimate foundation of mathematics that mathematicians
would, for the most part, not be very much concerned with it. But in
recent times I have seen to my surprise that so many mathematicians think
that these axioms of set theory provide the ideal foundation for
mathematics; therefore it seemed to me that the time had come for a
critique." – (Skolem)
--
Message posted using http://www.talkaboutscience.com/group/sci.logic/
More information at http://www.talkaboutscience.com/faq.html
David C. Ullrich
<shyma...@gmail.com> wrote:
>
>How do We Know that ZF is the Axiomatization that Proves everything
>provable?
>
>People say that ZF proves everything provable.
No, nobody says that. At least nobody who knows what
he's talking about.
Prople do say that ZFC is sufficient to do all the mathematics
that people usually want to do. The reason they say that is
that it's _true_. It's not something that can be proved
mathematically, because it's an _empirical_ truth, not
a logical one. But it's true.
>I wonder:
>
>1. How can we know that?
>
>2. Doesn't Godel say that there are true statements that are not
>provable in ZF but are provable in an extension of ZF?
Yes. (Well, actually I don't think you put this very accurately, but
for the purposes of this discussion "yes" seems closer to the truth
than "no".)
So what? That would show that the people who say ZFC proves
everything are wrong. Or it would if there were any such people.
The fact that there are statements independent of ZFC is very
well-known among mathematicians - there are even explicit
well-known statements that actually have to do with actual
mathematical problems, as opposed to things just constructed
by diagonal arguments with Godel numbers. (For example,
it's very well known among a certain subset of mathematicians
that the statement "Every homomorphism from one Banach
algebra to another" is independent of ZFC.)
>3. Why ZF and not the other axiomatizations of Set Theory (ZFC, NBG,
>KPU etc.)?
>
>4. If ZF proves all of Mathematics that can be proven, then these
>other axioms of Set Theory must be wrong. So why do we still use
>them?
>
>C-B
David C. Ullrich
Jan Burse
>> ZF proofs less than ZFC.
>
> Then ZFC is said to prove everything provable? Ok, then still the
> questions remain regarding ZFC.
No, CH is not proved.
Bye
Charlie-Boo
> "Charlie-Boo" <shymath...@gmail.com> ha scritto
So they just define "ordinary mathematics" to be the logic of ZF and
it's true by definition? I guess that's the only way you could try to
defend such a nebulous claim.
But who is so foolish as to think that there is a distinction to be
made within mathematics as to its principles being "ordinary" or not?
C-B
Charlie-Boo
> On Sat, 2 Feb 2008 05:58:56 -0800 (PST), Charlie-Boo
>
>
> >How do We Know that ZF is the Axiomatization that Proves everything
> >provable?
>
> >People say that ZF proves everything provable.
>
> No, nobody says that. At least nobody who knows what
> he's talking about.
A lot of people say to leave off the C.
> Prople do say that ZFC is sufficient to do all the mathematics
> that people usually want to do. The reason they say that is
> that it's _true_. It's not something that can be proved
> mathematically
If you are going to claim that a particular system can perform a
particular function, then to say that you cannot prove it means you
have not demonstrated (justified) your claim. That's what formal
systems are all about.
> because it's an _empirical_ truth, not
> a logical one. But it's true.
How has it been demonstrated that the majority of mathematics can be
done in ZF(C)?
> >I wonder:
>
> >1. How can we know that?
>
> >2. Doesn't Godel say that there are true statements that are not
> >provable in ZF but are provable in an extension of ZF?
>
> Yes. (Well, actually I don't think you put this very accurately, but
> for the purposes of this discussion "yes" seems closer to the truth
> than "no".)
I assume people know that what he did in PA can be done in ZF.
> So what? That would show that the people who say ZFC proves
> everything are wrong. Or it would if there were any such people.
> The fact that there are statements independent of ZFC is very
> well-known among mathematicians - there are even explicit
> well-known statements that actually have to do with actual
> mathematical problems, as opposed to things just constructed
> by diagonal arguments with Godel numbers. (For example,
> it's very well known among a certain subset of mathematicians
> that the statement "Every homomorphism from one Banach
> algebra to another" is independent of ZFC.)
So it's foolish to point out a mistake (or even pay attention to it)
if it is blatant? Actually, I think they figure that "provable"
excludes the various independencies from ZFC that have been
established.
> >3. Why ZF and not the other axiomatizations of Set Theory (ZFC, NBG,
> >KPU etc.)?
>
> >4. If ZF proves all of Mathematics that can be proven, then these
> >other axioms of Set Theory must be wrong. So why do we still use
> >them?
You miss the point - in the title: Why ZF and not the others?
Questions 1-2 were just a warm-up scrutiny of the very idea of making
such a claim. Being a nebulous one, it is debatable as to what it
even means.
But my point goes beyond that - and is made explicit in questions 3
and 4, which you skipped. How can we say that ZF(C) (any particular
axiomatization) is correct and the others aren't? And then why
continue using the incorrect ones?
The principle is not new. You take two big nebuouls things and say
they are equivalent. It's typically hard to say whether that is true
or not. But a problem occurs if someone points out that one of the
two nebulous things has siblings that are not the same. The question
becomes: Why this particular one?
The current topic concerns ZFC and "all of Mathematics", so people
like to say they're equivalent. Eureka! But why ZFC and not the
other axiomatizations?
Another example is when people say that Godel's 1st Incompleteness
Theorem is equivalent to Turing's unsolvability of the Halting
Problem. Again, there are siblings to both theorems (e.g. Rosser 1936
and the unsolvability of the Membership Problem.) This brings up the
point that we are declaring two things to be mathematically equivalent
in reality based on when they were discovered (each being the first
theorem developed in their respective domains.)
> David C. Ullrich
Chris Menzel
> People say that ZF proves everything provable. I wonder:
>
> 1. How can we know that?
Better question: Who the hell says that?
> 2. Doesn't Godel say that there are true statements that are not
> provable in ZF but are provable in an extension of ZF?
Sure. So what?
> 3. Why ZF and not the other axiomatizations of Set Theory (ZFC, NBG,
> KPU etc.)?
Why indeed. All four are used in various contexts.
> 4. If ZF proves all of Mathematics that can be proven,
Obviously it doesn't.
David C. Ullrich
<shyma...@gmail.com> wrote:
>On Feb 2, 9:40 am, David C. Ullrich <dullr...@sprynet.com> wrote:
>> On Sat, 2 Feb 2008 05:58:56 -0800 (PST), Charlie-Boo
>>
>> <shymath...@gmail.com> wrote:
>>
>> >How do We Know that ZF is the Axiomatization that Proves everything
>> >provable?
>>
>> >People say that ZF proves everything provable.
>>
>> No, nobody says that. At least nobody who knows what
>> he's talking about.
>
>A lot of people say to leave off the C.
>
>> Prople do say that ZFC is sufficient to do all the mathematics
>> that people usually want to do. The reason they say that is
>> that it's _true_. It's not something that can be proved
>> mathematically
>
>If you are going to claim that a particular system can perform a
>particular function, then to say that you cannot prove it means you
>have not demonstrated (justified) your claim. That's what formal
>systems are all about.
Sometimes you seem blind or deaf or something.
Suppose that I say that it's a nice day today here in Stillwater.
That's not something that I can demonstrate formally, since
it's not a logical proposition. It's also not something that I've
_claimed_ I can demonstrate logically. That doesn't mean it's
false - in fact it _is_ a nice day today here in Stillwater,
highs in the 50's during February.
>> because it's an _empirical_ truth, not
>> a logical one. But it's true.
>
>How has it been demonstrated that the majority of mathematics can be
>done in ZF(C)?
By _doing_ the mathematics in ZFC.
>> >I wonder:
>>
>> >1. How can we know that?
>>
>> >2. Doesn't Godel say that there are true statements that are not
>> >provable in ZF but are provable in an extension of ZF?
>>
>> Yes. (Well, actually I don't think you put this very accurately, but
>> for the purposes of this discussion "yes" seems closer to the truth
>> than "no".)
>
>I assume people know that what he did in PA can be done in ZF.
>
>> So what? That would show that the people who say ZFC proves
>> everything are wrong. Or it would if there were any such people.
>> The fact that there are statements independent of ZFC is very
>> well-known among mathematicians - there are even explicit
>> well-known statements that actually have to do with actual
>> mathematical problems, as opposed to things just constructed
>> by diagonal arguments with Godel numbers. (For example,
>> it's very well known among a certain subset of mathematicians
>> that the statement "Every homomorphism from one Banach
>> algebra to another" is independent of ZFC.)
>
>So it's foolish to point out a mistake (or even pay attention to it)
>if it is blatant?
Again, you really seem to have big problems reading simple
English. The point is not that it's foolish to point out a mistake
if it's blatant - the point is that it's foolish to make a big deal
out of refuting a statement that nobody is actually claiming
is true.
> Actually, I think they figure that "provable"
>excludes the various independencies from ZFC that have been
>established.
>
>> >3. Why ZF and not the other axiomatizations of Set Theory (ZFC, NBG,
>> >KPU etc.)?
>>
>> >4. If ZF proves all of Mathematics that can be proven, then these
>> >other axioms of Set Theory must be wrong. So why do we still use
>> >them?
>
>You miss the point - in the title: Why ZF and not the others?
Because the axioms seem reasonable to most people and they
do in fact suffice for most purposes.
>Questions 1-2 were just a warm-up scrutiny of the very idea of making
>such a claim. Being a nebulous one, it is debatable as to what it
>even means.
>
>But my point goes beyond that - and is made explicit in questions 3
>and 4, which you skipped. How can we say that ZF(C) (any particular
>axiomatization) is correct and the others aren't?
Uh, no, although I didn't type anything in the space below that
question I _did_ answer it. People _don't_ say that that
axiomatization is correct and the others are wrong.
>And then why
>continue using the incorrect ones?
>
>The principle is not new. You take two big nebuouls things and say
>they are equivalent. It's typically hard to say whether that is true
>or not. But a problem occurs if someone points out that one of the
>two nebulous things has siblings that are not the same. The question
>becomes: Why this particular one?
>
>The current topic concerns ZFC and "all of Mathematics", so people
>like to say they're equivalent.
You're just typing because it turns you on to see your words appear
on the internet, right? I mean you already conceded above that
people _don't_ say this - now you ask again about people saying this.
Have fun - if it ever happens that you have questions that you
actually want an answer to let us know.
>Eureka! But why ZFC and not the
>other axiomatizations?
>
>Another example is when people say that Godel's 1st Incompleteness
>Theorem is equivalent to Turing's unsolvability of the Halting
>Problem. Again, there are siblings to both theorems (e.g. Rosser 1936
>and the unsolvability of the Membership Problem.) This brings up the
>point that we are declaring two things to be mathematically equivalent
>in reality based on when they were discovered (each being the first
>theorem developed in their respective domains.)
>
>> David C. Ullrich
David C. Ullrich
Frederick Williams
> >How has it been demonstrated that the majority of mathematics can be
> >done in ZF(C)?
David C. Ullrich wrote:
> By _doing_ the mathematics in ZFC.
I doubt that the majority of mathematics has been done in ZFC.
(Actually, that is a horrible use of the word "majority", I would rather
say "most" than "the majority".)
There is, I suppose, a belief that most mathematics could be done in ZFC
given sufficient time and interest.
--
Going forward at this moment in time a raft of measures
have been put in place on the ground to target and
claw back the growth of cliché usage 24/7.
Remove "antispam" and ".invalid" for e-mail address.
Jan Burse
> Charlie-Boo wrote:
>
>>> How has it been demonstrated that the majority of mathematics can be
>>> done in ZF(C)?
>
> David C. Ullrich wrote:
>
>> By _doing_ the mathematics in ZFC.
>
> I doubt that the majority of mathematics has been done in ZFC.
> (Actually, that is a horrible use of the word "majority", I would rather
> say "most" than "the majority".)
>
> There is, I suppose, a belief that most mathematics could be done in ZFC
> given sufficient time and interest.
>
http://mizar.uwb.edu.pl/trybulec65/book.pdf
Probably they are not using ZFC, but some SOL.
Bye
Jan Burse
> Charlie-Boo wrote:
>
>>> How has it been demonstrated that the majority of mathematics can be
>>> done in ZF(C)?
>
> David C. Ullrich wrote:
>
>> By _doing_ the mathematics in ZFC.
>
> I doubt that the majority of mathematics has been done in ZFC.
> (Actually, that is a horrible use of the word "majority", I would rather
> say "most" than "the majority".)
>
> There is, I suppose, a belief that most mathematics could be done in ZFC
> given sufficient time and interest.
>
> Check out Mizar:
> http://mizar.uwb.edu.pl/trybulec65/book.pdf
>
> Probably they are not using ZFC, but some SOL.
Yes SOL is there. Here from its manual:
An Outline of PC Mizar
Michal Muzalewski
15. Schemes
Schemes are sentences of second order. First
comes the reserved word scheme and the identifier
of the schema. They are followed by the list
of parameters ...
... Finally one more example perfectly well known
to secondary school pupils, namely the schema
of induction:
scheme Ind { P[Nat] } : for k holds P[k] provided
B1 : P[0]
and
B2 : for k st P[k] holds P[k+1];
Jan Burse
> Charlie-Boo wrote:
>
>>> How has it been demonstrated that the majority of mathematics can be
>>> done in ZF(C)?
>
> David C. Ullrich wrote:
>
>> By _doing_ the mathematics in ZFC.
>
> I doubt that the majority of mathematics has been done in ZFC.
> (Actually, that is a horrible use of the word "majority", I would rather
> say "most" than "the majority".)
>
> There is, I suppose, a belief that most mathematics could be done in ZFC
> given sufficient time and interest.
>
Check out:
http://www.cs.ru.nl/~freek/100/
There exists a "top 100" of mathematical theorems on the web, which is a
rather arbitrary list (and most of the theorems seem rather elementary),
but still is nice to look at. On the current page I will keep track of
which theorems from this list have been formalized. Currently the
fraction that already has been formalized seems to be
80%
# Gödel's Incompleteness Theorem
* HOL Light, John Harrison
* Coq, contrib, Russell O'Connor
* nqthm, Natarajan Shankar
Aatu Koskensilta
> I doubt that the majority of mathematics has been done in ZFC.
Quoting myself in the thread /Can ZFC prove Addition is Associative?/:
What is at issue is the weaker, purely extensional claim, that
everything proved, to this day, in "ordinary mathematics" - which is
a sociological and statistical notion, as it happens - is formally
provable in ZFC, suitably formalised. Of course, this is not based on
anyone's actually producing formal proofs, though there are attempts
at that direction in the automated theorem proving circles, but
rather on observing that the basic modes of reasoning and the basic
mathematical principles used in proofs in ordinary mathematical
proofs have formal counterparts in ZFC, and then concluding on basis
of the completeness theorem -- and the observation, due to Kreisel,
that the informal notion of a statement following logically from a
set of axioms corresponds to formal provability of the formalisation
of the statement from the formalisation of those axioms -- that
there exists a formal proof in ZFC of any statement proved, in the
ordinary sense, using these modes of reasoning and principles.
(Message-ID: <1168866745....@38g2000cwa.googlegroups.com>)
--
Aatu Koskensilta (aatu.kos...@xortec.fi)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
Jan Burse
> On 2008-02-03, in sci.logic, Frederick Williams wrote:
>> I doubt that the majority of mathematics has been done in ZFC.
>
> Quoting myself in the thread /Can ZFC prove Addition is Associative?/:
>
> What is at issue is the weaker, purely extensional claim, that
> everything proved, to this day, in "ordinary mathematics" - which is
> a sociological and statistical notion, as it happens - is formally
> provable in ZFC, suitably formalised. Of course, this is not based on
> anyone's actually producing formal proofs, though there are attempts
> at that direction in the automated theorem proving circles, but
> rather on observing that the basic modes of reasoning and the basic
> mathematical principles used in proofs in ordinary mathematical
> proofs have formal counterparts in ZFC, and then concluding on basis
> of the completeness theorem -- and the observation, due to Kreisel,
> that the informal notion of a statement following logically from a
> set of axioms corresponds to formal provability of the formalisation
> of the statement from the formalisation of those axioms -- that
> there exists a formal proof in ZFC of any statement proved, in the
> ordinary sense, using these modes of reasoning and principles.
>
> (Message-ID: <1168866745....@38g2000cwa.googlegroups.com>)
>
but rather a "set" theory. So it doesn't really
refer to modes of reasoning, rather to classical
reasoning inside set theory.
Best Regards
Aatu Koskensilta
> With the problem that ZFC is not a "logical" theory,
> but rather a "set" theory. So it doesn't really
> refer to modes of reasoning, rather to classical
> reasoning inside set theory.
Here modes of reasoning does not refer to logical principles, but
rather to mathematical principles employed routinely in set theoretic
arguments, e.g. applying transfinite induction, defining operations on
sets by transfinite recursion, constructions involving the axiom of
choice and so on. It has been shown that using such principles all the
usual theorems of mathematics can be proved, and from this observation
it follows, by the completeness theorem and the conceptual analysis
due to Kreisel, that their formalisations are formally derivable in
ZFC which, directly or indirectly, contains the formlisations of these
principles. That is, on basis of the work of logicians, set theorists,
analysts and who not, we have a reduction of ordinary mathematics to
set theory, and, as ZFC is a formalisation of set theory, implicitly
to ZFC.
Charlie-Boo
Williams"@antispamhotmail.co.uk.invalid> wrote:
> Charlie-Boo wrote:
> > >How has it been demonstrated that the majority of mathematics can be
> > >done in ZF(C)?
> David C. Ullrich wrote:
> > By _doing_ the mathematics in ZFC.
>
> I doubt that the majority of mathematics has been done in ZFC.
> (Actually, that is a horrible use of the word "majority", I would rather
> say "most" than "the majority".)
>
> There is, I suppose, a belief that most mathematics could be done in ZFC
> given sufficient time and interest.
Are you saying using only ZFC's axioms?
Can you cite an on-line reference to anyone proving anything outside
of Arithmetic or Set Theory using ZFC?
If we can do most of Mathematics, then we can do a lot of simple
Mathematics. Then could you give a simple self-contained example of a
proof within ZFC that is outside of Arithmetic and Set Theory?
C-B
Charlie-Boo
That's just the cover of a book. Do you know of any on-line reference
containing a proof within ZFC outside of Arithmetic and Set Theory?
C-B
> Bye- Hide quoted text -
Charlie-Boo
> Frederick Williams schrieb:
>
>
>
> > Charlie-Boo wrote:
>
> >>> How has it been demonstrated that the majority of mathematics can be
> >>> done in ZF(C)?
>
> > David C. Ullrich wrote:
>
> >> By _doing_ the mathematics in ZFC.
>
> > I doubt that the majority of mathematics has been done in ZFC.
> > (Actually, that is a horrible use of the word "majority", I would rather
> > say "most" than "the majority".)
>
> > There is, I suppose, a belief that most mathematics could be done in ZFC
> > given sufficient time and interest.
>
> Jan Burse wrote:
>
> > Check out Mizar:
> >http://mizar.uwb.edu.pl/trybulec65/book.pdf
> >
> > Probably they are not using ZFC, but some SOL.
>
> Yes SOL is there. Here from its manual:
>
> An Outline of PC Mizar
> Michal Muzalewski
No, that defines its own axioms. The claim is that ZFC has all the
axioms needed for Mathematics. Do you know of any on-line proof
within ZFC outside of Arithmetic and Set Theory?
The assertion flies in the face of the history of Mathematics and
would mean that these axioms replace all of the work of Mathematicians
axiomatizing various branches of Mathematics.
I find it incredible that anyone would believe such a thing. And when
you consider the lack of any examples of ANYTHING outside of
Arithmetic and Set Theory (which is all that ZFC is), it becomes even
more dumbfounding.
(Chalk it up to inbreeding, I suppose.)
C-B
> 15. Schemes
> Schemes are sentences of second order. First
> comes the reserved word scheme and the identifier
> of the schema. They are followed by the list
> of parameters ...
>
> ... Finally one more example perfectly well known
> to secondary school pupils, namely the schema
> of induction:
>
> scheme Ind { P[Nat] } : for k holds P[k] provided
> B1 : P[0]
> and
> B2 : for k st P[k] holds P[k+1];
>
> See also:http://www.cs.ualberta.ca/~piotr/Mizar/Dagstuhl97/- Hide quoted text -
Aatu Koskensilta
> I find it incredible that anyone would believe such a thing. And when
> you consider the lack of any examples of ANYTHING outside of
> Arithmetic and Set Theory (which is all that ZFC is), it becomes even
> more dumbfounding.
>
> (Chalk it up to inbreeding, I suppose.)
Don't be too hard on yourself. I doubt inbreeding has anything to do
with your finding perfectly standard and straightforward stuff
'incredible' or 'dumbfounding'.
Charlie-Boo
wrote:
> On 2008-02-03, in sci.logic, Frederick Williams wrote:
>
> > I doubt that the majority of mathematics has been done in ZFC.
>
> Quoting myself in the thread /Can ZFC prove Addition is Associative?/:
>
> What is at issue is the weaker, purely extensional claim, that
> everything proved, to this day, in "ordinary mathematics" - which is
> a sociological and statistical notion, as it happens - is formally
> provable in ZFC, suitably formalised.
Do you mean that ZFC's axioms suffice (not just writing new axioms as
needed)?
If yes, then can you cite any examples of proving anything outside of
Arithmetic or Set Theory using only ZFC and its axioms?
If no, then can't that be done using any system whose expressions form
an r.e. set (and ZFC's axioms are irrelevant)?
C-B
> Of course, this is not based on
> anyone's actually producing formal proofs, though there are attempts
> at that direction in the automated theorem proving circles, but
> rather on observing that the basic modes of reasoning and the basic
> mathematical principles used in proofs in ordinary mathematical
> proofs have formal counterparts in ZFC, and then concluding on basis
> of the completeness theorem -- and the observation, due to Kreisel,
> that the informal notion of a statement following logically from a
> set of axioms corresponds to formal provability of the formalisation
> of the statement from the formalisation of those axioms -- that
> there exists a formal proof in ZFC of any statement proved, in the
> ordinary sense, using these modes of reasoning and principles.
>
> (Message-ID: <1168866745.894312.18...@38g2000cwa.googlegroups.com>)
>
> --
> Aatu Koskensilta (aatu.koskensi...@xortec.fi)
Charlie-Boo
wrote:
> On 2008-02-05, in sci.logic, Jan Burse wrote:
>
> > With the problem that ZFC is not a "logical" theory,
> > but rather a "set" theory. So it doesn't really
> > refer to modes of reasoning, rather to classical
> > reasoning inside set theory.
>
> Here modes of reasoning does not refer to logical principles, but
> rather to mathematical principles employed routinely in set theoretic
> arguments, e.g. applying transfinite induction, defining operations on
> sets by transfinite recursion, constructions involving the axiom of
> It has been shown that using such principles all the
> usual theorems of mathematics can be proved.
How do you define the "usual theorems of mathematics"? They are of
such a heterogeneous nature, how can you place any kind of a predicate
against the notion of an arbitrarty mathematical proof (other than
asking whether particular rules were used)? If you are saying that
all the rules are from a small given set, then you are simply defining
the systems that can be represented in ZFC. There is no inference.
> That is, on basis of the work of logicians, set theorists,
> analysts and who not, we have a reduction of ordinary mathematics to
> set theory, and, as ZFC is a formalisation of set theory, implicitly
> to ZFC.
Nonsense. Give an example of something outside of Arithmetic and Set
Theory being proven using only ZFC. How about a real simple, self-
contained example out of the countless number of theorems there are?
You can "model" any formal system with any other formal system just by
matching up the two r.e. sets of expressions. But that doesn't give
you the logical truth necessary to make deductions in that system.
You still need the axioms of that system. All you are doing is using
a different syntax for the expressions.
C-B
> --
> Aatu Koskensilta (aatu.koskensi...@xortec.fi)
Charlie-Boo
wrote:
> On 2008-02-05, in sci.logic, Charlie-Boo wrote:
>
> > I find it incredible that anyone would believe such a thing. And when
> > you consider the lack of any examples of ANYTHING outside of
> > Arithmetic and Set Theory (which is all that ZFC is), it becomes even
> > more dumbfounding.
>
> > (Chalk it up to inbreeding, I suppose.)
>
> Don't be too hard on yourself. I doubt inbreeding has anything to do
> with your finding perfectly standard
Politics!
> and straightforward stuff
> 'incredible' or 'dumbfounding'.
What's incredible is for people to believe something like "Most of ...
are ..." without any examples of any "..."! And even worse, claim to
be Mathematicians making a Mathematical statement while others watch
and even believe him or her!
Why don't you give an example? What Mathematics other than
Arithemetic and Set Theory has anyone ever proven in ZFC?
C-B
> --
> Aatu Koskensilta (aatu.koskensi...@xortec.fi)
>
Aatu Koskensilta
> And even worse, claim to be Mathematicians making a Mathematical
> statement while others watch and even believe him or her!
Positively repugnant!
> Why don't you give an example? What Mathematics other than
> Arithemetic and Set Theory has anyone ever proven in ZFC?
You'll find the set theoretic development of analysis in any decent
textbook on the subject.
--
Aatu Koskensilta (aatu.kos...@xortec.fi)
Charlie-Boo
Do any of the alternatives?
C-B
Charlie-Boo
wrote:
> On 2008-02-05, in sci.logic, Charlie-Boo wrote:
>
> > And even worse, claim to be Mathematicians making a Mathematical
> > statement while others watch and even believe him or her!
>
> Positively repugnant!
>
> > Why don't you give an example? What Mathematics other than
> > Arithemetic and Set Theory has anyone ever proven in ZFC?
>
> You'll find the set theoretic development of analysis in any decent
> textbook on the subject.
Where in "Classic Set Theory" by Derek Goldrei do I find a proof
within ZFC of something outside of Arithmetic and Set Theory?
C-B
> --
> Aatu Koskensilta (aatu.koskensi...@xortec.fi)
>
William Hale
<3785f951-0f6a-4620...@k39g2000hsf.googlegroups.com>,
Charlie-Boo <shyma...@gmail.com> wrote:
[cut]
> Why don't you give an example? What Mathematics other than
> Arithemetic and Set Theory has anyone ever proven in ZFC?
>
> C-B
Are you saying that Arithmetic Theory has been proven in ZFC?
If not, which I suspect, then why are you not discussing that instead of
trying to jump ahead to something more advance?
Bill Hale
Aatu Koskensilta
> Are you saying that Arithmetic Theory has been proven in ZFC?
What is "Arithmetic Theory" and what would it mean for it to be
"proven in ZFC"?
--
Aatu Koskensilta (aatu.kos...@xortec.fi)
MoeBlee
> If we can do most of Mathematics, then we can do a lot of simple
> Mathematics. Then could you give a simple self-contained example of a
> proof within ZFC that is outside of Arithmetic and Set Theory?
Such subjects as topology, analysis, abstract algebra, and graph
theory are not usually thought of as part of a set theory course
itself, but statements and theorems in such subjects can be given in
the language of and proven in ZFC.
Rather than me just list one theorem after another, a better question
would be for you to say what theorem of such subjects you think canNOT
be proven in ZFC.
MoeBlee
Aatu Koskensilta
> Where in "Classic Set Theory" by Derek Goldrei do I find a proof
> within ZFC of something outside of Arithmetic and Set Theory?
I have no idea -- I don't think I've read that book.
But why would you expect to find something "outside of Set Theory"
proven in ZFC? ZFC is a formal theory of sets, after all. When we
speak of some theorem in this or that field of mathematics being
provable in ZFC what is meant is that that theorem, expressed using
set theoretical notions as usual, and formalised in the language of
set theory, is formally provable in ZFC. The details of this business
of expressing theorems of analysis, topology, etc. in terms of sets,
you'll find in any decent textbook.
--
Aatu Koskensilta (aatu.kos...@xortec.fi)
Charlie-Boo
> On Sat, 2 Feb 2008 18:47:08 -0800 (PST), Charlie-Boo
> >> Prople do say that ZFC is sufficient to do all the mathematics
> >> that people usually want to do. The reason they say that is
> >> that it's _true_. It's not something that can be proved
> >> mathematically
> Sometimes you seem blind or deaf or something.
> Suppose that I say that it's a nice day today here in Stillwater.
I'm blind to irrelevant statements such as the above?
> >How has it been demonstrated that the majority of mathematics can be
> >done in ZF(C)?
>
> By _doing_ the mathematics in ZFC.
A majority of Mathematics has been done in ZFC? My goodness, there
should be hundreds of references to all sorts of theorems outside of
Arithmetic and Set Theory done in ZFC. But why have you no such
reference?
> it's foolish to make a big deal
> out of refuting a statement that nobody is actually claiming
> is true.
Who just said, "The reason they say that is that it's _true_."?
> Because the axioms seem reasonable to most people and they
> do in fact suffice for most purposes.
What are these purposes that the ZFC axioms serve that cover the
majority of Mathematical proof?
Just saying that you can "develop Mathematics in Set Theory" doesn't
mean that the ZFC axioms will give you the inferences that you need.
In fact, it doesn't really say anything about ZFC.
You can see all of Mathematics as being part of Set Theory simply
because everything can be the sole member of a set. So any time you
start to define a Mathematical system, any single thing or pair of
related things etc. constitutes a set.
But this also applies to lists, directed graphs etc. And in each
case, we are just adding a superstructure with no consequence. The
consequences come from the axioms and ZFC's axioms do not replace any
other system's (outside of Arithmetic and Set Theory.)
> >Questions 1-2 were just a warm-up scrutiny of the very idea of making
> >such a claim. Being a nebulous one, it is debatable as to what it
> >even means.
>
> >But my point goes beyond that - and is made explicit in questions 3
> >and 4, which you skipped. How can we say that ZF(C) (any particular
> >axiomatization) is correct and the others aren't?
>
> Uh, no, although I didn't type anything in the space below that
> question I _did_ answer it. People _don't_ say that that
> axiomatization is correct and the others are wrong.
How can they both be right?
> >And then why
> >continue using the incorrect ones?
>
> >The principle is not new. You take two big nebuouls things and say
> >they are equivalent. It's typically hard to say whether that is true
> >or not. But a problem occurs if someone points out that one of the
> >two nebulous things has siblings that are not the same. The question
> >becomes: Why this particular one?
>
> >The current topic concerns ZFC and "all of Mathematics", so people
> >like to say they're equivalent.
>
> You're just typing because it turns you on to see your words appear
> on the internet, right? I mean you already conceded above that
> people _don't_ say this - now you ask again about people saying this.
No I didn't. You're the one who said both - see above.
> Have fun - if it ever happens that you have questions that you
> actually want an answer to let us know.
Yeah, why people believe things without reason?
"The persistent and sustained reliance on falsehoods as the basis of
policy, even in the face of massive and well-understood evidence to
the contrary, seems to many Americans to have reached levels that were
previously unimaginable. Millions of us have been asking: 'Why do
reason, logic and truth seem to play a sharply diminished role in the
way America now makes important decisions?' " - Al Gore, The Assault
on Reason
> >Eureka! But why ZFC and not the
> >other axiomatizations?
>
> >Another example is when people say that Godel's 1st Incompleteness
> >Theorem is equivalent to Turing's unsolvability of the Halting
> >Problem. Again, there are siblings to both theorems (e.g. Rosser 1936
> >and the unsolvability of the Membership Problem.) This brings up the
> >point that we are declaring two things to be mathematically equivalent
> >in reality based on when they were discovered (each being the first
> >theorem developed in their respective domains.)
>
> >> David C. Ullrich
>
> David C. Ullrich- Hide quoted text -
>
> - Show quoted text -- Hide quoted text -
Charlie-Boo
How about the Pythagorean Theorem? Last I heard there were 20 or 30
different ways to prove it using "ordinary Mathematics".
C-B
> MoeBlee
Jan Burse
> On 2008-02-05, in sci.logic, Jan Burse wrote:
>> With the problem that ZFC is not a "logical" theory,
>> but rather a "set" theory. So it doesn't really
>> refer to modes of reasoning, rather to classical
>> reasoning inside set theory.
>
> Here modes of reasoning does not refer to logical principles, but
> rather to mathematical principles employed routinely in set theoretic
> arguments, e.g. applying transfinite induction, defining operations on
> sets by transfinite recursion, constructions involving the axiom of
> choice and so on. It has been shown that using such principles all the
> usual theorems of mathematics can be proved, and from this observation
> it follows, by the completeness theorem and the conceptual analysis
> due to Kreisel, that their formalisations are formally derivable in
> ZFC which, directly or indirectly, contains the formlisations of these
> principles. That is, on basis of the work of logicians, set theorists,
> analysts and who not, we have a reduction of ordinary mathematics to
> set theory, and, as ZFC is a formalisation of set theory, implicitly
> to ZFC.
>
formalize math.
And this is clearly a fallacy:
math_is_formalizable_in(ZFC) ->
(forall X (math_is_formalizable_in(X) -> X=ZFC))
And this is also a fallacy:
math_is_formalizable_in(X) ->
can_be_reduced_to(X,ZFC)
Isn't it?
Check out for example:
Appendix 4
Descriptive Set Theory
http://linas.org/mirrors/www.ltn.lv/2001.03.27/~podnieks/gtaa.html
If you think that trying of "experimental" axioms of set theory
(constructibility, determinateness, inaccessible cardinals etc.) is a
business not serious enough for a true mathematician, you may follow
prominent personalities like as H.Lebesgue, E.Borel, R.Baire,
P.S.Aleksandrov, N.N.Luzin and many others. Instead of quick postulating
new axioms that would allow solving of unsolvable problems (for example,
the continuum problem), these people prefer working in the classical set
theory (i.e. ZFC). If the axioms of ZFC do not allow proving the
continuum hypothesis, a true mathematician should not search for
additional axioms. Instead, he should ask: if I cannot prove the
continuum hypothesis, i.e. that there are no infinite sets of real
numbers with cardinality between (the countable) aleph0 and the
cardinality of the entire continuum, then, perhaps, I can prove that
there are no "simple" or "definable" sets of this kind?
[...]
The axiom of projective determinateness (PD)
Best Regards
Charlie-Boo
wrote:
> On 2008-02-05, in sci.logic, Charlie-Boo wrote:
>
> > Where in "Classic Set Theory" by Derek Goldrei do I find a proof
> > within ZFC of something outside of Arithmetic and Set Theory?
>
> I have no idea -- I don't think I've read that book.
Aren't you the fan who said that the answer to my question could be
found in any good book on Set Theory?
> But why would you expect to find something "outside of Set Theory"
> proven in ZFC? ZFC is a formal theory of sets, after all. When we
> speak of some theorem in this or that field of mathematics being
> provable in ZFC what is meant is that that theorem, expressed using
> set theoretical notions as usual, and formalised in the language of
> set theory, is formally provable in ZFC.
You know, you haven't even said if you are allowed to add axioms or
not. What is the claim - how about an example?
> The details of this business
> of expressing theorems of analysis, topology, etc. in terms of sets,
> you'll find in any decent textbook.
You already said that and when I gave you the title of a book you
wouldn't comment on it. It is a serious question. People's
livelihoods are at stake. Just think what would happen if the
Mathematical community discovered that ZFC is not the miracle cure-all
it's cracked up to be after all, and all this time all of these
professors were duped into believing it was from the lure of grant
money.
How about Axiomatic Set Thery (Bernays or Suppes), Set Theory for the
Working Mathematician, Cantorian Set Theory and Limitation of Size
(Hallett), Basic Set Theory (Levy), Notes on Set Theory (Moschovakis)
or Mathemartical Logic (Ebbinghaus et. al.)? What page number has a
proof of something outside of Arithmetic or Set Theory using only
ZFC's axioms?
C-B
> --
> Aatu Koskensilta (aatu.koskensi...@xortec.fi)
>
MoeBlee
There may be many ways of proving it in ordinary mathematics.
Meanwhile, such basic analytical geometry as a proof of the Pyhagorean
theorem can be formulated in ZFC. It would be a rather tedious
exercise to carry out such a formulation, but with sufficient
understanding of ZFC and of analytical geometry, one can see how such
a formulation would be accomplished.
By the way, the subjects I asked about were topology, analysis,
abstract algebra, and graph theory.
MoeBlee
Aatu Koskensilta
> I think there are some competing approach to formalize math.
Sure. I don't see what in my comments you took as implying
otherwise.
As to determinacy, large cardinals, descriptive set theory, and the
like, no claim has been made that all of the mathematics dealing with
such things can be formalised in ZFC. As an example, Kechris's proof
that determinacy holds in L(R) certainly doesn't go trough in
ZFC. These results are not currently considered part of "ordinary
mathematics".
You quote Podnieks on descriptive set theory
If you think that trying of "experimental" axioms of set theory
(constructibility, determinateness, inaccessible cardinals etc.) is a
business not serious enough for a true mathematician, you may follow
prominent personalities like as H.Lebesgue, E.Borel, R.Baire,
P.S.Aleksandrov, N.N.Luzin and many others. Instead of quick postulating
new axioms that would allow solving of unsolvable problems (for example,
the continuum problem), these people prefer working in the classical set
theory (i.e. ZFC). If the axioms of ZFC do not allow proving the
continuum hypothesis, a true mathematician should not search for
additional axioms. Instead, he should ask: if I cannot prove the
continuum hypothesis, i.e. that there are no infinite sets of real
numbers with cardinality between (the countable) aleph0 and the
cardinality of the entire continuum, then, perhaps, I can prove that
there are no "simple" or "definable" sets of this kind?
but I fail to see in support of what. In this excerpt he describes the
attitude of many "ordinary mathematicians" -- that if such problems as
the continuum hypothesis are too general and abstract, unsolvable
using the ordinary principles of set theory, we should concentrate on
their, possibly more manageable, special cases obtained by restricting
our attention to some simpler, more structured class of sets (of
reals, say). What does this have to do with any claim that ordinary
mathematics is formalisable in ZFC?
(By the reduction of ordinary mathematics to set theory I had in mind
nothing but what you'll find in textbooks on analysis, topology and
such like, where ordered pairs, functions, relations, naturals,
rationals, reals, etc. are defined using set theoretic notions. To
this reduction we need not attach any such ideas as that mathematics
is only about sets, or that reals are "really" sets or what not. That
is, it is possible to view it as a purely mathematical device.)
billh04
[cut]
> You can see all of Mathematics as being part of Set Theory simply
> because everything can be the sole member of a set.
No. Euclid's points and lines are things in Mathematics, but they are
not part of set theory: they are part of Euclid's theory of geometry.
There is no set theory in Euclid's geometry.
You first accept the axiomatic system that you are going to work in,
and then you start proving the statements of your theory. You appear
to be saying that if I am doing mathematics, then I can take an item
and consider the set that has it as its sole member. This is not true:
I might not have any axioms that say thinds are sets.
> So any time you
> start to define a Mathematical system, any single thing or pair of
> related things etc. constitutes a set.
No. I may not be working in set theory.
> But this also applies to lists, directed graphs etc. And in each
> case, we are just adding a superstructure with no consequence.
The consequences would follow from the definitions of list, directed
graph, etc.
> The consequences come from the axioms and ZFC's axioms do not replace any
> other system's (outside of Arithmetic and Set Theory.)
Well, if you have axioms for those items, then of course ZFC's axioms
do not replace those axioms.
But, you could try to start with just the axioms of ZFC and define
items like lists, directed graphs, etc and then use the axioms of ZFC
to deduce additional properties of those items.
-- Bill Hale
Jan Burse
> but I fail to see in support of what. In this excerpt he describes the
> attitude of many "ordinary mathematicians" -- that if such problems as
> the continuum hypothesis are too general and abstract, unsolvable
> using the ordinary principles of set theory, we should concentrate on
> their, possibly more manageable, special cases obtained by restricting
> our attention to some simpler, more structured class of sets (of
> reals, say). What does this have to do with any claim that ordinary
> mathematics is formalisable in ZFC?
>
One could also proclaim, the more simpler and more
structured classes are the "ordinary" mathematis object,
and thus cannot be handled in ZFC.
Bye
Aatu Koskensilta
> You know, you haven't even said if you are allowed to add axioms or
> not.
You can freely add broccoli for all anyone cares. Do you think
repeating the same idiotic ramble every now and then will actually
achieve something?
--
Aatu Koskensilta (aatu.kos...@xortec.fi)
Jan Burse
> How do We Know that ZF is the Axiomatization that Proves everything
> provable?
>
> People say that ZF proves everything provable. I wonder:
>
> 1. How can we know that?
>
> provable in ZF but are provable in an extension of ZF?
>
> KPU etc.)?
>
> other axioms of Set Theory must be wrong. So why do we still use
> them?
>
> C-B
>
ZFC hardly proves all of Math, see for example:
http://en.wikipedia.org/wiki/List_of_statements_undecidable_in_ZFC
Gaps in:
Functional analysis
Measure theory
Axiomatic set theory
Set theory of the real line
Group theory
Order theory
Aatu Koskensilta
> structured classes are the "ordinary" mathematis object,
> and thus cannot be handled in ZFC.
You seem to be misunderstanding Podniek's point. There's nothing
problematic in formalising ordinary mathematical reasoning about
constructible sets, projective sets of reals, etc. in ZFC.
Jan Burse
> On 2008-02-05, in sci.logic, Jan Burse wrote:
>> One could also proclaim, the more simpler and more
>> structured classes are the "ordinary" mathematis object,
>> and thus cannot be handled in ZFC.
>
> You seem to be misunderstanding Podniek's point. There's nothing
> problematic in formalising ordinary mathematical reasoning about
> constructible sets, projective sets of reals, etc. in ZFC.
>
why isn't it already there?
Aatu Koskensilta
> But why do I have to add so much to ZFC,
> why isn't it already there?
I have no idea what you're on about. In order to formalise ordinary
mathematical reasoning about constructible sets, projective sets of
reals, etc. there is no need to "add" anything to ZFC.
Frederick Williams
>
> On Feb 5, 12:13 pm, Charlie-Boo <shymath...@gmail.com> wrote:
> > On Feb 5, 2:53 pm, MoeBlee <jazzm...@hotmail.com> wrote:
> >
> > > On Feb 5, 11:06 am, Charlie-Boo <shymath...@gmail.com> wrote:
> >
> > > > If we can do most of Mathematics, then we can do a lot of simple
> > > > Mathematics. Then could you give a simple self-contained example of a
> > > > proof within ZFC that is outside of Arithmetic and Set Theory?
> >
> > > Such subjects as topology, analysis, abstract algebra, and graph
> > > theory are not usually thought of as part of a set theory course
> > > itself, but statements and theorems in such subjects can be given in
> > > the language of and proven in ZFC.
> >
> > > Rather than me just list one theorem after another, a better question
> > > would be for you to say what theorem of such subjects you think canNOT
> > > be proven in ZFC.
> >
> > How about the Pythagorean Theorem? Last I heard there were 20 or 30
> > different ways to prove it using "ordinary Mathematics".
>
> There may be many ways of proving it in ordinary mathematics.
> Meanwhile, such basic analytical geometry as a proof of the Pyhagorean
> theorem can be formulated in ZFC. It would be a rather tedious
> exercise to carry out such a formulation,
My point was: _so_ tedious that no one will bother. A computer may do
it some day (perhaps already has) and when it happens it may be of
interest to computer folk but will it be of interest to mathematicians?
> but with sufficient
> understanding of ZFC and of analytical geometry, one can see how such
> a formulation would be accomplished.
>
> By the way, the subjects I asked about were topology, analysis,
> abstract algebra, and graph theory.
>
> MoeBlee
--
Going forward at this moment in time a raft of measures
have been put in place on the ground to target and
claw back the growth of cliche usage 24/7.
Remove "antispam" and ".invalid" for e-mail address.
Frederick Williams
>
> On 2008-02-03, in sci.logic, Frederick Williams wrote:
> > I doubt that the majority of mathematics has been done in ZFC.
>
> Quoting myself in the thread /Can ZFC prove Addition is Associative?/:
>
> What is at issue is the weaker, purely extensional claim, that
> everything proved, to this day, in "ordinary mathematics" - which is
> a sociological and statistical notion, as it happens - is formally
> anyone's actually producing formal proofs, though there are attempts
> at that direction in the automated theorem proving circles, but
> rather on observing that the basic modes of reasoning and the basic
> mathematical principles used in proofs in ordinary mathematical
> proofs have formal counterparts in ZFC, and then concluding on basis
> of the completeness theorem -- and the observation, due to Kreisel,
> that the informal notion of a statement following logically from a
> set of axioms corresponds to formal provability of the formalisation
> of the statement from the formalisation of those axioms -- that
> there exists a formal proof in ZFC of any statement proved, in the
> ordinary sense, using these modes of reasoning and principles.
>
I should probably explain that by "done in ZFC" I meant "formalized in
ZFC". That most of it _could_ be formalized I don't doubt.
Aatu Koskensilta
> I should probably explain that by "done in ZFC" I meant "formalized in
> ZFC". That most of it _could_ be formalized I don't doubt.
Then we're in heartening agreement. Formal derivations would in any
case be of no particular mathematical interest, and, as you note,
would be quite dreadfully tedious. Machine checkable proofs in some
more suitable system might conceivably be of interest, of course.
MoeBlee
Williams"@antispamhotmail.co.uk.invalid> wrote:
> MoeBlee wrote:
>
> > On Feb 5, 12:13 pm, Charlie-Boo <shymath...@gmail.com> wrote:
> > > On Feb 5, 2:53 pm, MoeBlee <jazzm...@hotmail.com> wrote:
>
> > > > On Feb 5, 11:06 am, Charlie-Boo <shymath...@gmail.com> wrote:
>
> > > > > If we can do most of Mathematics, then we can do a lot of simple
> > > > > Mathematics. Then could you give a simple self-contained example of a
> > > > > proof within ZFC that is outside of Arithmetic and Set Theory?
>
> > > > Such subjects as topology, analysis, abstract algebra, and graph
> > > > theory are not usually thought of as part of a set theory course
> > > > itself, but statements and theorems in such subjects can be given in
> > > > the language of and proven in ZFC.
>
> > > > Rather than me just list one theorem after another, a better question
> > > > would be for you to say what theorem of such subjects you think canNOT
> > > > be proven in ZFC.
>
> > > How about the Pythagorean Theorem? Last I heard there were 20 or 30
> > > different ways to prove it using "ordinary Mathematics".
>
> > There may be many ways of proving it in ordinary mathematics.
> > Meanwhile, such basic analytical geometry as a proof of the Pyhagorean
> > theorem can be formulated in ZFC. It would be a rather tedious
> > exercise to carry out such a formulation,
>
> My point was: _so_ tedious that no one will bother. A computer may do
> it some day (perhaps already has) and when it happens it may be of
> interest to computer folk but will it be of interest to mathematicians?
Do what? Write full formalizations in the language of set theory of
virtually all the ordinary mathematics as currently published in
textbooks and journal articles? I can't say what will interest
mathematicians. But I guess that most mathematicians wouldn't be
interested in reading very many of those formalizations but that some
number of mathematicians would be interested that such a thing had
been done.
MoeBlee
David C. Ullrich
<shyma...@gmail.com> wrote:
>On Feb 5, 2:42 pm, Aatu Koskensilta <aatu.koskensi...@xortec.fi>
>wrote:
>> On 2008-02-05, in sci.logic, Charlie-Boo wrote:
>>
>> > And even worse, claim to be Mathematicians making a Mathematical
>> > statement while others watch and even believe him or her!
>>
>> Positively repugnant!
>>
>> > Why don't you give an example? What Mathematics other than
>> > Arithemetic and Set Theory has anyone ever proven in ZFC?
>>
>> You'll find the set theoretic development of analysis in any decent
>> textbook on the subject.
>
>Where in "Classic Set Theory" by Derek Goldrei do I find a proof
>within ZFC of something outside of Arithmetic and Set Theory?
He meant in any decent text on _analysis_.
The last time this came up you simply refused to look at
a text that was suggested. It's certainly true that if you don't
look at examples then you won't see any.
>
>C-B
>
>> --
>> Aatu Koskensilta (aatu.koskensi...@xortec.fi)
>>
>> "Wovon man nicht sprechen kann, daruber muss man schweigen"
>> - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
David C. Ullrich
David C. Ullrich
Williams"@antispamhotmail.co.uk.invalid> wrote:
>Charlie-Boo wrote:
>
>> >How has it been demonstrated that the majority of mathematics can be
>> >done in ZF(C)?
>
>David C. Ullrich wrote:
>
>> By _doing_ the mathematics in ZFC.
>
>I doubt that the majority of mathematics has been done in ZFC.
>(Actually, that is a horrible use of the word "majority", I would rather
>say "most" than "the majority".)
>
>There is, I suppose, a belief that most mathematics could be done in ZFC
>given sufficient time and interest.
If you mean that most mathematics has not actually been formalized
in ZFC then no, I suppose it hasn't. But I'm puzzled by your use of
the word "belief" here. Maybe the problem is that I just don't know
much math - all the math I know has the property that it's incredibly
clear that it could be formalized in ZFC.
(Define the natural numbers to be the finite ordinals, use them
to define the integers, then the rationals, then the reals in standard
ways - then go ahead and prove all the standard theorems of
analysis. What part of that seems non-trivial?
(Of course here I'm using the word "trivial" in a precise
sense, such that for example playing perfect chess is
trivial, since it's trivial to give an algorithm that plays
perfect chess, although calculuating an acceptable
first move would probably take longer than the age
of the universe.))
David C. Ullrich
David C. Ullrich
Williams"@antispamhotmail.co.uk.invalid> wrote:
>Aatu Koskensilta wrote:
>>
>> On 2008-02-03, in sci.logic, Frederick Williams wrote:
>> > I doubt that the majority of mathematics has been done in ZFC.
>>
>> Quoting myself in the thread /Can ZFC prove Addition is Associative?/:
>>
>> What is at issue is the weaker, purely extensional claim, that
>> everything proved, to this day, in "ordinary mathematics" - which is
>> a sociological and statistical notion, as it happens - is formally
>> provable in ZFC, suitably formalised. Of course, this is not based on
>> anyone's actually producing formal proofs, though there are attempts
>> at that direction in the automated theorem proving circles, but
>> rather on observing that the basic modes of reasoning and the basic
>> mathematical principles used in proofs in ordinary mathematical
>> proofs have formal counterparts in ZFC, and then concluding on basis
>> of the completeness theorem -- and the observation, due to Kreisel,
>> that the informal notion of a statement following logically from a
>> set of axioms corresponds to formal provability of the formalisation
>> of the statement from the formalisation of those axioms -- that
>> there exists a formal proof in ZFC of any statement proved, in the
>> ordinary sense, using these modes of reasoning and principles.
>>
>> (Message-ID: <1168866745....@38g2000cwa.googlegroups.com>)
>
>I should probably explain that by "done in ZFC" I meant "formalized in
>ZFC". That most of it _could_ be formalized I don't doubt.
Fine - then ignore the question I just posted elsewhere in the thread.
It _sounded_ as though you doubted this, or at least that you doubted
that it was obviously true.
David C. Ullrich
David C. Ullrich
<shyma...@gmail.com> wrote:
>On Feb 2, 8:15 pm, Chris Menzel <cmen...@remove-this.tamu.edu> wrote:
>> On Sat, 2 Feb 2008 05:58:56 -0800 (PST), Charlie-Boo
>> <shymath...@gmail.com> said:
>>
>> > People say that ZF proves everything provable. I wonder:
>>
>> > 1. How can we know that?
>>
>> Better question: Who the hell says that?
>>
>> > 2. Doesn't Godel say that there are true statements that are not
>> > provable in ZF but are provable in an extension of ZF?
>>
>> Sure. So what?
>>
>> > 3. Why ZF and not the other axiomatizations of Set Theory (ZFC, NBG,
>> > KPU etc.)?
>>
>> Why indeed. All four are used in various contexts.
>>
>> > 4. If ZF proves all of Mathematics that can be proven,
>>
>> Obviously it doesn't.
>
>Do any of the alternatives?
You really _haven't_ heard of Godel's theorems, eh? Huh.
>
>C-B
>
David C. Ullrich
David C. Ullrich
<shyma...@gmail.com> wrote:
>On Feb 3, 8:53 am, David C. Ullrich <dullr...@sprynet.com> wrote:
>> On Sat, 2 Feb 2008 18:47:08 -0800 (PST), Charlie-Boo
>> >On Feb 2, 9:40 am, David C. Ullrich <dullr...@sprynet.com> wrote:
>
>> >> Prople do say that ZFC is sufficient to do all the mathematics
>> >> that people usually want to do. The reason they say that is
>> >> that it's _true_. It's not something that can be proved
>> >> mathematically
>
>> Sometimes you seem blind or deaf or something.
>> Suppose that I say that it's a nice day today here in Stillwater.
>
>I'm blind to irrelevant statements such as the above?
When you repeatedly ask for a formal proof of something after
it's been pointed out that it's simply not the sort of thing that
one can give a formal proof for then yes, you seem blind or deaf.
David C. Ullrich
Jan Burse
> On 2008-02-05, in sci.logic, Jan Burse wrote:
>> But why do I have to add so much to ZFC,
>> why isn't it already there?
>
> I have no idea what you're on about. In order to formalise ordinary
> mathematical reasoning about constructible sets, projective sets of
> reals, etc. there is no need to "add" anything to ZFC.
>
There is a difference between:
ZFC + X |- Y or my favorite base BASE, with BASE |- Y (1)
And:
ZFC |- X -> Y (2)
It depends whether you see X as a general principle or
part of the thing that is proved. You can always
say no, its not a principle, and proclaim what a
mathematician is doing, is (2).
But I am sure some mathematicians feel that it should
be (1), i.e. that there should be a better base. I
am sure that there are some principles X that maybe
somebody would like to see in the base.
Best Regards
Charlie-Boo
How do you define formalizing in ZFC? Are you allowed to add axioms?
If not, I'd like to see how the Pythagorean Theorem is proven. If you
do have to add axioms, then I would still like to see it, and
especially how the ZFC axioms played any actual role in that
particular proof (as opposed to saying "You need ZFC to state that
there is an infinite set." in response to every theorem, which is not
the point.)
The ZFC axioms have nothing to do with anything but an attempt to
avoid Russell's Paradox. The Peano Axioms are another matter, as they
guarantee that Addition, Multiplication and the Universal Set are
representable, which has many implications. But that's all you have -
some theorems of Set Theory from ZFC proper, and much of simple
Arithmetic from the Peano Axioms that are added to that.
The fact that there are no references to anything outside of sets and
integers (not e.g. Geometry or Trigonometry) ever being carried out in
ZFC shows that.
And it's not the ability to "represent" the objects of the system that
is at issue. You can pair up any two r.e. sets of expressions, and
call the first set or graph or list or wff or program zero. And the
next one you call one.
The problem is that you need to know specific relationships among
those objects, and the ZFC axioms state only certain relationships.
You could probably prove the following interesting question: For any
r.e. set, use Peano's Axioms to prove that it is r.e. But this
doesn't give you things like the Pythagorean Theorem and the rest of
common Mathematics.
C-B
> And this is also a fallacy:
>
> math_is_formalizable_in(X) ->
> can_be_reduced_to(X,ZFC)
>
> Isn't it?
>
> Check out for example:
> Appendix 4
> Descriptive Set Theoryhttp://linas.org/mirrors/www.ltn.lv/2001.03.27/~podnieks/gtaa.html
>
> If you think that trying of "experimental" axioms of set theory
> (constructibility, determinateness, inaccessible cardinals etc.) is a
> business not serious enough for a true mathematician, you may follow
> prominent personalities like as H.Lebesgue, E.Borel, R.Baire,
> P.S.Aleksandrov, N.N.Luzin and many others. Instead of quick postulating
> new axioms that would allow solving of unsolvable problems (for example,
> the continuum problem), these people prefer working in the classical set
> theory (i.e. ZFC). If the axioms of ZFC do not allow proving the
> continuum hypothesis, a true mathematician should not search for
> additional axioms. Instead, he should ask: if I cannot prove the
> continuum hypothesis, i.e. that there are no infinite sets of real
> numbers with cardinality between (the countable) aleph0 and the
> cardinality of the entire continuum, then, perhaps, I can prove that
> there are no "simple" or "definable" sets of this kind?
>
> [...]
>
> The axiom of projective determinateness (PD)
>
> Best Regards- Hide quoted text -
Charlie-Boo
What would be the informal proof that was being formalized? You start
out with a right triangle. Then how does the logic go?
At some point you'll need axioms. Do you have to add your own axioms
to ZFC's? Are you claiming that it can be done using ZFC, or that we
could present new axioms in ZFC that would do it?
C-B
> By the way, the subjects I asked about were topology, analysis,
> abstract algebra, and graph theory.
and quite well
> MoeBlee- Hide quoted text -
Charlie-Boo
wrote:
> but I fail to see in support of what. In this excerpt he describes the
> attitude of many "ordinary mathematicians" -- that if such problems as
> the continuum hypothesis are too general and abstract, unsolvable
> using the ordinary principles of set theory, we should concentrate on
> their, possibly more manageable, special cases obtained by restricting
> our attention to some simpler, more structured class of sets (of
> reals, say).
That's true of any problem. You start with simpler special cases and
work your way up to the general case. I used that to solve one of the
problems in C. S. Ogilvy's "Tomorrow's Math: Unsolved Problems for the
Amateur" when I was in high school (he acknowledged.) How many rounds
on the average does it take before someone has all the money when 3
players start with A, B and C coins, after each round one randomly
chosen player receives one coin for each of the other players, and
drop out when they reach 0? Divide and conquer.
C-B
> What does this have to do with any claim that ordinary
> mathematics is formalisable in ZFC?
>
> (By the reduction of ordinary mathematics to set theory I had in mind
> nothing but what you'll find in textbooks on analysis, topology and
> such like, where ordered pairs, functions, relations, naturals,
> rationals, reals, etc. are defined using set theoretic notions. To
> this reduction we need not attach any such ideas as that mathematics
> is only about sets, or that reals are "really" sets or what not. That
> is, it is possible to view it as a purely mathematical device.)
>
> --
> Aatu Koskensilta (aatu.koskensi...@xortec.fi)
Charlie-Boo
> On Feb 5, 2:05 pm, Charlie-Boo <shymath...@gmail.com> wrote:
> [cut]
>
> > You can see all of Mathematics as being part of Set Theory simply
> > because everything can be the sole member of a set.
>
> No. Euclid's points and lines are things in Mathematics, but they are
> not part of set theory: they are part of Euclid's theory of geometry.
> There is no set theory in Euclid's geometry.
I am being the Devil's Advocate. People say that you start every
mathematical system by declaring sets and so they are all defined in
or using set theory. Or something like that. But the point is that
because any thing X that is described can be described as the set
{ X } people sometimes say that it is carried out in Set Theory, or
ZFC in particular. The real problem is that the ZFC Axioms must give
you the theorems of that theory.
C-B
BTW The fact that for every X there is a set { X } is an axiom in CBL,
in more general terms. For every number there is a Turing Machine
that outputs only it and halts. There is a wff w(x) that is provable
iff x is replaced by that number. There is a sentence in English that
is true etc. These are all the same expression in CBL.
In CBL it is the expression wr(I) # I and it is used e.g. to prove the
Double Recursion Theorem: For any two recursive functions f(x) and
g(x) there are numbers M and N such that program M outputs f(N) and
program N outputs g(M). In CBL that's M # f(N) + N # g(M) and takes
about 5-10 steps to prove constructively - i.e. there is a recursive
function (applied to the numbers for f and g) to give us M and N and
that function is given. (This formally solves one of Smullyan's
puzzles.) No theorem anywhere nearly as complex in Recursion Theory
has even been formalized in print AFAIK.
> You first accept the axiomatic system that you are going to work in,
> and then you start proving the statements of your theory. You appear
> to be saying that if I am doing mathematics, then I can take an item
> and consider the set that has it as its sole member. This is not true:
> I might not have any axioms that say thinds are sets.
>
> > So any time you
> > start to define a Mathematical system, any single thing or pair of
> > related things etc. constitutes a set.
>
> No. I may not be working in set theory.
>
> > But this also applies to lists, directed graphs etc. And in each
> > case, we are just adding a superstructure with no consequence.
>
> The consequences would follow from the definitions of list, directed
> graph, etc.
>
> > The consequences come from the axioms and ZFC's axioms do not replace any
> > other system's (outside of Arithmetic and Set Theory.)
>
> Well, if you have axioms for those items, then of course ZFC's axioms
> do not replace those axioms.
>
> But, you could try to start with just the axioms of ZFC and define
> items like lists, directed graphs, etc and then use the axioms of ZFC
> to deduce additional properties of those items.
You can produce that about as much as you were able to produce the
last claim of yours regarding what has been formalized. ZFC gives you
nothing outside of sets and arithmetic. If you tried to apply its
rules to the analogies of the Set Theory expressions in another
system, heavens knows what a meaningless mess you'd get!
C-B
> -- Bill Hale
Charlie-Boo
wrote:
> On 2008-02-05, in sci.logic, Charlie-Boo wrote:
>
> > You know, you haven't even said if you are allowed to add axioms or
> > not.
>
> You can freely add broccoli for all anyone cares. Do you think
> repeating the same idiotic ramble every now and then will actually
> achieve something?
Broccoli isn't idiotic?
Do you see the big difference between adding and not adding axioms?
Is it that you are just quoting a passage from somewhere and don't
really know what it means, so you can't say whether you have to add to
ZFC's axioms?
If you add to ZFC then you are not using its axioms for anything but
are just using a different syntax, and you don't know if the resulting
statements in the corresponding system will be meaningful or useful.
If you don't add to ZFC, then I am amazed and would love to see that
example even more.
C-B
> --
> Aatu Koskensilta (aatu.koskensi...@xortec.fi)
>
Charlie-Boo
> Charlie-Boo schrieb:
>
>
>
> > How do We Know that ZF is the Axiomatization that Proves everything
> > provable?
>
> > People say that ZF proves everything provable. I wonder:
>
> > 1. How can we know that?
>
> > 2. Doesn't Godel say that there are true statements that are not
> > provable in ZF but are provable in an extension of ZF?
>
> > 3. Why ZF and not the other axiomatizations of Set Theory (ZFC, NBG,
> > KPU etc.)?
>
> > 4. If ZF proves all of Mathematics that can be proven, then these
> > other axioms of Set Theory must be wrong. So why do we still use
> > them?
>
> > C-B
>
> ZFC hardly proves all of Math, see for example:
>
> http://en.wikipedia.org/wiki/List_of_statements_undecidable_in_ZFC
Yes, Prof. Ullrich pointed that out, and as I informed him, I believe
that is excluded by the term "provable" in conventional academic
wisdom.
That really shows that ZFC is just a dinky little nothing that many
people are stupid enough to believe is great because somebody wrote it
in a book or magazine with a slick cover. Can you believe that? Of
all the stupid nonsense. Just like Al Gore wrote, Americans are
getting more and more stupider.
C-B
> Gaps in:
> Functional analysis
> Measure theory
> Axiomatic set theory
> Set theory of the real line
> Group theory
> Order theory- Hide quoted text -
Charlie-Boo
wrote:
> On 2008-02-05, in sci.logic, Jan Burse wrote:
>
> > One could also proclaim, the more simpler and more
> > structured classes are the "ordinary" mathematis object,
> > and thus cannot be handled in ZFC.
>
> You seem to be misunderstanding Podniek's point. There's nothing
> problematic in formalising ordinary mathematical reasoning about
> constructible sets, projective sets of reals, etc. in ZFC.
Do you mean representing just the statements or that the proofs are
carried out using only ZFC's axioms as the axioms and rules?
Or is that just a quote from somewhere?
C-B
> --
> Aatu Koskensilta (aatu.koskensi...@xortec.fi)
>
Charlie-Boo
Because you can't put all that stuff in one place. Simply saying
"This ain't so." proves that no system is Self-representing,
Negatable, Substitution representing, Complete and Consistent
(including English.)
C-B
Charlie-Boo
wrote:
> On 2008-02-05, in sci.logic, Jan Burse wrote:
>
> > But why do I have to add so much to ZFC,
> > why isn't it already there?
>
> I have no idea what you're on about. In order to formalise ordinary
> mathematical reasoning about constructible sets, projective sets of
> reals, etc. there is no need to "add" anything to ZFC.
Can you give one tiny little self-contained example? Every system has
its 0+1=1 and P&~P == FALSE and other tiny little constructions. In
Program Synthesis I synthesize BETW(I,J,K) decide if one number is
between two others - it takes 2 simple steps.
Just one example of a self-contained, nothing hidden, all out in the
open, instance of something clearly not arithmetic or sets, being
proven using only ZFC's axioms?
C-B
> --
> Aatu Koskensilta (aatu.koskensi...@xortec.fi)
>
Charlie-Boo
wrote:
> On 2008-02-05, in sci.logic, Frederick Williams wrote:
>
> > I should probably explain that by "done in ZFC" I meant "formalized in
> > ZFC". That most of it _could_ be formalized I don't doubt.
>
> Then we're in heartening agreement. Formal derivations would in any
> case be of no particular mathematical interest,
Hilbert? Godel?
> and, as you note,
> would be quite dreadfully tedious.
Oh f*** no. Are you referring to the BS obfuscation smoke screens
that people throw up when claiming to be formal? The complexity of
anything depends on the level of abstraction it is at. It has nothing
to do with whether it is formal or not. ("Informal" really means the
theorems are not r.e.)
CBL is completely formal and generated the world's shortest proof of
Rosser's 1936 extension to Godel: "If the system is complete and
consistent, then by propositional calculus the non-theorems coincide
with the refutable sentences, but only the latter is r.e." Is this
not a proof of Rosser 1936? Is it not about 100 times shorter than
his proof? Have you seen it elsewhere? Have you seen anything
shorter?
Tell me your favorite theorem from CS. (The last time I said this
someone said "There is a UTM." That took about 5-10 steps.
> Machine checkable proofs in some
> more suitable system might conceivably be of interest, of course.
How is that so different from formal derivations (that are of no
mathematical interest)?
C-B
> --
> Aatu Koskensilta (aatu.koskensi...@xortec.fi)
>
Charlie-Boo
> On Tue, 5 Feb 2008 11:46:09 -0800 (PST), Charlie-Boo
> <shymath...@gmail.com> wrote:
> >On Feb 5, 2:42 pm, Aatu Koskensilta <aatu.koskensi...@xortec.fi>
> >wrote:
> >> On 2008-02-05, in sci.logic, Charlie-Boo wrote:
>
> >> > And even worse, claim to be Mathematicians making a Mathematical
> >> > statement while others watch and even believe him or her!
>
> >> Positively repugnant!
>
> >> > Why don't you give an example? What Mathematics other than
> >> > Arithemetic and Set Theory has anyone ever proven in ZFC?
>
> >> You'll find the set theoretic development of analysis in any decent
> >> textbook on the subject.
>
> >Where in "Classic Set Theory" by Derek Goldrei do I find a proof
> >within ZFC of something outside of Arithmetic and Set Theory?
>
> He meant in any decent text on _analysis_.
Oh, he covers the representation of other systems in ZFC. Section 3.2
is the Construction of the Natural Numbers. Isn't that what we're
really looking for?
The problem is that, as always, he is only talking about arithmetic
and sets. Isn't that so?
> The last time this came up you simply refused to look at
> a text that was suggested. It's certainly true that if you don't
> look at examples then you won't see any.
What text was that and what is represented without additional axioms
outside of arithmetic and sets?
> >C-B
>
> >> --
> >> Aatu Koskensilta (aatu.koskensi...@xortec.fi)
>
> >> "Wovon man nicht sprechen kann, daruber muss man schweigen"
> >> - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
>
> David C. Ullrich- Hide quoted text -
Charlie-Boo
Williams"@antispamhotmail.co.uk.invalid> wrote:
> MoeBlee wrote:
>
> > On Feb 5, 12:13 pm, Charlie-Boo <shymath...@gmail.com> wrote:
> > > On Feb 5, 2:53 pm, MoeBlee <jazzm...@hotmail.com> wrote:
>
> > > > On Feb 5, 11:06 am, Charlie-Boo <shymath...@gmail.com> wrote:
>
> > > > > If we can do most of Mathematics, then we can do a lot of simple
> > > > > Mathematics. Then could you give a simple self-contained example of a
> > > > > proof within ZFC that is outside of Arithmetic and Set Theory?
>
> > > > Such subjects as topology, analysis, abstract algebra, and graph
> > > > theory are not usually thought of as part of a set theory course
> > > > itself, but statements and theorems in such subjects can be given in
> > > > the language of and proven in ZFC.
>
> > > > Rather than me just list one theorem after another, a better question
> > > > would be for you to say what theorem of such subjects you think canNOT
> > > > be proven in ZFC.
>
> > > How about the Pythagorean Theorem? Last I heard there were 20 or 30
> > > different ways to prove it using "ordinary Mathematics".
>
> > There may be many ways of proving it in ordinary mathematics.
> > Meanwhile, such basic analytical geometry as a proof of the Pyhagorean
> > theorem can be formulated in ZFC. It would be a rather tedious
> > exercise to carry out such a formulation,
>
> My point was: _so_ tedious that no one will bother.
No no no. You don't get away with your bullshit that easy. Ever if
it were incredibly complex, one could give a high level description
that explains the logic of the proof and the construction. Saying it
is too complex to give any details at all to substantiate it won't
work. "That dog won't hunt." as Bill Clinton once said.
> A computer may do
> it some day (perhaps already has) and when it happens it may be of
> interest to computer folk but will it be of interest to mathematicians?
How interested are you in Turing's work?
C-B
> > but with sufficient
> > understanding of ZFC and of analytical geometry, one can see how such
> > a formulation would be accomplished.
>
> > By the way, the subjects I asked about were topology, analysis,
> > abstract algebra, and graph theory.
>
> > MoeBlee
>
> --
> Going forward at this moment in time a raft of measures
> have been put in place on the ground to target and
> claw back the growth of cliche usage 24/7.
> Remove "antispam" and ".invalid" for e-mail address.- Hide quoted text -
Charlie-Boo
> On Sun, 03 Feb 2008 14:26:52 GMT, Frederick Williams <"Frederick
>
>
>
>
>
> Williams"@antispamhotmail.co.uk.invalid> wrote:
> >Charlie-Boo wrote:
>
> >> >How has it been demonstrated that the majority of mathematics can be
> >> >done in ZF(C)?
>
> >David C. Ullrich wrote:
>
> >> By _doing_ the mathematics in ZFC.
>
> >I doubt that the majority of mathematics has been done in ZFC.
> >(Actually, that is a horrible use of the word "majority", I would rather
> >say "most" than "the majority".)
>
> >There is, I suppose, a belief that most mathematics could be done in ZFC
> >given sufficient time and interest.
>
> If you mean that most mathematics has not actually been formalized
> in ZFC then no, I suppose it hasn't. But I'm puzzled by your use of
> the word "belief" here. Maybe the problem is that I just don't know
> much math - all the math I know has the property that it's incredibly
> clear that it could be formalized in ZFC.
>
> (Define the natural numbers to be the finite ordinals, use them
> to define the integers, then the rationals, then the reals in standard
> ways - then go ahead and prove all the standard theorems of
> analysis. What part of that seems non-trivial?
Can the proofs be carried out using only ZFC's axioms?
Yes, I agree that all of the systems above have recursively enumerable
sets of theorems.
> (Of course here I'm using the word "trivial" in a precise
> sense, such that for example playing perfect chess is
> trivial, since it's trivial to give an algorithm that plays
> perfect chess, although calculuating an acceptable
> first move would probably take longer than the age
> of the universe.))
>
> David C. Ullrich- Hide quoted text -
William Hale
<3ac510dd-8272-46da...@c23g2000hsa.googlegroups.com>,
Charlie-Boo <shyma...@gmail.com> wrote:
> On Feb 5, 3:49 pm, billh04 <h...@tulane.edu> wrote:
> > On Feb 5, 2:05 pm, Charlie-Boo <shymath...@gmail.com> wrote:
> > [cut]
> >
> > > You can see all of Mathematics as being part of Set Theory simply
> > > because everything can be the sole member of a set.
> >
> > No. Euclid's points and lines are things in Mathematics, but they are
> > not part of set theory: they are part of Euclid's theory of geometry.
> > There is no set theory in Euclid's geometry.
>
> I am being the Devil's Advocate. People say that you start every
> mathematical system by declaring sets and so they are all defined in
> or using set theory. Or something like that. But the point is that
> because any thing X that is described can be described as the set
> { X } people sometimes say that it is carried out in Set Theory, or
> ZFC in particular.
But I am saying Euclid's theory for geometry is not carried out in Set
Theory. I am definitely not saying that Euclid's theory for geometry
would be carried out in Set Theory if I started describing a triangle
ABC as the set {ABC}. I am not going to add the axioms of Set Theory to
the axioms of Euclid for geometry and then say that I am doing geometry
in Set Theory.
> The real problem is that the ZFC Axioms must give
> you the theorems of that theory.
It is impossible that the ZFC Axioms could ever give you the theorems of
Euclid's theory of geometry. A Euclid line is not a set, and Set Theory
is only about sets, so Set Theory cannot even state a theorem in
Euclid's theory of geometry, much less even prove it true.
-- Bill Hale
Jan Burse
> Yes, Prof. Ullrich pointed that out, and as I informed him, I believe
> that is excluded by the term "provable" in conventional academic
> wisdom.
There no such thing as a term "provable" in conventional academic
wisdom. In mathematical logic the game is to view the term "provable"
from all angles. See proof theory.
"undecidable" means probably with the help of FOL from ZFC. But maybe
there are some other cases. But clearly there are other approaches
than FOL. But we do not want to make matters complicated for the moment.
So lets stick to FOL and ZFC, eh voila there are *already* some gaps.
> That really shows that ZFC is just a dinky little nothing that many
> people are stupid enough to believe is great because somebody wrote it
> in a book or magazine with a slick cover. Can you believe that? Of
> all the stupid nonsense. Just like Al Gore wrote, Americans are
> getting more and more stupider.
Like a car that every body believes he needs to buy? Or what?
But ZFC is nevertheless no non-sense. Its quite good on the contrary.
It has even some self reflection capabilities, via comprehension. I
wouldn't call it a dinky little nothing, I would rather call it
a little bit out dated.
Bye
Charlie-Boo
> On Tue, 5 Feb 2008 12:05:45 -0800 (PST), Charlie-Boo
>
> >On Feb 3, 8:53 am, David C. Ullrich <dullr...@sprynet.com> wrote:
> >> On Sat, 2 Feb 2008 18:47:08 -0800 (PST), Charlie-Boo
> >> >On Feb 2, 9:40 am, David C. Ullrich <dullr...@sprynet.com> wrote:
>
> >> >> Prople do say that ZFC is sufficient to do all the mathematics
> >> >> that people usually want to do. The reason they say that is
> >> >> that it's _true_. It's not something that can be proved
> >> >> mathematically
>
> >> Sometimes you seem blind or deaf or something.
> >> Suppose that I say that it's a nice day today here in Stillwater.
>
> >I'm blind to irrelevant statements such as the above?
>
> When you repeatedly ask for a formal proof of something after
> it's been pointed out that it's simply not the sort of thing that
> one can give a formal proof for then yes, you seem blind or deaf.
Representing a proof in ZFC is a formal process and is exactly what is
most amenable to mathematical proof.
A proof would begin with a demonstration of how the simplest theorems
of a large number of branches of mathematics can be developed. Then
it could give more complex ones, either in complete detail or describe
the exact procedure. It could point out the various types of theorems
there are in each system and say something about how each type would
be represented.
The idea that ZFC contains all the "principles of logic" and so it can
do most all of Mathematics is not well-founded. The problem is the
exact relationship of the objects within each system, and whether or
not ZFC's axioms reflect that same relationship when applied to the
corresponding objects within ZFC. That is, if you use only ZFC's
axioms.
And what if you change the representation e.g. N is {}, {{}},
{{{}}}, . . . ? That will make the theorems all different. How do
you know which representation will work?
This is like the subject of this thread: What difference between ZFC
and all the other axiomatizations makes it the one and not the
others? These are all formal systems. The only differences are
formal. What is that formal difference that says ZFC is the one?
Nobody has even answered the question of whether additional axioms
will be needed or not. That sounds just like people are quoting from
something published but don't know how it works, and so can't say.
Why can't you say whether additional axioms will be needed?
If ZFC's axioms sufficed, then you have the almost automatic creation
of axioms for all branches of Mathematics, despite the efforts
required for individual branches in the past. Simply match up the
expressions of the system with set definitions, and see what the ZFC
axioms correspond to in that other system. Then hide ZFC and say,
"Look, I just came up with axioms for this branch of Mathematics!"
I wonder if you could win any prizes with the axiomatizations that you
create? It'd be like having a Mathematical formula to write The
Beatle's songs. Gone are the days of toil.
William Hale
<4460d6da-0f83-406b...@l16g2000hsh.googlegroups.com>,
Charlie-Boo <shyma...@gmail.com> wrote:
[cut]
> Nobody has even answered the question of whether additional axioms
> will be needed or not.
No additional axioms will be needed.
Charlie-Boo
> Charlie-Boo schrieb:
>
> > Yes, Prof. Ullrich pointed that out, and as I informed him, I believe
> > that is excluded by the term "provable" in conventional academic
> > wisdom.
>
> There no such thing as a term "provable" in conventional academic
> wisdom.
I am referring to its use in the assertion to wit, ZFC can prove
essentially everything provable.
> In mathematical logic the game is to view the term "provable"
> from all angles. See proof theory.
No, is better to see the other concepts that are siblings to provable
(true, r.e., well-founded set, etc.) and the angle each of these comes
from. See Computationally Based Logics.
(It creates a much better system. The Peano requirements are both
shorter (by a factor of about 10) and more general! Thus studying CBL
is easier (the proofs are so short) and produces more general
theorems. For example, the axioms of PA prove facts about arithmetic,
but so do others - but they have in common that these relations are
representable. This includes both PA's formulation and others. For
example, after we say that a+b=c => a+b'=c' we can say a'+b=c' or a
+b=c => b+a=c or a+b=b+a etc. for alternatives. But CBL includes all
of them because it is at a higher level and the net result is the same
in terms of what can be proven. (This can be shown formally.)
When I helped develop and standardize a programming language, I told
the people writing it up that they are describing the language at a
too low level of abstraction, and that stifles implementation of
better algorithms - and even the understanding of what it is doing.
There is an analogy here. Peano's Axioms are kept at the low level of
abstraction, and so are also messy and complex, and they exclude
alternate formulations of logic that share their properties by having
the same theorems.
That is, theorems about formal systems that include "If Peano's Axioms
are included . . ." (e.g. Q) would be more general because the higher
level formalism (besides being shorter) includes that and other sets
of axioms that produce the same theorems.
> "undecidable" means probably with the help of FOL from ZFC. But maybe
> there are some other cases. But clearly there are other approaches
> than FOL. But we do not want to make matters complicated for the moment.
> So lets stick to FOL and ZFC, eh voila there are *already* some gaps.
>
> > That really shows that ZFC is just a dinky little nothing that many
> > people are stupid enough to believe is great because somebody wrote it
> > in a book or magazine with a slick cover. Can you believe that? Of
> > all the stupid nonsense. Just like Al Gore wrote, Americans are
> > getting more and more stupider.
>
> Like a car that every body believes he needs to buy? Or what?
About the 9/11 attacks and how they believe in ZFC: "This lack of
logical thought extends even to research and academia, where fraud and
abuse are routine revelations in the news."
> But ZFC is nevertheless no non-sense. Its quite good on the contrary.
> It has even some self reflection capabilities, via comprehension. I
> wouldn't call it a dinky little nothing, I would rather call it
> a little bit out dated.
How about "over-hyped"?
In a nutshell (higher LOA):
ZFC is no better suited to represent Mathematics than any other formal
system. It's axioms are not used, and the constructions described can
be applied to the expressions of any formal system.
ZFC IS NOTHING! Show me a proof in ZFC of a theorem in another system
and I'll show you how to do it in any other system using its Nth
expression every time we use the Nth ZFC expression.
C-B
> Bye
Charlie-Boo
> In article
> <4460d6da-0f83-406b-8a8c-6ac421d15...@l16g2000hsh.googlegroups.com>, Charlie-Boo <shymath...@gmail.com> wrote:
>
> [cut]
>
> > Nobody has even answered the question of whether additional axioms
> > will be needed or not.
>
> No additional axioms will be needed.
Oh No! And what of systems that have statements known to be
indepenent of ZFC?
So, if we know the axioms already, does the chosen correspondence
between expressions in ZFC and expressions in the other system
matter? Would they all produce the same set of theorems in the other
system?
C-B
William Hale
<785910fb-ce98-4653...@j20g2000hsi.googlegroups.com>,
Charlie-Boo <shyma...@gmail.com> wrote:
> On Feb 7, 6:01 pm, William Hale <h...@tulane.edu> wrote:
> > In article
> > <4460d6da-0f83-406b-8a8c-6ac421d15...@l16g2000hsh.googlegroups.com>, Charlie
> > -Boo <shymath...@gmail.com> wrote:
> >
> > [cut]
> >
> > > Nobody has even answered the question of whether additional axioms
> > > will be needed or not.
> >
> > No additional axioms will be needed.
>
> Oh No! And what of systems that have statements known to be
> indepenent of ZFC?
What of them? There is no requirement that all systems lead to the same
set of theorems. I really don't understand what you are trying to get at.
> So, if we know the axioms already, does the chosen correspondence
> between expressions in ZFC and expressions in the other system
> matter?
Matter to whom? There may not even be a correspondence between
expressions in ZFC and expressions in the other system. I really don't
understand what you are trying to get at.
> Would they all produce the same set of theorems in the other
> system?
Probably not. Why would you think so?
-- Bill Hale
Chris Menzel
<shyma...@gmail.com> said:
> ...
> That really shows that ZFC is just a dinky little nothing that many
> people are stupid enough to believe is great because somebody wrote it
> in a book or magazine with a slick cover. Can you believe that? Of
> all the stupid nonsense.
Yes, yes, I myself remember being *dazzled* by the cover of Drake's Set
Theory. All that...yellow. And those large, friendly letters. So
enticing. It wasn't until I got to the chapter on the constructible
universe that I began to realize that set theory is itself a
"constructed" universe, "a dinky little nothing" (as you so eloquently
put it), a conceptual house of cards foisted on naive and trusting
students by unscrupulous professors interested only in retaining their
power and rank in the Academy! So keep up the good work, sir! A
benighted world needs to heed your clarion call! Only ignorance and
misery await those taken in by books with slick covers!
Charlie-Boo
>
> Charlie-Boo <shymath...@gmail.com> wrote:
> > On Feb 7, 6:01 pm, William Hale <h...@tulane.edu> wrote:
> > > In article
> > > <4460d6da-0f83-406b-8a8c-6ac421d15...@l16g2000hsh.googlegroups.com>, Charlie
> > > -Boo <shymath...@gmail.com> wrote:
>
> > > [cut]
>
> > > > Nobody has even answered the question of whether additional axioms
> > > > will be needed or not.
>
> > > No additional axioms will be needed.
>
> > Oh No! And what of systems that have statements known to be
> > indepenent of ZFC?
>
> What of them? There is no requirement that all systems lead to the same
> set of theorems. I really don't understand what you are trying to get at.
What does not require additional axioms and how do you formally define
that?
C-B
William Hale
<5a66b45c-dcbc-48f8...@v4g2000hsf.googlegroups.com>,
Charlie-Boo <shyma...@gmail.com> wrote:
> On Feb 7, 7:38 pm, William Hale <h...@tulane.edu> wrote:
> > In article
> > <785910fb-ce98-4653-94d9-977ecc532...@j20g2000hsi.googlegroups.com>,
> >
> > Charlie-Boo <shymath...@gmail.com> wrote:
> > > On Feb 7, 6:01 pm, William Hale <h...@tulane.edu> wrote:
> > > > In article
> > > > <4460d6da-0f83-406b-8a8c-6ac421d15...@l16g2000hsh.googlegroups.com>, Cha
> > > > rlie
> > > > -Boo <shymath...@gmail.com> wrote:
> >
> > > > [cut]
> >
> > > > > Nobody has even answered the question of whether additional axioms
> > > > > will be needed or not.
> >
> > > > No additional axioms will be needed.
> >
> > > Oh No! And what of systems that have statements known to be
> > > indepenent of ZFC?
> >
> > What of them? There is no requirement that all systems lead to the same
> > set of theorems. I really don't understand what you are trying to get at.
>
> What does not require additional axioms and how do you formally define
> that?
Nothing requires additional axioms. I am free to choose what axioms I
want.
What does the last "that" refer to?
Charlie-Boo
Are you serious, or are you just playing games with definitions? Are
you aware of proofs within ZFC that concern things other than sets? I
mean, the natural numbers, anyway.
C-B
> -- Bill Hale- Hide quoted text -
Charlie-Boo
> On Thu, 7 Feb 2008 10:14:32 -0800 (PST), Charlie-Boo
> <shymath...@gmail.com> said:
>
> > ...
> > That really shows that ZFC is just a dinky little nothing that many
> > people are stupid enough to believe is great because somebody wrote it
> > in a book or magazine with a slick cover. Can you believe that? Of
> > all the stupid nonsense.
>
> Yes, yes, I myself remember being *dazzled* by the cover of Drake's Set
> Theory. All that...yellow. And those large, friendly letters.
Not to mention the cryptic symbols and terminology inside. And all
the references to prestigious universities! Where the AI department
has a walking . . . dog?
> So enticing. It wasn't until I got to the chapter on the constructible
> universe that I began to realize that set theory is itself a
> "constructed" universe, "a dinky little nothing" (as you so eloquently
> put it),
Where did I say that Set Theory is a dinky little nothing? The only
thing negative I can think about Set Theory is that it should have
been merged with Logic, as they are the same system except for a minor
difference in interpretation.
> a conceptual house of cards
How many popular systems of logic have been proven inconsistent?
No, the claim that ZFC can prove anything outside of arithmetic and
sets has no foundation in the first place. When you look at the real
variety in proof methods, it's silly to think they are all encompassed
in one system of axioms. But then, simply ask for examples. What is
the high level proof of the Pythagorean Theorem expressed in ZFC?
> foisted on naive and trusting
> students by unscrupulous professors interested only in retaining their
> power and rank in the Academy!
Oh hell yes. Are you aware of the politics involved? Are you aware
of Cantor's personal life?
> So keep up the good work, sir! A
> benighted world needs to heed your clarion call! Only ignorance and
> misery await those taken in by books with slick covers!
These books claim that the theory of computation, recursion theory,
program synthesis, and other branches of Computer Science have been
formalized and automated. But outside of general Logic (which is
formal by its nature), where are the Axioms and Rules of Inference for
all of these systems?
C-B
Charlie-Boo
> Charlie-Boo <shymath...@gmail.com> wrote:
> > On Feb 7, 7:38 pm, William Hale <h...@tulane.edu> wrote:
> > > In article
> > > <785910fb-ce98-4653-94d9-977ecc532...@j20g2000hsi.googlegroups.com>,
>
> > > Charlie-Boo <shymath...@gmail.com> wrote:
> > > > On Feb 7, 6:01 pm, William Hale <h...@tulane.edu> wrote:
> > > > > In article
> > > > > <4460d6da-0f83-406b-8a8c-6ac421d15...@l16g2000hsh.googlegroups.com>, Cha
> > > > > rlie
> > > > > -Boo <shymath...@gmail.com> wrote:
>
> > > > > [cut]
>
> > > > > > Nobody has even answered the question of whether additional axioms
> > > > > > will be needed or not.
>
> > > > > No additional axioms will be needed.
>
> > > > Oh No! And what of systems that have statements known to be
> > > > indepenent of ZFC?
>
> > > What of them? There is no requirement that all systems lead to the same
> > > set of theorems. I really don't understand what you are trying to get at.
>
> > What does not require additional axioms and how do you formally define
> > that?
>
> Nothing requires additional axioms. I am free to choose what axioms I
> want.
But can ZFC then prove so much of common mathematical theorems?
> What does the last "that" refer to?
Your answer to the first part of that question.
C-B
> > C-B
>
> > > > So, if we know the axioms already, does the chosen correspondence
> > > > between expressions in ZFC and expressions in the other system
> > > > matter?
>
> > > Matter to whom? There may not even be a correspondence between
> > > expressions in ZFC and expressions in the other system. I really don't
> > > understand what you are trying to get at.
>
> > > > Would they all produce the same set of theorems in the other
> > > > system?
>
> > > Probably not. Why would you think so?
>
> > > -- Bill Hale- Hide quoted text -
William Hale
<e0642dc4-7180-4cc5...@s8g2000prg.googlegroups.com>,
Charlie-Boo <shyma...@gmail.com> wrote:
I am serious.
> Are you aware of proofs within ZFC that concern things other than sets?
> I mean, the natural numbers, anyway.
A natural number, within ZFC, is a set.
I am losing the point that you are trying to make.
I thought that your point was that there are these mathematical entities
like natural numbers with Peano axioms, lines and points with Euclid
axioms, and real numbers with complete ordered field axioms, etc and
that these entities are brought into ZFC Set Theory by making each
entity into a set, eg {37} or {3.14}, so that we can say that we are
then working in the ZFC axiom system and thus reduced all standard
mathematics to ZFC.
But, my point that I am trying to make is that this is not what we are
claiming. You have it backwards. We are not starting with Peano natural
numbers, or Euclid lines and points, etc. and then going to ZFC. Rather,
we are starting with ZFC and its axioms and singling out certain sets
from those available within ZFC and then using those sets to say what a
natural number is, what a line is, what a real number is.
-- Bill Hale
Aatu Koskensilta
> Or is that just a quote from somewhere?
It's from Pynchon's _Mason & Dixie_.
--
Aatu Koskensilta (aatu.kos...@xortec.fi)
Aatu Koskensilta
> How many popular systems of logic have been proven inconsistent?
None.
> Oh hell yes. Are you aware of the politics involved? Are you aware
> of Cantor's personal life?
It's well known Cantor tortured kittens for a hobby. So much worse for
set theory!
David C. Ullrich
<shyma...@gmail.com> wrote:
>On Feb 6, 5:59 am, David C. Ullrich <dullr...@sprynet.com> wrote:
>> On Tue, 5 Feb 2008 11:46:09 -0800 (PST), Charlie-Boo
>
>> <shymath...@gmail.com> wrote:
>> >On Feb 5, 2:42 pm, Aatu Koskensilta <aatu.koskensi...@xortec.fi>
>> >wrote:
>> >> On 2008-02-05, in sci.logic, Charlie-Boo wrote:
>>
>> >> > And even worse, claim to be Mathematicians making a Mathematical
>> >> > statement while others watch and even believe him or her!
>>
>> >> Positively repugnant!
>>
>> >> > Why don't you give an example? What Mathematics other than
>> >> > Arithemetic and Set Theory has anyone ever proven in ZFC?
>>
>> >> You'll find the set theoretic development of analysis in any decent
>> >> textbook on the subject.
>>
>> >Where in "Classic Set Theory" by Derek Goldrei do I find a proof
>> >within ZFC of something outside of Arithmetic and Set Theory?
>>
>> He meant in any decent text on _analysis_.
>
>Oh, he covers the representation of other systems in ZFC. Section 3.2
>is the Construction of the Natural Numbers. Isn't that what we're
>really looking for?
>
>The problem is that, as always, he is only talking about arithmetic
>and sets. Isn't that so?
It's really hard to decide whether you're just blind or what.
No, when someone says that you can find a set-theoretic development
of analysis somewhere he's not talking about developing the natural
numbers, he's talking about developing _analysis_.
Ok, maybe it's just ignorance. The term "analysis" refers to things
like the real numbers, continuous functions, differentiable functions,
etc.
>> The last time this came up you simply refused to look at
>> a text that was suggested. It's certainly true that if you don't
>> look at examples then you won't see any.
>
>What text was that
Rudin, "Principles of Mathematical Analysis".
>and what is represented without additional axioms
>outside of arithmetic and sets?
Analysis.
No, there's no mention of ZF or ZFC in the book. But it's
clear that everything there can easily be formalized in ZFC:
You start with the naturals, then you "construct" the integers
and then the rationals. Then a real number is defined to be a
set of rationals with certain properties, etc.
>> >C-B
>>
>> >> --
>> >> Aatu Koskensilta (aatu.koskensi...@xortec.fi)
>>
>> >> "Wovon man nicht sprechen kann, daruber muss man schweigen"
>> >> - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
>>
>> David C. Ullrich- Hide quoted text -
>>
>> - Show quoted text -
David C. Ullrich
David C. Ullrich
And people _do_ that. The fact that you refuse to look at places
where this is done doesn't prove it doesn't happen.
"Giving a high-level description that explains the logic and
the construction" is exactly what people do. That's not the
same as giving formal proofs inside ZFC.
> Saying it
>is too complex to give any details at all to substantiate it won't
>work. "That dog won't hunt." as Bill Clinton once said.
>
>> A computer may do
>> it some day (perhaps already has) and when it happens it may be of
>> interest to computer folk but will it be of interest to mathematicians?
>
>How interested are you in Turing's work?
>
>C-B
>
>> > but with sufficient
>> > understanding of ZFC and of analytical geometry, one can see how such
>> > a formulation would be accomplished.
>>
>> > By the way, the subjects I asked about were topology, analysis,
>> > abstract algebra, and graph theory.
>>
>> > MoeBlee
>>
>> --
>> Going forward at this moment in time a raft of measures
>> have been put in place on the ground to target and
>> claw back the growth of cliche usage 24/7.
>> Remove "antispam" and ".invalid" for e-mail address.- Hide quoted text -
>>
>> - Show quoted text -
David C. Ullrich
David C. Ullrich
Why do you continue to ask the same question? Do you think
that the answer's going to change if you ask enough times?
>Yes, I agree that all of the systems above have recursively enumerable
>sets of theorems.
>
>> (Of course here I'm using the word "trivial" in a precise
>> sense, such that for example playing perfect chess is
>> trivial, since it's trivial to give an algorithm that plays
>> perfect chess, although calculuating an acceptable
>> first move would probably take longer than the age
>> of the universe.))
>>
>> David C. Ullrich- Hide quoted text -
>>
>> - Show quoted text -
David C. Ullrich
David C. Ullrich
<shyma...@gmail.com> wrote:
>On Feb 6, 6:09 am, David C. Ullrich <dullr...@sprynet.com> wrote:
>> On Tue, 5 Feb 2008 12:05:45 -0800 (PST), Charlie-Boo
>>
>> <shymath...@gmail.com> wrote:
>> >On Feb 3, 8:53 am, David C. Ullrich <dullr...@sprynet.com> wrote:
>> >> On Sat, 2 Feb 2008 18:47:08 -0800 (PST), Charlie-Boo
>> >> >On Feb 2, 9:40 am, David C. Ullrich <dullr...@sprynet.com> wrote:
>>
>> >> >> Prople do say that ZFC is sufficient to do all the mathematics
>> >> >> that people usually want to do. The reason they say that is
>> >> >> that it's _true_. It's not something that can be proved
>> >> >> mathematically
>>
>> >> Sometimes you seem blind or deaf or something.
>> >> Suppose that I say that it's a nice day today here in Stillwater.
>>
>> >I'm blind to irrelevant statements such as the above?
>>
>> When you repeatedly ask for a formal proof of something after
>> it's been pointed out that it's simply not the sort of thing that
>> one can give a formal proof for then yes, you seem blind or deaf.
>
>Representing a proof in ZFC is a formal process and is exactly what is
>most amenable to mathematical proof.
And that's not what we were talking about above! You omitted the
context.
My assertion was that most ordinary mathematics _can_ be done in
set theory, and the evidence I offered was that people _do_ it.
That's _not_ a statement in a formal system, it's an empirical
fact.
>A proof would begin with a demonstration of how the simplest theorems
>of a large number of branches of mathematics can be developed. Then
>it could give more complex ones, either in complete detail or describe
>the exact procedure. It could point out the various types of theorems
>there are in each system and say something about how each type would
>be represented.
Indeed. And since you refuse to look at places where people do exactly
this it follows that people don't do this.
Tell me: Exactly what is the first place in Rudin "Principles of
Mathematical Analysis" where it's not clear to you that the
argument _could_ be formalized in ZFC if we wished to do so?
David C. Ullrich
David C. Ullrich
<shyma...@gmail.com> wrote:
>On Feb 7, 6:01 pm, William Hale <h...@tulane.edu> wrote:
>> In article
>> <4460d6da-0f83-406b-8a8c-6ac421d15...@l16g2000hsh.googlegroups.com>, Charlie-Boo <shymath...@gmail.com> wrote:
>>
>> [cut]
>>
>> > Nobody has even answered the question of whether additional axioms
>> > will be needed or not.
>>
>> No additional axioms will be needed.
>
>Oh No! And what of systems that have statements known to be
>indepenent of ZFC?
The question was whether "most mathematics" can be done in ZFC.
Most mathematics that people actually do doesn't involve such
statements.
A largely irrelevant point:
One doesn't need to, for example, add CH to discuss things that
depend on CH (which of course most mathematics does not).
One simply discusses theorems of the form "If CH holds then ...".
>So, if we know the axioms already, does the chosen correspondence
>between expressions in ZFC and expressions in the other system
>matter? Would they all produce the same set of theorems in the other
>system?
What an utterly stupid question.
>C-B
David C. Ullrich
Charlie-Boo
You mention the 3 sets of axioms, but they are not involved. The
claim is only that certain theorems are proven, not that some
unspecified axioms are to be used.
The question of what axioms are used - ZFC's or any axioms you want -
is what nobody here can answer - apparently because they are all just
quoting from some title or abstract (or passage somewhere in an
article) and don't really know what it means.
The reason that I say this is that using ZFC vs whatever axioms you
want are two totally different processes. If you don't know which it
is, then you really know nothing about what the books are describing
at all. You can quote jargon (like Atta) but the Reality Check asks:
Are you using only ZFC's axioms? The actual definition is not even
being given - just generic buzzwords. It doesn't even express what is
happening, much less formally.
Possibility A. If only ZFC is used: Then the axioms in ZFC, which
refer to sets, are translated into statements about some other
mathematical objects (natural number, line, real number, etc.) and so
we have statements about them. And from those statements (axioms) we
are to deduce common mathematical theorems. So ZFC is the cure-all
that automatically creates axioms for us! It takes the place of many
years of effort, past and future.
(What does that remind you of? The Halting Problem! It's as if
someone said they solved it - and now we'll learn the truth concerning
so many mysteries about Mathematics! I wonder if what is claimed
about ZFC is even possible?)
Possibility B. If you can use your own axioms: Then we are not saying
anything mathematical about ZFC. We are saying that ZFC + WHATEVER WE
WANT proves all of these common Mathematical theorems. But WHATEVER
WE WANT includes the ZFC axioms, so that is equivalent to WHATEVER WE
WANT. We are not using ZFC's axioms at all (only casual
coincidences.)
So we are only using the syntax of ZFC expressions. How? By standard
Godel Numbering. For example, for arithmetIc we have:
PA: 0, 0', 0'', 0''', . . .
ZFC: {} , {{}} , {{},{{}}}, . . .
and we pair them up by sequence number (Godel Number).
But can't we do that with any formal system that has an r.e. set of
expressions?
So we are not using ZFC!!! We are not using the ZFC axioms. We are
simply using only Set Theory expressions for symbols.
Isn't it really just saying that Set Theory provides a model for PA
i.e. values for 0, x', N?
And so we get no actual examples of anything proven outside of
arithmetic and sets.
C-B
But in all fairness, look closely at the papers that talk about "doing
Math in ZFC". From what I see they don't formally define what they
are doing. They are just a hodge podge of assertions from ZFC or from
PA or from general logical terms or definitions. It's not saying what
is really proving what. It's just a bunch of statements. So maybe
that's why nobody here knows how it really works.
> But, my point that I am trying to make is that this is not what we are
> claiming. You have it backwards. We are not starting with Peano natural
> numbers, or Euclid lines and points, etc. and then going to ZFC. Rather,
> we are starting with ZFC and its axioms and singling out certain sets
> from those available within ZFC and then using those sets to say what a
> natural number is, what a line is, what a real number is.
>
Charlie-Boo
wrote:
> On 2008-02-07, in sci.logic, Charlie-Boo wrote:
>
> > Or is that just a quote from somewhere?
>
> It's from Pynchon's _Mason & Dixie_.
Do you use axioms outside of ZFC's?
C-B
>
> --
> Aatu Koskensilta (aatu.koskensi...@xortec.fi)
>
Charlie-Boo
wrote:
> On 2008-02-08, in sci.logic, Charlie-Boo wrote:
>
> > How many popular systems of logic have been proven inconsistent?
>
> None.
How about Naive Set Theory, Church's Lambda Calculus, Curry's
Combinatory Logic and Quine's ML?
C-B
> > Oh hell yes. Are you aware of the politics involved? Are you aware
> > of Cantor's personal life?
>
> It's well known Cantor tortured kittens for a hobby. So much worse for
> set theory!
>
> --
> Aatu Koskensilta (aatu.koskensi...@xortec.fi)
Aatu Koskensilta
> How about Naive Set Theory, Church's Lambda Calculus, Curry's
> Combinatory Logic and Quine's ML?
These are by no stretch of the imagination "popular systems of logic".
--
Aatu Koskensilta (aatu.kos...@xortec.fi)
Charlie-Boo
> On Thu, 7 Feb 2008 10:44:04 -0800 (PST), Charlie-Boo
>
> >On Feb 6, 5:59 am, David C. Ullrich <dullr...@sprynet.com> wrote:
> >> On Tue, 5 Feb 2008 11:46:09 -0800 (PST), Charlie-Boo
>
> >> <shymath...@gmail.com> wrote:
> >> >On Feb 5, 2:42 pm, Aatu Koskensilta <aatu.koskensi...@xortec.fi>
> >> >wrote:
> >> >> On 2008-02-05, in sci.logic, Charlie-Boo wrote:
>
> >> >> > And even worse, claim to be Mathematicians making a Mathematical
> >> >> > statement while others watch and even believe him or her!
>
> >> >> Positively repugnant!
>
> >> >> > Why don't you give an example? What Mathematics other than
> >> >> > Arithemetic and Set Theory has anyone ever proven in ZFC?
>
> >> >> You'll find the set theoretic development of analysis in any decent
> >> >> textbook on the subject.
>
> >> >Where in "Classic Set Theory" by Derek Goldrei do I find a proof
> >> >within ZFC of something outside of Arithmetic and Set Theory?
>
> >> He meant in any decent text on _analysis_.
>
> >Oh, he covers the representation of other systems in ZFC. Section 3.2
> >is the Construction of the Natural Numbers. Isn't that what we're
> >really looking for?
>
> >The problem is that, as always, he is only talking about arithmetic
> >and sets. Isn't that so?
>
> It's really hard to decide whether you're just blind or what.
Blind? What do you call someone who gives a reference and then admits
it doesn't contain what he claimed (as you do below)?
> No, when someone says that you can find a set-theoretic development
> of analysis somewhere he's not talking about developing the natural
> numbers, he's talking about developing _analysis_.
That's right. But the books only give the development of arithmetic
and sets. This is just one more example of that. Can you show
otherwise?
Do you need more than just ZFC's axioms? Why can't you say? You have
not defined what you are doing - are you using your own axioms or
ZFC's? Or are you just quoting a passage from somewhere and don't
really know what it means?
You're debating a point that has never been defined. Are you talking
about proofs using only ZFC's axioms or any axioms you can express?
> Ok, maybe it's just ignorance.
Are you answering my question about why you don't know if axioms other
than ZFC are used?
>The term "analysis" refers to things
> like the real numbers, continuous functions, differentiable functions,
> etc.
>
> >> The last time this came up you simply refused to look at
> >> a text that was suggested. It's certainly true that if you don't
> >> look at examples then you won't see any.
>
> >What text was that
>
> Rudin, "Principles of Mathematical Analysis".
No.
> >and what is represented without additional axioms
> >outside of arithmetic and sets?
>
> Analysis.
>
> No, there's no mention of ZF or ZFC in the book.
Can you come up with a reference that does use ZF or ZFC?
Charlie-Boo
> On Thu, 7 Feb 2008 10:48:22 -0800 (PST), Charlie-Boo
>
>
>
>
>
What does that have to do with the above exchange?
> David C. Ullrich- Hide quoted text -
>
> - Show quoted text -- Hide quoted text -
Charlie-Boo
Because it shows who doesn't know what the f*** they're talking
about. It also allows a discussion of a question if that question
were defined - if anybody does know the answer.
1. ZFC has nothing to do with performing general Mathematics.
2. You talk about proving almost all of Mathematics (outside of
arithmetic and sets), but have no examples of anything at all being
proven. Just one example would help answer the question above and
define the assertion.
How could there be a greater disconnect between reality and what you
are saying?
CLAIM: Almost all of Mathematics. Almost as big as you can imagine.
REALITY: 0 examples. As small as you can get.
Charlie-Boo
> On Thu, 7 Feb 2008 14:48:57 -0800 (PST), Charlie-Boo
>
>
>
>
>
> <shymath...@gmail.com> wrote:
> >On Feb 6, 6:09 am, David C. Ullrich <dullr...@sprynet.com> wrote:
> >> On Tue, 5 Feb 2008 12:05:45 -0800 (PST), Charlie-Boo
>
> >> <shymath...@gmail.com> wrote:
> >> >On Feb 3, 8:53 am, David C. Ullrich <dullr...@sprynet.com> wrote:
> >> >> On Sat, 2 Feb 2008 18:47:08 -0800 (PST), Charlie-Boo
> >> >> >On Feb 2, 9:40 am, David C. Ullrich <dullr...@sprynet.com> wrote:
>
> >> >> >> Prople do say that ZFC is sufficient to do all the mathematics
> >> >> >> that people usually want to do. The reason they say that is
> >> >> >> that it's _true_. It's not something that can be proved
> >> >> >> mathematically
>
> >> >> Sometimes you seem blind or deaf or something.
> >> >> Suppose that I say that it's a nice day today here in Stillwater.
>
> >> >I'm blind to irrelevant statements such as the above?
>
> >> When you repeatedly ask for a formal proof of something after
> >> it's been pointed out that it's simply not the sort of thing that
> >> one can give a formal proof for then yes, you seem blind or deaf.
>
> >Representing a proof in ZFC is a formal process and is exactly what is
> >most amenable to mathematical proof.
>
> And that's not what we were talking about above! You omitted the
> context.
What's not what we're talking about - representing a proof in ZFC?
> My assertion was that most ordinary mathematics _can_ be done in
> set theory, and the evidence I offered was that people _do_ it.
> That's _not_ a statement in a formal system, it's an empirical
> fact.
Then how about a reference? You finally admitted that your last one
says nothing about ZF(C).
> >A proof would begin with a demonstration of how the simplest theorems
> >of a large number of branches of mathematics can be developed. Then
> >it could give more complex ones, either in complete detail or describe
> >the exact procedure. It could point out the various types of theorems
> >there are in each system and say something about how each type would
> >be represented.
>
> Indeed. And since you refuse to look at places where people do exactly
> this it follows that people don't do this.
>
> Tell me: Exactly what is the first place in Rudin "Principles of
> Mathematical Analysis" where it's not clear to you that the
> argument _could_ be formalized in ZFC if we wished to do so?
It's not a question about C-B, it's about what has been published and
demonstrated. I haven't demonstrated it because it's either false (if
only ZFC's axioms are used) or true of any formal system (the ability
to use its expressions instead of sets and apply ZFC or whatever) and
inconsequential if we can supply our own.
Charlie-Boo
> On Thu, 7 Feb 2008 16:14:18 -0800 (PST), Charlie-Boo
>
The answer is yes, it depends on your coding scheme. So how do we
know which one to use? This is like the question of why ZF and not
some other axiomatization. This is a formal question - the difference
between ZF and the other axiomatizations that make it the one that
produces the truth.
The reality is these various axiomatizations are all more or less
random, trying to avoid Russell's Paradox and being patched along the
way. But like any software development project, after the users make
numerous critical changes, we ask, "Why do you think the system you
have right now is the correct one?"
> >C-B
>
> David C. Ullrich
MoeBlee
> How do you define formalizing in ZFC?
I don't think it is intended that the notion that ordinary mathematics
can be formalized in ZFC is itself a formal notion. To me, the
question of whether ZFC can formalize ordinary mathematics is that of
whether the theorems of ordinary informal mathematics have
representations as theorems of ZFC such that as we "read off" the
representation we arrive at a reasonable version of the informal
English.
For example, read off the formula
An(neN -> (On v En))
where (N reads as 'the set of natural numbers' and 'O' reads as 'is
odd' and 'E' reads as 'is even'.
The formula (a theorem of ZFC) reads off as the ordinary theorem of
informal mathematics:
For all n, if n is a member of the set of natural numbers, then n is
odd or n is even.
Personally, I don't venture to propose anything more rigorous than
that in terms of what I mean by the informal notion of ZFC "capturing"
or "representing" or "formalizing" ordinary mathematics. And if lack
of further rigor in this informal notion is brought to issue, then I
make no contest. My point would be only that one may check for oneself
whether a given theorem of ordinary informal mathematics is or is not
captured (by one's own standard of "capturing") by a counterpart that
is a ZFC formula.
> Are you allowed to add axioms?
Only definitional axioms, which extend the language and theory via
definitions, but do not add substantively to the theory.
If other axioms are added to prove a theorem of ordinary informal
mathematics, then we should note that that theorem was proven by ZFC
+S, where 'S' stands for the added axiom.
> If not, I'd like to see how the Pythagorean Theorem is proven.
Even with ZFC as it is found in working mathematics, there are two
kinds of presentations of a proof: One is stricly formal and is a
sequence of formulas in the first order language of set theory. The
other is an informal presentation, a rendering, that indicates how one
would, given enough time and patience, put togehter a stricly formal
proof. Ordinarily, we use the latter, the rendering approach, since
our purpose is to see that a strictly formal proof does exist but
without wasting our time and patience poring over exasperatingly long
and complicated pure symbolisms.
So such a rendering of a ZFC proof of the Pythagorean theorem may look
virtually the same as one of your favorite ordinary proofs of the
Pythagorean theorem. The crucial point though is that all the
terminology in the ordinary proof will have previously been defined in
set theoretic terms that can ultimately be traced back to the set
theoretic primitive. Also, all the lemmas in the ordinary proof will
have been previously proven in ZFC. And the ordinary geometric axioms
will not be taken as axioms in the sense of axioms for a theory, but
rather as clauses in a definition of a certain kind of system (as a
tuple) - a certain geometric system - much the same as the axioms for
a frist order theory of, say, groups, may be taken in ZFC as clauses
in a definition of a certain kind of algebra, or a system in the form
of a certain tuple.
> If you
> do have to add axioms, then I would still like to see it, and
> especially how the ZFC axioms played any actual role in that
> particular proof (as opposed to saying "You need ZFC to state that
> there is an infinite set." in response to every theorem, which is not
> the point.)
Take a look at 'Elementary Geometry From An Advanced Standpoint' (2nd
edition) by Moise. There you will find the specification of a geometry
using the notion of 'member of' and with a certain system (a triple to
begin with, to get started with an incidence geometry).
But you said "how ZFC axioms play a crucial role...". The claim about
ZFC formalizing mathematics is NOT that only ZFC can do so nor that in
any given proof all the ZFC axioms are used. Rather, the claim is that
the ZFC axioms are an adequate axiomatization. Not a claim that the
ZFC axioms are necessary for the task, but rather that the ZFC axioms
alone are sufficient for the task.
And it is NOT the case that ZFC is needed to state there is an
infinite set. (Who said that in what exact context?). We of course
recognize that there are theories other than ZFC that have the theorem
that there exist infinite sets.
> The ZFC axioms have nothing to do with anything but an attempt to
> avoid Russell's Paradox.
No, that is incorrect. The ZFC axioms prove theorems as well as
avoiding Russell's paradox.
> The Peano Axioms are another matter, as they
> guarantee that Addition, Multiplication and the Universal Set are
> representable,
I have no idea what you think PA does regarding "representing the
universal set".
> which has many implications. But that's all you have -
> some theorems of Set Theory from ZFC proper, and much of simple
> Arithmetic from the Peano Axioms that are added to that.
No, if you read nearly any introductory textbook in set theory, you'll
see how just Z set theory alone provides for a construction of not
just the naturals, but then the integers, rationals, reals, complex
numbers; and in ZFC we can do topology, analysis, and a whole bunch of
mathematics.
> The fact that there are no references to anything outside of sets and
> integers (not e.g. Geometry or Trigonometry) ever being carried out in
> ZFC shows that.
In ZFC every object is a set (as 'set' is suitably defined), but all
kinds of mathematical objects are then represented as a certain set or
kind of set.
> And it's not the ability to "represent" the objects of the system that
> is at issue. You can pair up any two r.e. sets of expressions, and
> call the first set or graph or list or wff or program zero. And the
> next one you call one.
You can do that. But it doesn't diminish that ZFC can represent
various mathematics.
> The problem is that you need to know specific relationships among
> those objects, and the ZFC axioms state only certain relationships.
From the axioms we infer theorems that state certain relations.
> You could probably prove the following interesting question: For any
> r.e. set, use Peano's Axioms to prove that it is r.e. But this
> doesn't give you things like the Pythagorean Theorem and the rest of
> common Mathematics.
I won't argue with you that whatever you have in mind about PA and
recursively enumerable sets doesn't provide for much of ordinary
mathematics. But that doesn't diminish that ZFC can provide for much
of ordinary mathematics, in the sense of 'provide' as in to capture or
represent, as mentioned earlier in this post.
MoeBlee
MoeBlee
> On Feb 5, 3:30 pm, MoeBlee <jazzm...@hotmail.com> wrote:
> > On Feb 5, 12:13 pm, Charlie-Boo <shymath...@gmail.com> wrote:
>
> > > On Feb 5, 2:53 pm, MoeBlee <jazzm...@hotmail.com> wrote:
>
> > > > On Feb 5, 11:06 am, Charlie-Boo <shymath...@gmail.com> wrote:
>
> > > > > If we can do most of Mathematics, then we can do a lot of simple
> > > > > Mathematics. Then could you give a simple self-contained example of a
> > > > > proof within ZFC that is outside of Arithmetic and Set Theory?
>
> > > > Such subjects as topology, analysis, abstract algebra, and graph
> > > > theory are not usually thought of as part of a set theory course
> > > > itself, but statements and theorems in such subjects can be given in
> > > > the language of and proven in ZFC.
>
> > > > Rather than me just list one theorem after another, a better question
> > > > would be for you to say what theorem of such subjects you think canNOT
> > > > be proven in ZFC.
>
> > > How about the Pythagorean Theorem? Last I heard there were 20 or 30
> > > different ways to prove it using "ordinary Mathematics".
>
> > There may be many ways of proving it in ordinary mathematics.
> > Meanwhile, such basic analytical geometry as a proof of the Pyhagorean
> > theorem can be formulated in ZFC. It would be a rather tedious
> > exercise to carry out such a formulation, but with sufficient
> > understanding of ZFC and of analytical geometry, one can see how such
> > a formulation would be accomplished.
>
> What would be the informal proof that was being formalized? You start
> out with a right triangle.
(I'm goint to strike now my mention of 'analytical geometry' there.
It's not needed for me to complicate the main matter here with the
question of whether something fits in analytical geomertry
specifically).
As I mentioned in my previous post, you can see Moise's approach in
his book. And he gives three different proofs of the Pythagorean
theorem. I haven't worked out the details that a far in the book, but
I don't see what cannot be done in set theory. Especially, as I
mentioned, we specify certain systems (certain kinds of tuples) that
are geometric systems and prove various things about them.
Meanwhile, to get a better sense of where we stand in this
conversation, a while ago you claimed not to see that we can prove, in
ZFC, that if S is recursively enumerable and the complement of S is
recursively enumerable then S is recursive. I pointed you right to the
proof in Boolos, et. al (a textbook you do consult). I would still
like to know whether you still can't see that that proof is
formalizable in ZFC.
> Then how does the logic go?
Only first order reasoning is permitted. Once the terminology of an
ordinary proof of the Pythagorean theorem has been defined in ZFC and
each of the lemmas used in the proof are proven in ZFC, then the logic
used for proving the theorem itself should still be only first order
logic. Sure, you may find certain proofs of the Pythagorean theorem
that seem to be so greatly involved in illustrations that abstract
logic alone might seem inadequate. But that doesn't diminish that
there are OTHER proofs (including the Euclid proof itself) that can be
formalized; moreover, it is not precluded that with more work we could
even formalize in ZFC those proofs that are so greatly involved in
illustrations.
> At some point you'll need axioms. Do you have to add your own axioms
> to ZFC's?
No. Perphas the best approach is the one I mentioned in the previous
post: Take the geometric axioms as clauses in a definition of a
certain kind of system, as is done in Moise, and is done routinely in
abstract algebra.
> Are you claiming that it can be done using ZFC, or that we
> could present new axioms in ZFC that would do it?
My claim is that I don't see anything to prevent us from doing it in
ZFC without added axioms. I admit I've not worked through the details
of specifically the Pythagorean theorem in ZFC so that I can't (YET)
personally confirm to an absolute certitude that it can be done; but I
have worked through a certain amount of geometry as within ZFC, and
I've read over enough of the text in, for example Moise, including his
proofs of the Pythagorean to have confidence that when I get that far
in the ZFC formalization, a proof of the Pythagorean theorem will
conform too.
> > By the way, the subjects I asked about were topology, analysis,
> > abstract algebra, and graph theory.
>
> and quite well
What does that mean? My point is that you've not shown anything in
those subjects that can't be done in ZFC.
MoeBlee