Cartesian product dilemma
Daniel J. Greenhoe
appreciate it if someone could help me out.
1. I define the Cartesian product of AxB as
AxB = { (a,b) | a in A <and> b in B }
where (a,b) is an ordered pair and
<and> is the "logical AND" operator
2. But this definition of the Cartesian product depends on the binary
operator <and>. A binary operator (or a function), is a special case
of a relation. So before I can define the Cartesian product, I need to
first define a relation.
3. But the definition of a relation R is that it is a subset of a
Cartesian product.
4. So, in the end, my definition of Cartesian product depends on the
definition of the Cartesian product --- hence my dilemma.
Stated another way,
AxB = { (a,b) | a in A <and> b in B }
but <and> is of the form
<and>:{T,F}x{TxF} --> {T,F}
... and so the definition of the Cartesian product depends on an
operator defined with the help of a Cartesian product.
Is there any standard way out of this dilemma? Any classic references
would be especially appreciated.
Many thanks in advance,
Dan Greenhoe
Stephen J. Herschkorn
Daniel J. Greenhoe wrote:
Halmos, Kunen, and Suppes all give careful constructions of the
Cartesian product. Halmos and Suppes invoke the Axiom of Power and the
Scheme of Comprehension, Kunen the Scheme of Replacement
--
Stephen J. Herschkorn sjher...@netscape.net
Math Tutor on the Internet and in Central New Jersey and Manhattan
William Elliot
> I have run into a dilemma defining the Cartesian product. I would
> appreciate it if someone could help me out.
>
> 1. I define the Cartesian product of AxB as
> AxB = { (a,b) | a in A <and> b in B }
> where (a,b) is an ordered pair and
> <and> is the "logical AND" operator
>
> 2. But this definition of the Cartesian product depends on the binary
> operator <and>. A binary operator (or a function), is a special case
> of a relation. So before I can define the Cartesian product, I need to
> first define a relation.
>
> 3. But the definition of a relation R is that it is a subset of a
> Cartesian product.
>
> 4. So, in the end, my definition of Cartesian product depends on the
> definition of the Cartesian product --- hence my dilemma.
>
Though you see <and> as a set theory relation, in the actual definition
AxB = { (a,b) | a in A and b in B }
'and' is a logical connective which is prior to and underlying any set
theory developed from a first order logic (FOL) by the addition of a
binary relation 'in' (written as small epsilon) and axioms involving 'in'.
Daniel J. Greenhoe
> Halmos, Kunen, and Suppes all give careful constructions of the
> Cartesian product. Halmos and Suppes invoke the Axiom of Power and the
> Scheme of Comprehension, Kunen the Scheme of Replacement
Thank you very much for your response. Could you give titles for these
books? I have used Google book search to search inside the English
version of Halmos' "Naive Set Theory", and there were no search
results for either "Axiom of Power" or the "Scheme of Comprehension".
However, I realize that you may be referencing the same work in a
different language with slightly different terminology. Or maybe you
were referencing another one of Halmos' many books.
On Aug 13, 2:00 pm, "Stephen J. Herschkorn" <sjhersc...@netscape.net>
wrote:
> Stephen J. Herschkorn sjhersc...@netscape.net
Daniel J. Greenhoe
> Though you see <and> as a set theory relation, ...
> 'and' is a logical connective which is prior to and underlying any set
> theory ...
Thank you very much for your response. But all logical operators (or
"connectives" as you call them), "functions", "functionals, vector
space operators, ordering relations, equivalence relations, etc... are
_all_ relations --- that is they are all some subset of the Cartesian
product XxY where X is the domain of the operation and Y is the range.
_None_ are prior to the concept of the set and _none_ underly the
concept of the set.
On the other hand, logical operators _do_ underly and _are_ prior to
set _operators_ such as union, intersection, symmetric difference, etc
--- but again, logical operators do not underly the concept of the set
itself.
One might note that it is important to not use the rules of grammar
from the English language (in this case, the use of conjunctions or
"connectives") as an axiomatic foundation for mathematics.
Dan Greenhoe
Brian VanPelt
Daniel:
Why not try to make up something that works?
If the theory is needing, mabyse it's time to make something
consistent.
Good luck.
Brian
Daniel J. Greenhoe
> Why not try to make up something that works?
> If the theory is needing, mabyse it's time to make something
> consistent.
> Good luck.
> Brian
Thanks for the encouragement. I think a lot of people would very much
like to find a "Russell's Paradox" type oversight and find a way to
fix it. But in this case, it would be hard to believe that no one has
yet addressed the problem and found a good solution. And one might
expect that that solution is already out there in a published set
theory book somewhere.
Dan
"Your discovery of the contradiction caused me the greatest surprise
and, I would
almost say, consternation, since it has shaken the basis on which I
intend to build
arithmetic.. . . In any case your discovery is very remarkable and
will perhaps result in a great advance in logic, unwelcomed as it may
seem at first glance."
---Friedrich Ludwig Gottlob Frege (1848-1925), German mathematical
logic pioneer in his 1902 June 22 letter of reply to Bertrand
Russell's 1902 June 16 letter which introduced "Russell's paradox"
showing that naive set theory is flawed
David C. Ullrich
<dgre...@yahoo.com> wrote:
>I have run into a dilemma defining the Cartesian product. I would
>appreciate it if someone could help me out.
>
>1. I define the Cartesian product of AxB as
> AxB = { (a,b) | a in A <and> b in B }
> where (a,b) is an ordered pair and
> <and> is the "logical AND" operator
>
>2. But this definition of the Cartesian product depends on the binary
>operator <and>. A binary operator (or a function), is a special case
>of a relation. So before I can define the Cartesian product, I need to
>first define a relation.
>
>3. But the definition of a relation R is that it is a subset of a
>Cartesian product.
>
>4. So, in the end, my definition of Cartesian product depends on the
>definition of the Cartesian product --- hence my dilemma.
This is silly.
We don't need to regard "and" as a Cartesian product.
"And" is simply an English word with a well-known
meaning.
I'm not going to attempt to give a definition of
"and" - I'd end up with something like
"A and B" is true if and only if A is true and B is true
which is clearly circular, or with something that just
hides the circularity with a synonym.
You can't possible define _everything_ in terms of
previously defined terms. If the reader does not
understand ordinary English then no amount of
explanation, in English, is going to make the
reader understand.
In fact your readers _do_ know what "and" means,
so there's no problem.
>Stated another way,
> AxB = { (a,b) | a in A <and> b in B }
>but <and> is of the form
> <and>:{T,F}x{TxF} --> {T,F}
>... and so the definition of the Cartesian product depends on an
>operator defined with the help of a Cartesian product.
>
>
>Is there any standard way out of this dilemma? Any classic references
>would be especially appreciated.
>
>Many thanks in advance,
>Dan Greenhoe
************************
David C. Ullrich
Stephen J. Herschkorn
>On Aug 13, 2:00 pm, "Stephen J. Herschkorn" wrote:
>
>
>>Halmos, Kunen, and Suppes all give careful constructions of the
>>Cartesian product. Halmos and Suppes invoke the Axiom of Power and the
>>Scheme of Comprehension, Kunen the Scheme of Replacement
>>
>>
> <>
> Thank you very much for your response. Could you give titles for these
> books? I have used Google book search to search inside the English
> version of Halmos' "Naive Set Theory", and there were no search
> results for either "Axiom of Power" or the "Scheme of Comprehension".
> However, I realize that you may be referencing the same work in a
> different language with slightly different terminology. Or maybe you
> were referencing another one of Halmos' many books.
Halmos, Paul R. Naive Set Theory. (Remarkably, without a diaeresis.)
Section 6. He calls them the Axiom of Powers and the Axiom of
Specification. If you are studying set theory anew, this slim text is a
classic. I highly recommend it. (I have encountered disparagement of
this book from others on USENET. Of course, since they disagree with
me, such posters must be jerks. :-))
Kunen, Kenneth. Set Theory: An Introduction to Independence Proofs.
Page 13.
Suppes, Patrick. Axiomatic Set Theory. Page 49. He calls them the
Power Set Axiom and the Axiom Schema of Separation.
--
Stephen J. Herschkorn sjher...@netscape.net
Stephen J. Herschkorn
To put it another way, even the careful set-theoretic constructions of
the Cartesian product assume the word "and" as part of the logical
vocabulary. So you end up constructing sets like
{x in y: f(x) and g(x)}, where f(x) and g(x) are logical
formulae wherein the variable x is free.
The weakness of your presentation (as opposed to rigorous constructions)
is that you need to prove somehow (via comprehension or replacement)
that such a set of (a,b) exists.
george
> I have run into a dilemma defining the Cartesian product. I would
> appreciate it if someone could help me out.
>
> 1. I define the Cartesian product of AxB as
> AxB = { (a,b) | a in A <and> b in B }
> where (a,b) is an ordered pair and
> <and> is the "logical AND" operator
>
> 2. But this definition of the Cartesian product depends on the binary
> operator <and>.
Not exactly. "And" IS NOT a "binary operator" in the set-theoretical
sense, in this context. And is part of a whole 'nother LANGUAGE.
And is a logical operator in the UNDERlying LOGICAL language of the
paradigm.
You are making a fairly important mistake with your use of the verb
"depends". There is simply no such thing as depending around here.
There are, instead, simply things that are defined (or postulated)
earlier and
things that are defined later in terms of them. Put another way, yes,
"and" must have gotten a definition FROM SOMEwhere before you can use
it in set theory, but it does NOT get THAT definition FROM set theory.
It gets it from propositional logic. Where propositional logic got
it from
is not reall your concern. THAT and is just defined as the binary
operator
on ordered pairs OF BOOLEAN values -- NOT OF SETS -- that returns
true iff both arguments are true. It ALREADY HAS that definition and
YOU
do NOT have to define it.
george
> appreciate it if someone could help me out.
>
> 1. I define the Cartesian product of AxB as
> AxB = { (a,b) | a in A <and> b in B }
> where (a,b) is an ordered pair and
> <and> is the "logical AND" operator
>
> 2. But this definition of the Cartesian product depends on the binary
> operator <and>. A binary operator (or a function), is a special case
> of a relation. So before I can define the Cartesian product, I need to
> first define a relation.
The previous lesson was about your not needing to define "and".
This one is about your not needing to define "relation".
A relation is defined as a set of ordered pairs, usually, in ZFC
terms.
But that has nothing to do with subsetting a cartesian product.
Definable subsets ALWAYS exist according to the Axiom[schema]of
Separation. So there is simply no work for you to do there.
You can take any subset of any cartesian product (OR any OTHER set)
that you can describe, WITHOUT having to first define what a relation
is, or what ANYthing
is, OR EVEN WHAT A SUBSET IS. JUST USE the axiom.
Daniel J. Greenhoe
Thank you for your reply. I do genuinely appreciate it when people
take time to respond to my posts.
> This is silly.
> We don't need to regard "and" as a Cartesian product.
In mathematics, it is not what we regard something to be that is
significant, but rather what an object is as determined by the utility
it performs. And in this case, the utility that "and" performs is the
_operation_ of the <logical and>. And the <logical and> is indeed a
relation. And a relation is indeed a subset of a Cartesian product.
And as such, how can we not take care to rigorously define it? Because
"and" performs a very real mathematical binary operation, of course we
must define it rigorously.
> "And" is simply an English word with a well-known
> meaning.
"And" indeed is an English word and its meaning is well-known. But
this is not surprising, because propositional logic is fundamental to
the human thought process; and so one would expect that every one of
the 6000 or so human languages would contain vocabulary for the
<logical and> and <logical or>. In conversational languages, the
meaning of logical operators is understood by intuition partly because
we use them since early childhood. But in mathematics, understanding
by intuition is not sufficient. At some point every concept must be
well defined based on whatever axiomatic system we are in.
> I'm not going to attempt to give a definition of
> "and" - I'd end up with something like
> "A and B" is true if and only if A is true and B is true
> which is clearly circular, or with something that just
> hides the circularity with a synonym.
Actually, your definition here is essentially the same as the standard
definition of the <logical and> used in classical logic:
A B | A and B
--------------
F F | F
F T | F
T F | F
T T | T
That is, "A and B" is T(rue) if and only if A is true and B is true...
which is clearly not circular (meaning that the definition of <logical
and> in no way depends on itself).
> You can't possible define _everything_ in terms of
> previously defined terms. If the reader does not
> understand ordinary English then no amount of
> explanation, in English, is going to make the
> reader understand.
> In fact your readers _do_ know what "and" means,
> so there's no problem.
It would seem that such an argument is based on pedagogical
considerations rather than considerations regarding mathematical
construction. That is, such an argument is concerned with what is
necessary to make someone understand the concept of the Cartesian
product. And if one was preparing class notes for a high school class,
this may be a reasonable approach --- most all high school students
know what "and" means and so that is that. But in mathematics,
especially in analysis, mathematicians are concerned with
"constructions" --- how can we construct this or that starting with
just a few axioms such as the notion of a set?
And it is in the spirit of construction, not pedagogy, that I try to
define the Cartesian product --- that is, I try to construct it
starting from the simple notion of the set and not from the notion of
concepts pre-built into this or that European language that drifts
about every few hundred years.
And I am certainly not alone in this thinking. Would one also tell all
the mathematicians who worked on constructing the real number system
from integers, ring operations, and metric spaces/topology concepts
that their work is "silly" because after all, everyone knows what a
real number is? Or would Frege tell Russell that his "Russell's
Paradox" argument was "silly", because, after all, every one know what
we mean when we talk about a set?
Kind Regards,
Dan Greenhoe
Daniel J. Greenhoe
wrote:
> Halmos, Paul R. Naive Set Theory. Section 6.
> Kunen, Kenneth.
> Set Theory: An Introduction to Independence Proofs. Page 13.
> Suppes, Patrick. Axiomatic Set Theory. Page 49.
Thank you very much for information about these excellent references.
I see on the web that a not too distant library has all three books so
I hope I can be able to take a look this week.
I think I might be able to find the answers I have been looking for
there --- thank you very much!
dan
On Aug 13, 9:11 pm, "Stephen J. Herschkorn" <sjhersc...@netscape.net>
wrote:
> Stephen J. Herschkorn sjhersc...@netscape.net
Denis Feldmann
> On Aug 13, 8:24 pm, David C. Ullrich <ullr...@math.okstate.edu>
>
> Thank you for your reply. I do genuinely appreciate it when people
> take time to respond to my posts.
Sure, as all good trols do
>
>> This is silly.
>> We don't need to regard "and" as a Cartesian product.
>
> In mathematics, it is not what we regard something to be that is
> significant,
<cut of stupid aznd pompour tries to make David Ulrich understand what
ids mathematics and mathematical rigor>
You are a troll, making a lot of better people than you are waste their
time. Now go away.
george
> > previously defined terms.
Clearly. Importantly. Endorsably. Commendably. Necessarily.
> > If the reader does not
> > understand ordinary English then no amount of
> > explanation, in English, is going to make the
> > reader understand.
> > In fact your readers _do_ know what "and" means,
> > so there's no problem.
On Aug 13, 10:11 am, "Daniel J. Greenhoe" <dgreen...@yahoo.com> wrote:
> It would seem that such an argument is based on pedagogical
> considerations rather than considerations regarding mathematical
> construction.
NO, it would not and does NOT seem that way.
You have paid too much attention to the second part of the argument
and not enough to the first. You are thwarted by the "no amount of
explanation"
part. That is NOT the problem. FORGET that part. The IMPORTANT part
is,
precisely as DCU said, that you canNOT POSSIBLY define EVERYthing in
terms
of previously defined terms. That ABSOLUTELY IS based on
considerations
of mathematical construction (standardly, we require constructions to
be well-
FOUNDED) AND NOT "mere pedagogical considerations".
> That is, such an argument is concerned with what is
> necessary to make someone understand the concept of the Cartesian
> product.
You sound too arrogant here. You yourself personally have not clearly
understood what it means to "define" an operator,whether it be the
Cartesian cross-product or anything else. You need to LEARN this,
yourself, before you start dismissing anything as too much about
"teaching" it.
> And if one was preparing class notes for a high school class,
> this may be a reasonable approach --- most all high school students
> know what "and" means and so that is that. But in mathematics,
> especially in analysis, mathematicians are concerned with
> "constructions" --- how can we construct this or that starting with
> just a few axioms such as the notion of a set?
In the case of the cartestian product, that is trivially easy and well-
answered.
But "the notion of a set" DOES NOT involve "just a few" axioms.
In the usual formulation it involves between 8 and 12, at least one of
which
is a schema with infinitely many instances. There is a finite
axiomatization but
it is in some ways MORE complex than the one with the schema.
> And it is in the spirit of construction, not pedagogy, that I try to
> define the Cartesian product
You have no idea how stupid you sound.
Let me try to give you an idea: you sound really stupid.
It IS NOT UP to YOU to "define the Cartesian product"!
The Cartesian product of two sets ALREADY HAS a clear
definition! You have practically already stated it! And you think
that because that definition contains and "and", you have to define
"and"?? Well, guess what: long before YOU came along "And" ALREADY
HAD a clear definition: ITS TRUTH TABLE is its definition.
That really is just all there is to it.
There is NOTHING LEFT for YOU to define or add!
george
> I have run into a dilemma defining the Cartesian product. I would
> appreciate it if someone could help me out.
>
> 1. I define the Cartesian product of AxB as
> AxB = { (a,b) | a in A <and> b in B }
> where (a,b) is an ordered pair and
> <and> is the "logical AND" operator
>
> 2. But this definition of the Cartesian product depends on the binary
> operator <and>. A binary operator (or a function), is a special case
> of a relation. So before I can define the Cartesian product, I need to
> first define a relation.
This is far more idiotic than you realize.
Rather than the Cartesian Product Operator, you should focus on a
DIFFERENT
binary operator between sets: THE INTERSECTION operator. This has
the
following definition:
A ^ B = { x | xeA <and> xeB }.
Now you are going to tell us, again, that <and>, being a binary
operator, is a special
case of a relation, so you must first define relation? That is just
ridiculous.
Conjunction ("and") is NOT defined HERE in terms of set theory! IT
was defined BEFORE,
in terms of logic.
The <and> operators operates on ordered pairs of booleans, and no set
is actually
"equal" to "true" or "false". That is just a completely separate and,
most importantly,
PRIOR *realm*.
george
> Thank you very much for your response. But all logical operators (or
> "connectives" as you call them), "functions", "functionals, vector
> space operators, ordering relations, equivalence relations, etc... are
> _all_ relations
You are flaunting a WHOLE lot of ignorance here.
You are in a realm (set theory) where "relation" is a DEFINED
term. It DOES NOT MATTER what YOU think you know "relation"
means. THE DEFINITION of "relation" in this context is "a set of
ordered pairs". Although the usual set theory doesn't recognize
proper classes, one can legitimately extend this to "a class of
ordered pairs". But you are simply LYING about the status
of logical operators and connectives. They belong to a PRIOR LEVEL
OF LANGUAGE, precisely as Elliott said, in this paradigm.
No, you MAY not disagree. This is NOT a matter of opinion.
This is a fact about how people (literally millions of people,
most of them smarter than you) use mathematical language.
If you want to participate coherently in a discussion with the
mathematical community then you simply ARE going to have
to speak THEIR dialect, on some matters. THIS IS one of them.
Angus Rodgers
<gre...@cs.unc.edu> wrote:
>The Cartesian product of two sets ALREADY HAS a clear
>definition! You have practically already stated it!
The relationship between informal concepts of binary
relations, functions, and binary operations (on the
one hand) and set-theoretic reconstructions of these
concepts (on the other) is far from clear. Even if
there does exist a fully clear explication of this
relationship, it is not one easily accessible in
standard textbooks. There is plenty of room for a
beginning student (or even a more experienced student)
to be intelligently puzzled by such philosophical
matters.
The fact that the OP has asked a reasonable question,
in a polite manner, has not resorted to abuse even
when called "silly" (note: I'm not saying that David's
comment was over the top), and has then been shouted
at and told that he is (1) a troll, (2) arrogant, (3)
stupid, (4) really stupid, (5) ignorant, (6) a liar,
and that what he thinks doesn't matter, and that he
has no right to disagree, is clear evidence that his
question is one which provokes anxiety, and is being
met with a crudely irrational social process instead
of reasoned argument.
In all areas of thought there are such foundational
questions. Trolls and cranks are two categories of
people who exploit such uncertainties; but they are
not the only people who ask questions about them.
I wanted very much to stay out of this thread, until
provoked to enter it by one really abusive response.
My own advice to the OP (as a fellow-student, not an
expert) would be not to worry /too/ much, even about
what seem to be real difficulties in the foundations,
but (in Cauchy's words) to "carry on, and faith will
come". For, it seems to be a fact that foundational
questions, although encountered by the beginner, have
to wait until much later in one's education before
they have a hope of being answered. For what it's
worth, I think that your question makes sense; but
I don't know how to answer it.
--
Angus Rodgers
(twirlip@ eats spam; reply to angusrod@)
Contains mild peril
george
> On Mon, 13 Aug 2007 09:02:24 -0700, george
>
> >The Cartesian product of two sets ALREADY HAS a clear
> >definition! You have practically already stated it!
>
> The relationship between informal concepts of binary
> relations, functions, and binary operations (on the
> one hand) and set-theoretic reconstructions of these
> concepts (on the other) is far from clear.
It is equally far from relevant.
Precisely because the informal concepts ARE INformal,
they are also going to be INclear.
More to the point,
if the informal concepts actually are clear to you to begin with,
then the relationships are well-defined.
The truth-table for and, for example, looks like
_ & _
T T T
T F F
F F F
F F F
The first important thing to note as that as a FORMAL construct,
this stands ALONE and simply does not NEED any relationship
to any INformal ANYthing. But if you WANT to relate it to something
informal, then the statement "The operator spelled '&' in this table
is
properly thought of as translating the natural-language or INformal
concept 'and' BECAUSE it returns true if and only if its 1st argument
AND its 2nd are both true; this is analogous to the truth-behvavior of
natural-language sentences combined with 'and'."
This is basically as clear as any translation of something
WITH parts -- as opposed to a translation of 1 noun as 1 thing --
Could Possibly Ever get. The formal thing is a binary operator
but the point is that the allegedly INformal natural-language thing
IS A BINARY OPERATOR, TOO. The translation is unusually exact &
accurate, almost so much so as to reduce the whole enterprise to
little more than a re-SPELLing of the operator(s) involved.
george
> relations, functions, and binary operations (on the
> one hand) and set-theoretic reconstructions of these
> concepts (on the other) is far from clear.
This is borderline ridiculous.
It is in fact borderline BACKwards.
The set-theoretical reconstructions, BECAUSE THEY ARE ADEQUATE
*become* DEFINITIVE. The only possible grounds for DENYING them
that status would be some DIScrepancy between the way they behave and
the way some "informal"(HAH!) analogue behaved. The guffaw of
dismissive
contempt here is because "binary relations, functions, and binary
operations"
are about as FORMAL as formal CAN get, EVEN when they're not being
phrased in terms of a 1st-order formal theory like ZFC. If the
objects are
inherently abstract TO BEGIN with, well, there just isn't anything you
can
do about it. And if they necessarily have PARTS that have to be in
CERTAIN
RELATIONSHIPS to each other, then they have FORM and therefore ARE
formal. INherently! THERE IS NO SUCH THING as an informal binary
operator!
Brian VanPelt
Don't you wonder, just a little bit, why people are afraid of
mathematics? Thinkers certainly don't seem to be encouraged, do they?
Brian
george
> in a polite manner, has not resorted to abuse even
> when called "silly" (note: I'm not saying that David's
> comment was over the top), and has then been shouted
> at and told that he is (1) a troll, (2) arrogant, (3)
> stupid, (4) really stupid, (5) ignorant, (6) a liar,
> and that what he thinks doesn't matter, and that he
> has no right to disagree, is clear evidence that his
> question is one which provokes anxiety, and is being
> met with a crudely irrational social process instead
> of reasoned argument.
What UTTER bs.
Just for the record, there certainly is no reasoned argument,
mathemtical or otherwise, in the above. Alleging that because
some language is abusive, the people speaking it are therefore
not arguing rationally
IS A LOGICAL FALLACY,
dumbass. Most of the articles posted here are MORE THAN ONE line long
and it is ENTIRELY possible for you -- ESPECIALLY you -- to get abused
on one line AND REFUTED ON THE NEXT, though I expect the argument
might be more effective if the lines were in the other order.
> In all areas of thought there are such foundational
> questions. Trolls and cranks are two categories of
> people who exploit such uncertainties; but they are
> not the only people who ask questions about them.
The question as phrased simply WAS NOT foundational.
If you want to ask a foundational question, then you ask,
"How do we define a propositional binary connective like "and"?"
THAT is the foundational question. That was NOT the question asked.
Kudos to DCU for cutting to the chase, on that one.
The questioner was fool enough to ask about the cartesian cross-
product, when, if "and" was what he actually cared about, the CORRECT
binary operator on sets would've been THE MUCH SIMPLER
intersection, as *I* (kudos to me) pointed out.
The fact that you didn't notice any of this going on just proves
YOU don't know what's going on. The fact that, IN SPITE of this
relevant analysis going on, you condemned those of us condemning the
OP
as not arguing rationally, is just one more piece of evidence that you
wouldn't
know a rational argument if you got whacked upside the head with one.
Which definitely needs to happen a few more times than it has so far.
> My own advice to the OP (as a fellow-student, not an
> expert) would be not to worry /too/ much, even about
> what seem to be real difficulties in the foundations,
> but (in Cauchy's words) to "carry on, and faith will
> come".
Yeah, that's really helpful.
What would be more helpful is if you would stop worrying about what
is abusive and start worrying about what is logical.
I almost wish I had let DCU continue to handle this by himself.
Some people deserve tha.
> For, it seems to be a fact that foundational
> questions, although encountered by the beginner, have
> to wait until much later in one's education before
> they have a hope of being answered.
Most beginners don't think of them. This one certainly hasn't.
He actually thought that "relations" came first.
Jeezus.
> For what it's
> worth, I think that your question makes sense;
Well, you're wrong.
The question that WOULD'VE made sense would've, as I said,
been about Intersection, NOT cross-product. And that question
would've
led directly, precisely as DCU suggested, to the question of how to
define
"and" (or "or", or "not", or "implies") without being circular.
But that is a question that the OP not only still won't ask, but still
tries to EVADE by talking about "relations" and things that are
"informal".
Here's the other basic foundational question (after truth-tables for
propositional
operators). We present all these arguments using finite strings. Of
characters.
But What Is a string? How do you define "string"? How do you define
"character"
or symbol or "letter"? Far more importantly, how do you define
"finite"?
If you want circularity, THAT is where you get it. You DON'T have to
go all the
way to "functions" and "relations" and absurd claims that THOSE are
"informal" or
"foundational".
george
> Though you see <and> as a set theory relation, in the actual definition
> AxB = { (a,b) | a in A and b in B }
> 'and' is a logical connective which is prior to and underlying any set
> theory developed from a first order logic (FOL) by the addition of a
This is the single best smart answer to a dumb question that I have
seen in this forum all year. If I were rich enough to be giving out
awards,
I would give one for this answer.
MoeBlee
> On Aug 13, 9:11 pm, "Stephen J. Herschkorn" <sjhersc...@netscape.net>
> wrote:
>
> > Halmos, Paul R. Naive Set Theory. Section 6.
> > Kunen, Kenneth.
> > Set Theory: An Introduction to Independence Proofs. Page 13.
> > Suppes, Patrick. Axiomatic Set Theory. Page 49.
>
> Thank you very much for information about these excellent references.
> I see on the web that a not too distant library has all three books so
> I hope I can be able to take a look this week.
>
> I think I might be able to find the answers I have been looking for
> there --- thank you very much!
Suppes would be good for a start. But (since you've shown confusion on
certain more basic logical matters) I suggest you learn the predicate
calculus even before taking on Suppes (which has only a cursory
section on the logic), from a good textbook on it, such as 'Logic:
Techniques Of Formal Reasoning' by Kalish, Montague, and Mar.
You are confused as to the differences among such matters as
sentential connectives (such as 'and') and relations. A good textbook
on symbolic logic (the predicate calculus, in particular), such as
'Logic: Techniques Of Formal Reasoning', will put you on the way to
clearing up your confusions.
MoeBlee
Angus Rodgers
<gre...@cs.unc.edu> wrote:
>Here's the other basic foundational question (after truth-
>tables for propositional operators). We present all these
>arguments using finite strings. Of characters.
>But What Is a string? How do you define "string"? How do
>you define "character" or symbol or "letter"? Far more
>importantly, how do you define "finite"?
>
>If you want circularity, THAT is where you get it.
Nice to find something that I can agree with. How would
you answer these questions yourself? (I've brooded over
them for years, but have sworn off thinking about them
for the moment!)
You might be interested to see a brief extract from my
first response to the OP (which I thought wasn't worth
posting - no doubt you'd agree!):
From twi...@bigfoot.com Mon Aug 13 10:02:03 2007
[...]
Because functions, binary relations and binary operations
are, so to speak, naturally occurring objects outside the
set-theoretic formalisation of mathematics, a problem
seems to me to arise as soon as one tries to apply the
set-theoretic formalisation of these concepts to such
instances in the real world, including those concepts
involved in: (i) the formalisation of set theory itself
(e.g. the membership relation); and (ii) more generally,
the concepts involved in the construction of any formal
theory at all (e.g. finite strings of characters, which
are what, exactly?).
It's not a debate I care to get into at the moment (not
only does thinking about it drive me nuts, but I never
understand any of the answers I get when I ask people
about it, however patiently they explain their thinking
- I did try in sci.math a few times, long ago), but nor
can I keep silent when I see someone else struggling
in what seems to be the same way.
[...]
MoeBlee
> On Mon, 13 Aug 2007 12:13:41 -0700, george
>
> >Here's the other basic foundational question (after truth-
> >tables for propositional operators). We present all these
> >arguments using finite strings. Of characters.
> >But What Is a string? How do you define "string"? How do
> >you define "character" or symbol or "letter"? Far more
> >importantly, how do you define "finite"?
>
> >If you want circularity, THAT is where you get it.
>
> Nice to find something that I can agree with. How would
> you answer these questions yourself?
Definitions I use (anyone can devise whatever else may work better for
them):
Pn = {x | x subset of n}
m is a subset-minimal element of b <-> (meb & Axeb ~x subset of m)
n is finite <-> Ab((b subset of Pn & ~b=0) -> Em m is a subset-minimal
element of b)
w = {n | n is a finite ordinal}
s is a sequence <-> (s is a function & dom(s) is an ordinal)
'string' is used as another word for 'sequence'.
s is a denumerable sequence <-> (s is a sequence & dom(s) = w)
s is a finite sequence <-> (s is a sequence & Enew dom(s) = n)
s is a positive finite sequence <-> (s is a function & En>1 dom(s) = n
\{0})
/
For 'symbol', a more general definition can be given than the one I
give here, as one can more generally define 'is a language' while here
I only define 'is a first order language':
L is a first order language
<->
Ecqvfp(L = <c q v f p> &
v is a 1-1 function & dom(v) = w\{0} &
f is a function & range(f) subset of w &
p is a function & range p subset of w &
~dom(p) = 0 &
{{c} {q} range(v) dom(f) dom(p)} is pairwise disjoint)
That is a modification of the definition found in 'Mathematical Logic'
by J. Donald Monk.
>From that definition, we go on to define (I won't include those
definitions here):
connective(L)
quantifier(L)
variables(L)
function-symbols(L)
predicate-symbols(L)
Then
symbols(L) = {connective(L)} u {quantifier(L)} u variables(L) u
function-symbols(L) u predicate-symbols(L)
Then
expressions(L) = {s | s is a positive finite sequence & range(s)
subset of symbols(L)}
That set exists even without the axiom of replacement, by the way.
MoeBlee
'x is a symbol of L' <-> (L is a language &
(I've brooded over
> them for years, but have sworn off thinking about them
> for the moment!)
>
> You might be interested to see a brief extract from my
> first response to the OP (which I thought wasn't worth
> posting - no doubt you'd agree!):
>
> From twir...@bigfoot.com Mon Aug 13 10:02:03 2007
MoeBlee
Here it is corrected:
Hero
> I have run into a dilemma defining the Cartesian product. I would
> appreciate it if someone could help me out.
>
> 1. I define the Cartesian product of AxB as
> <and> is the "logical AND" operator
>
> 2. But this definition of the Cartesian product depends on the binary
> operator <and>. A binary operator (or a function), is a special case
> of a relation. So before I can define the Cartesian product, I need to
> first define a relation.
>
> Cartesian product.
>
> definition of the Cartesian product --- hence my dilemma.
>
> Stated another way,
> <and>:{T,F}x{TxF} --> {T,F}
> ... and so the definition of the Cartesian product depends on an
> operator defined with the help of a Cartesian product.
>
> Is there any standard way out of this dilemma? Any classic references
> would be especially appreciated.
>
What about AxB := { (a,b) | a in A , b in B }?
Well ...hmm , may be not
And AxB := { (a,b) | a in A as well as b in B }?
Or AxB := { (a,b) | a in A at the same time as b in B }?
Or is it just a problem from writing? So imagine You make a
holographic picture.
Looked upon from one angle You see
AxB := { (a,b) | a in A } and from another perspective You
see
AxB := { (a,b) | b in B }?
Or AxB := { (a,b) | under the two conditions: 1) a in A
2) b
in B }.
Now wonder, axioms are at the end of long chains of mathematical
developements.(Thales did his proofs before they found the axioms for
it).
Getting simpler and simpler becomes more and more difficult.
With friendly greetings
Hero
Angus Rodgers
<jazz...@hotmail.com> wrote:
>On Aug 13, 12:55 pm, Angus Rodgers <twir...@bigfoot.com> wrote:
>
>> On Mon, 13 Aug 2007 12:13:41 -0700, george
>>
>> <gree...@cs.unc.edu> wrote:
>> >[...] What Is a string? How do you define "string"? How do
>> >you define "character" or symbol or "letter"? Far more
>> >importantly, how do you define "finite"? [...]
>
>>[...]
>
>Definitions I use (anyone can devise whatever else may work
>better for them):
[huge snip - mostly read, some only skimmed]
(BTW, was that a version of Tarski's definition of
finiteness that you used? Neat.)
>s is a sequence <-> (s is a function & dom(s) is an ordinal)
>
>'string' is used as another word for 'sequence'.
[more snipped]
>> From twir...@bigfoot.com Mon Aug 13 10:02:03 2007
>> [...]
>> Because functions, binary relations and binary operations
>> are, so to speak, naturally occurring objects outside the
>> set-theoretic formalisation of mathematics, [...]
For practical purposes, do you regard a language, as you
have defined it here, as a construct within set theory?
If so, then which set theory (since there seem to be
several)?
If not, then could you please explain (in whatever seems
to you to be reasonable detail) the meanings of the words
"function" and "ordinal" which you used in the definition
of "sequence"?
For instance, this message I am typing is (roughly speaking)
a finite sequence of characters: does your definition apply
to it, and in particular would you say that the typed word
"sequence" is a function defined on a finite ordinal?
(I know this is possibly a particularly silly question
because the definition of "ordinal" is likely to be
completely standard; but I haven't got to ordinals yet
in the book I'm reading, and I've forgotten how they were
defined the last time I looked at a book on set theory.
I'm just guessing an ordinal is some particular kind of
well-ordered set.)
And as for a "function", what is that? A functional
relation (or an ordered triple, one of whose members
is a relation), where a relation is a set of "ordered
pairs", where an ordered pair is a particular kind of set?
Or do you mean something else? (This is still conditional
on your answer to my very first question above being "no" -
whereas of course it may very well be "yes", in which case
you can ignore most of this post.)
(These questions are not intended as rhetorical, but
please excuse me if they still don't make sense. As I
keep protesting, I really don't want to think about any
of it, but I suppose I have to, now that I've foolishly
got myself into this - having very much wanted to stay
out of it! As I said, I have been trying for years not
to think about any of this stuff, so now I find myself
in the awkward position of feeling obliged to try to
remember what used to bother me, while still wishing
earnestly not to be bothered by it any more ... If
the thread continues, perhaps I will be able to think
myself back into the frame of mind where I knew what
questions bothered me and why I was bothered by them.
Meanwhile, I am only expressing a general kind of
approval of the way the OP was asking /his/ questions.)
Chris Menzel
> On Aug 13, 7:24 am, "Daniel J. Greenhoe" <dgreen...@yahoo.com> wrote:
>> On Aug 13, 9:11 pm, "Stephen J. Herschkorn" <sjhersc...@netscape.net>
>> wrote:
>>
>> > Halmos, Paul R. Naive Set Theory. Section 6.
>> > Kunen, Kenneth. Set Theory: An Introduction to Independence Proofs. Page 13.
>> > Suppes, Patrick. Axiomatic Set Theory. Page 49.
Kunen's book is an advanced text. Not recommended for learning the
subject (though it is a very nice book indeed).
>> Thank you very much for information about these excellent references.
>> I see on the web that a not too distant library has all three books
>> so I hope I can be able to take a look this week.
>>
>> I think I might be able to find the answers I have been looking for
>> there --- thank you very much!
>
> Suppes would be good for a start. But (since you've shown confusion on
> certain more basic logical matters) I suggest you learn the predicate
> calculus even before taking on Suppes (which has only a cursory
> section on the logic), from a good textbook on it, such as 'Logic:
> Techniques Of Formal Reasoning' by Kalish, Montague, and Mar.
>
> You are confused as to the differences among such matters as
> sentential connectives (such as 'and') and relations.
And, perhaps most notably (and most helpfully for the OP), the
difference between object language and metalanguage. The easiest
solution to his original puzzle is that "and", unlike "cartesian
product", is a logical primitive of the language of set theory and hence
is not defined (in the object language) to be a relation.
Daniel J. Greenhoe
> ...
> Definable subsets ALWAYS exist according to the Axiom[schema]of
I think this may be along the lines of what I am looking for. The
Axiom of Separation was also mentioned by Stephen J. Herschkorn in his
helpful reply. I will take a look at this axiom. Thank you so much for
your help.
Dan
Daniel J. Greenhoe
> Kunen's book is an advanced text. Not recommended for learning the
> subject (though it is a very nice book indeed).
Thank you for providing information about this reference. I greatly
appreciate suggestions of rigorous texts with a sufficient history to
be considered classics.
Dan (OP)
On Aug 14, 4:17 am, Chris Menzel <cmen...@remove-this.tamu.edu> wrote:
MoeBlee
> (BTW, was that a version of Tarski's definition of
> finiteness that you used? Neat.)
Yes, Tarski's definition, as given in Suppes's 'Axiomatic Set Theory'.
> For practical purposes, do you regard a language, as you
> have defined it here, as a construct within set theory?
I don't know what you mean by 'practical purposes', but I do define
'first order language' in Z set theory and take first order languages
to be objects in Z set theory (and if the language is for Z set theory
itself, then the language is formulated in a Z set theory that is a
META-theory for the language of Z set theory as an OBJECT language;
I.e., the the meta-theory is a "clone" of the object theory). But I
don't claim that only formulations in Z set theory may be
countenanced. It's just that I do choose to work in Z set theory.
> If so, then which set theory (since there seem to be
> several)?
Personally, I use Z set theory without the axiom of regularity as my
"default" mathematical context, and I mention specifically if I go
past Z set theory wihout regularity by using the axiom of regularity,
axiom schema of replacement, or a particular choice axiom; and I use
the numeration theorem as a "temporary axiom" to develop cardinals so
that I don't have to adopt both choice and replacement (or regularity
and replacement if Scott's method is used) for that portion of the
study. Notice, though, that when I say Z set theory without regularity
is my "default" context, that doesn't mean that I don't also perform
deductions in and study other languages and theories. It's just that Z
set theory without regularity is my "default" theory for studying
languages and theories and my "default" object theory for deriving
theorems of ordinary matheamtics.
> If not, then could you please explain (in whatever seems
> to you to be reasonable detail) the meanings of the words
> "function" and "ordinal" which you used in the definition
> of "sequence"?
I use ordinary set theoretic definitions of 'function' and
'ordinal' (see later in this post).
> For instance, this message I am typing is (roughly speaking)
> a finite sequence of characters: does your definition apply
> to it, and in particular would you say that the typed word
> "sequence" is a function defined on a finite ordinal?
In my own "view" of things, in this particular regard, I distinguish
between four kinds of things:
(1) symbols, which are mathematical objects, and are formal and
abstractions.
(2) typographic shapes, which are abstractions, but are informal
abstractions.
(3) the concrete instances of typographic shapes, such as the ink on a
page, as the typographic shapes are informal abstractions from the
concretes.
(4) my (or a person's) experience of seeing concrete instance of
typographic shapes, as such experiences are the basis for a positing
of the existence of concretes such as ink on a page.
I consider concerns about (2), (3), and (4) to be philosophical
concerns, and concerns about (1) to be mathematical concerns.
I don't expect anyone else to adopt that view, and I don't claim it is
necessarily the best view, but it is one that works for me as an
explanatory view to help me keep clear what I'm dealing with when I
think about mathematics.
> (I know this is possibly a particularly silly question
> because the definition of "ordinal" is likely to be
> completely standard; but I haven't got to ordinals yet
> in the book I'm reading, and I've forgotten how they were
> defined the last time I looked at a book on set theory.
> I'm just guessing an ordinal is some particular kind of
> well-ordered set.)
Roughtly put:
n is an ordinal <-> (union of n is a subset of n & n is well ordered
by the membership relation on n)
> And as for a "function", what is that? A functional
> relation (or an ordered triple, one of whose members
> is a relation), where a relation is a set of "ordered
> pairs", where an ordered pair is a particular kind of set?
No, the standard set theoretic definition is:
f is a function <-> (f is a relation & Axyz((<x y> e f & <x z> e f) ->
y=z)
The notion you mentioned (a triple of a relation that is "functional"
with a domain and a codomain) is a definition of 'function' also used
in certain mathematics but is not the standard set theoretic
definition. However, it is also definable in set theory, though I
don't know of a standard word for it when 'function' is already taken,
so I just call it a 'func-sys'.
MoeBlee
J. Burse
> I have run into a dilemma defining the Cartesian product. I would
> appreciate it if someone could help me out.
>
> 1. I define the Cartesian product of AxB as
> AxB = { (a,b) | a in A <and> b in B }
> where (a,b) is an ordered pair and
> <and> is the "logical AND" operator
>
> 2. But this definition of the Cartesian product depends on the binary
> operator <and>. A binary operator (or a function), is a special case
> of a relation. So before I can define the Cartesian product, I need to
> first define a relation.
>
> Cartesian product.
>
> 4. So, in the end, my definition of Cartesian product depends on the
> definition of the Cartesian product --- hence my dilemma.
>
>
> Stated another way,
> <and>:{T,F}x{TxF} --> {T,F}
> ... and so the definition of the Cartesian product depends on an
> operator defined with the help of a Cartesian product.
>
>
> Is there any standard way out of this dilemma? Any classic references
> would be especially appreciated.
>
> Dan Greenhoe
>
there the cartesian product, is some diagramm
that commutes according to some abracadabra
(universal/existencial attributions).
http://en.wikipedia.org/wiki/Product_%28category_theory%29
And the good thing is you have also a coproduct!
Bye
J. Burse
>> Is there any standard way out of this dilemma? Any classic references
>> would be especially appreciated.
>>
>> Many thanks in advance,
>> Dan Greenhoe
>>
> Why not use abstract nonsense (category theory),
> there the cartesian product, is some diagramm
> that commutes according to some abracadabra
> (universal/existencial attributions).
>
> http://en.wikipedia.org/wiki/Product_%28category_theory%29
>
> And the good thing is you have also a coproduct!
>
> Bye
Just stepped over the following:
Yvon Gauthier, Internal Logic. Foundations of Mathematics
from Kronecker to Hilbert, Dordrecht, Kluwer "Synthese Library",
vol.310, 2002.
There is a chapter: Mother-theories.
Daniel J. Greenhoe
> My own advice to the OP (as a fellow-student, not an
> expert) would be not to worry /too/ much, even about
> what seem to be real difficulties in the foundations,
> but (in Cauchy's words) to "carry on, and faith will
> come". For, it seems to be a fact that foundational
> questions, although encountered by the beginner, have
> to wait until much later in one's education before
> they have a hope of being answered. For what it's
> worth, I think that your question makes sense; but
> I don't know how to answer it.
Thank you very much for your encouragement and tolerance and advice. I
very much appreciate it.
dan
On Aug 14, 12:42 am, Angus Rodgers <twir...@bigfoot.com> wrote:
> On Mon, 13 Aug 2007 09:02:24 -0700, george
>
Daniel J. Greenhoe
for granted. I am sorry if I annoyed some --- that was not my
intention in asking this question. I have had the question tagged as a
problem for a long long time.
Many thanks to Brian VanPelt and Angus Rodgers who, despite the
measure of disdain received by the topic, offered encouragement even
at the risk of using their real names.
Many thanks also to those who, although finding themselves in a
position in which the original post and original poster sounded
ludicrous, were honest enough with me to honestly express their views
in a very clear and open manner. I have benefited very much from this.
Kind Regards to all,
Daniel J. Greenhoe
Angus Rodgers
<jazz...@hotmail.com> wrote:
>On Aug 13, 3:02 pm, Angus Rodgers <twir...@bigfoot.com> wrote:
>
>> For practical purposes, do you regard a language, as you
>> have defined it here, as a construct within set theory?
>
>I don't know what you mean by 'practical purposes',
Sorry, I knew as I was writing it that it must sound vague
(but I was being pressed to read more of the latest Harry
Potter - and I risked a Cruciatus curse if I delayed any
longer!); what I meant was, when you use formal languages
in your mathematical work, are they constructs within some
set theory? And you have answered this, affirmatively:
>but I do define
>'first order language' in Z set theory and take first order languages
>to be objects in Z set theory (and if the language is for Z set theory
>itself, then the language is formulated in a Z set theory that is a
>META-theory for the language of Z set theory as an OBJECT language;
>I.e., the the meta-theory is a "clone" of the object theory). But I
>don't claim that only formulations in Z set theory may be
>countenanced. It's just that I do choose to work in Z set theory.
(Reminds me of when I had to write a parser generator
in C++, and I was trying to get it to generate itself.
So many distinctions to make!) <shudder>
>> For instance, this message I am typing is (roughly speaking)
>> a finite sequence of characters: does your definition apply
>> to it, and in particular would you say that the typed word
>> "sequence" is a function defined on a finite ordinal?
>
>In my own "view" of things, in this particular regard, I distinguish
>between four kinds of things:
>
>(1) symbols, which are mathematical objects, and are formal and
>abstractions.
>
>(2) typographic shapes, which are abstractions, but are informal
>abstractions.
>
>(3) the concrete instances of typographic shapes, such as the ink on a
>page, as the typographic shapes are informal abstractions from the
>concretes.
>
>(4) my (or a person's) experience of seeing concrete instance of
>typographic shapes, as such experiences are the basis for a positing
>of the existence of concretes such as ink on a page.
>
>I consider concerns about (2), (3), and (4) to be philosophical
>concerns, and concerns about (1) to be mathematical concerns.
Might it also be defensible to say that (1) belongs to pure
mathematics (which is timeless), and that (2), (3) and (4)
belong to applied mathematics (where time and/or space almost
always seem to be considerations)?
(Anything related to applied maths confuses me, and I think
that formal logic may be a case in point. The confusion is
interesting, but also frightening.)
>I don't expect anyone else to adopt that view, and I don't claim it is
>necessarily the best view, but it is one that works for me as an
>explanatory view to help me keep clear what I'm dealing with when I
>think about mathematics.
As far as it goes, it makes good sense to me, also. However,
I lack any real understanding of what the apparently Platonic
objects of pure mathematics are. I'm happy enough doing pure
maths, so long as it doesn't come into collision with applied
maths (like this) - and then I stop feeling that I understand
what any of it means, and it all threatens to dissolve into
meaningless character strings ...
>Roughtly put:
>
>n is an ordinal <-> (union of n is a subset of n & n is well ordered
>by the membership relation on n)
I seem to recall that these are often called "von Neumann"
ordinals - is that right?
>> And as for a "function", what is that? A functional
>> relation (or an ordered triple, one of whose members
>> is a relation), where a relation is a set of "ordered
>> pairs", where an ordered pair is a particular kind of set?
>
>No, the standard set theoretic definition is:
>
>f is a function <-> (f is a relation & Axyz((<x y> e f & <x z> e f) ->
>y=z)
That's what I meant by functional relation. (The ordered
triple thing was an alternative suggestion.)
>The notion you mentioned (a triple of a relation that is "functional"
>with a domain and a codomain) is a definition of 'function' also used
>in certain mathematics but is not the standard set theoretic
>definition.
OK.
>However, it is also definable in set theory, though I
>don't know of a standard word for it when 'function' is already taken,
>so I just call it a 'func-sys'.
Anyway, the funny thing (to me) is that things like character
strings and formal languages still seem to exist even when
one has no rooted belief in any form of set theory (such as
ZF, which I assume is what you denote by Z). So there seems
(to me) to be a question about how these things - with their
funny form of existence in the "real world", outside pure
maths - relate to those apparently very similar Platonic
entities which you construct as denotations (in some model
or other) of expressions in some formal language. Can some
version of set theory have a model in the "real world outside
pure maths"? If so, then such a theory becomes subject to
some kind of empirical, correspondence-theory type of criterion
of validity, as well as the more usual rational, coherence-
theory type of criteria which pure mathematicians seem to apply.
(Although I make no effort to follow the relevant threads in
sci.math, it has sometimes seemed to me that some demand such
as this is part of what motivates at least some of the "anti-
Cantor cranks"; and, in principle at least, it does not seem
particularly cranky to me. But, cranky or not, I prefer not
to think about it at all these days, and just to get on with
learning some pure maths, and then perhaps some applied maths
and physics as well. I find I quite enjoy using "naive" set
theory like a "working mathematician"; for me, reflecting on
the justification for it all is a task for the future, which
I know I'm not nearly ready for yet.)
(Thanks for your clear answers, and sorry that I've been led
to ramble a bit. Perhaps tomorrow I'll be able to condense
some of these cloudy ramblings into a clear question or two.
For the moment, my only real question seems to be whether
the pure/applied maths distinction is a meaningful one to
make here, as I have suggested.)
William Elliot
> On Aug 13, 5:08 pm, Brian VanPelt <bvanp...@neo.rr.com> wrote:
> > Why not try to make up something that works?
> > If the theory is needing, mabyse it's time to make something
> > consistent.
> > Good luck.
> > Brian
>
> Thanks for the encouragement. I think a lot of people would very much
> like to find a "Russell's Paradox" type oversight and find a way to
> fix it. But in this case, it would be hard to believe that no one has
> yet addressed the problem and found a good solution. And one might
> expect that that solution is already out there in a published set
> theory book somewhere.
>
Oh yes, it's due for publication in a few years after the unified field
theory solution has been published. To get warmed up for these Herculean
brain chores, I suggest you first tackle squaring the circle. There's a
nice solution involving a third ancient construction tool. If you can be
happy with that, then you can be happy with, not one but three (your
choice), of settled set theories (post Russell's rustling) and thence
proceed thee forth post haste, to the big surprise of all, an unified
field theory.
Brian VanPelt
Yeah, Einstein tried to work the unified field theory - poor idiot
that he was. He mentioned that, at his age (I don't remember what it
was), people like him could waste their time in such endeavors since
they didn't produce anything worthwhile anyways. Can you believe such
stupidity?
Brian
herbzet
Hero wrote:
> Getting simpler and simpler becomes more and more difficult.
Right. Simplicity should not be confused with a lack of sophistication.
Simplicity is often a mark of sophistication.
--
hz
Han de Bruijn
>
> What UTTER bs.
> Just for the record, there certainly is no reasoned argument,
> some language is abusive, the people speaking it are therefore
> not arguing rationally IS A LOGICAL FALLACY, dumbass.
Mathematics a part of human civilization?
Maybe it HAS BEEN a part of human civilization ..
The Original Poster is completely right: the mainstream definition of
the Cartesian product is circular. And name calling won't help anyone.
What helps is the insight that there is computer hardware implementing
the AND and there is computer software implementing functions. And there
are Cartesian products with database applications. Starting from there
is anyways consistent, otherwise it won't work.
Han de Bruijn
...
Indeed.
“Everything should be made as simple as possible, but not simpler.”
- A. Einstein
"Simplicity is the ultimate sophistication."
- Leonardo da Vinci
This one is OT but I like it:
Good, fast, cheap ... pick any two.
8-)
george
> On Mon, 13 Aug 2007 12:16:49 -0700, MoeBlee <jazzm...@hotmail.com> said:
> > You are confused as to the differences among such matters as
> > sentential connectives (such as 'and') and relations.
On Aug 13, 4:17 pm, Chris Menzel <cmen...@remove-this.tamu.edu> wrote:
> And, perhaps most notably (and most helpfully for the OP), the
> difference between object language and metalanguage. The easiest
> solution to his original puzzle is that "and", unlike "cartesian
> product", is a logical primitive of the language of set theory and hence
> is not defined (in the object language) to be a relation.
Wm.Elliott made a similar point.
I pray this will help the OP but I fear it won't.
I fear he is thinking in terms of "real" and "informal" notions of
"relations". The fact that "and" is not defined-IN-the-OBJECT-
language
TO be a relation will not, in the OP's mind, change the fact that
"and" IS
a relation.
As he defines "relation".
The fundamental issue here really is circularity.
The main reason why set theory works as a foundation is that
you can define functions in it. But if you do that using a language
that has binary operators, well, binary propositional connectives, for
example, ARE binary boolean FUNCTIONS. It becomes hard to
claim that the set-theoretical definition of a function has any
"explanatory"
power, if it is defined using a logical language whose basic elements
are
themselves (boolean) FUNCTIONS. You have to ALREADY know what a
function is before you can understand this "definition".
I think what the OP was trying to ask for was a cure for THAT.
MoeBlee
> On Mon, 13 Aug 2007 16:31:52 -0700, MoeBlee
> >between four kinds of things:
>
> >(1) symbols, which are mathematical objects, and are formal and
> >abstractions.
>
> >(2) typographic shapes, which are abstractions, but are informal
> >abstractions.
>
> >(3) the concrete instances of typographic shapes, such as the ink on a
> >page, as the typographic shapes are informal abstractions from the
> >concretes.
>
> >(4) my (or a person's) experience of seeing concrete instance of
> >typographic shapes, as such experiences are the basis for a positing
> >of the existence of concretes such as ink on a page.
>
> >I consider concerns about (2), (3), and (4) to be philosophical
> >concerns, and concerns about (1) to be mathematical concerns.
>
> Might it also be defensible to say that (1) belongs to pure
> mathematics (which is timeless), and that (2), (3) and (4)
> belong to applied mathematics (where time and/or space almost
> always seem to be considerations)?
If that works for you, then I have no objection, but I think of
'applied' in 'applied mathematics' in a different sense, and qualify
that my own sense is rather general and not informed by any work in
the philosophy of mathematics regarding 'applied' beyond the
introductory level as found in ordinary introductory texts and
anthologies on philosophy of mathematics.
> As far as it goes, it makes good sense to me, also. However,
> I lack any real understanding of what the apparently Platonic
> objects of pure mathematics are.
I don't regard them as platonic.
> >n is an ordinal <-> (union of n is a subset of n & n is well ordered
> >by the membership relation on n)
>
> I seem to recall that these are often called "von Neumann"
> ordinals - is that right?
Right.
> Anyway, the funny thing (to me) is that things like character
> strings and formal languages still seem to exist even when
> one has no rooted belief in any form of set theory (such as
> ZF, which I assume is what you denote by Z).
Indeed the ordinary intuitive notions of a string of characters and
things like that are in our minds before we get to the stage of
formalizing.
> So there seems
> (to me) to be a question about how these things - with their
> funny form of existence in the "real world", outside pure
> maths - relate to those apparently very similar Platonic
> entities which you construct as denotations (in some model
> or other) of expressions in some formal language.
I don't regard the formalizations as platonic. As to how the
formalizations relate to the pre-formal notions, yes, there are all
kind of interesting questions. But those questions don't interfere
with just carrying out the mathematics, which, given my limited time
and talent, is what I choose to concentrate on in my own current
studies (which are merely a hobby).
> Can some
> version of set theory have a model in the "real world outside
> pure maths"? If so, then such a theory becomes subject to
> some kind of empirical, correspondence-theory type of criterion
> of validity, as well as the more usual rational, coherence-
> theory type of criteria which pure mathematicians seem to apply.
But what is the "real world"? Are abstractions members of its
universe?
> (Although I make no effort to follow the relevant threads in
> sci.math, it has sometimes seemed to me that some demand such
> as this is part of what motivates at least some of the "anti-
> Cantor cranks"; and, in principle at least, it does not seem
> particularly cranky to me.
The concern you raise is not in itself crankish. What is crankish are
the particular ways the cranks proceed upon the concern.
MoeBlee
herbzet
george wrote:
> Put another way, yes,
> "and" must have gotten a definition FROM SOMEwhere before you can use
> it in set theory, but it does NOT get THAT definition FROM set theory.
> It gets it from propositional logic. Where propositional logic got
> it from is not really your concern. THAT "and" is just defined as the
> binary operator on ordered pairs OF BOOLEAN values -- NOT OF SETS --
> that returns true iff both arguments are true.
I basically agree with you, but it is interesting that Boole himself
in "A Mathematical Analysis of Logic" (1847) first interpreted
the values of his variables as ranging over sets of things. In
this interpretation "A & B" represents the intersection of sets
A and B, and "A v B" represents the union of _disjoint_ sets A
and B. (This restriction to disjoint sets was later relaxed in
the work of Jevons and Schroder.) "~A" represents the set of things
that are not elements of A.
It is a second interpretation of the calculus in the same work
that the variables are interpreted as representing the truth
values of propositions.
Boole himself considered the ability to recognize distinct
classes of things as the fundamental intellectual attribute
that allows logical reasoning to occur, preceding reasoning
by means of propositions.
In yet a third interpretation in the same book the variables
are allowed to range over the numbers in the interval from 0 to 1,
in order to use the calculus to calculate probabilities.
As it turns out, there are lots of possible interpretations of
the Boolean algebra.
Of course when you use the phrase "Boolean values" you intend
what has become standard usage -- the two values "true" and "false".
But this is somewhat different than what Boole intended!
The seminal work "The Mathematical Analysis of Logic" is quite
a short read, with a certain charm -- worth looking up.
--
hz
Angus Rodgers
<jazz...@hotmail.com> wrote:
>I don't regard the formalizations as platonic. As to how the
>formalizations relate to the pre-formal notions, yes, there are all
>kind of interesting questions. But those questions don't interfere
>with just carrying out the mathematics, which, given my limited time
>and talent, is what I choose to concentrate on in my own current
>studies (which are merely a hobby).
>
>> [...]
>
>But what is the "real world"? Are abstractions members of its
>universe?
Thereby hangs the whole philosophy of mathematics, of course.
I live with these questions every day, but, for the moment,
much like you, I have to devote more of my limited time and
energy to doing mathematics than to reflecting on what it is
(even though, ultimately, my main reason for doing it may be
to be able to reflect on what it is).
I just want to acknowledge that the questions are important,
that I didn't intend to imply that mathematics was not part
of the "real world" (a phrase which I nearly always put in
scare quotes), and that "Whereof one cannot speak ...".
I.N. Galidakis
[snip]
> But what is the "real world"? Are abstractions members of its
> universe?
[snip]
I haven't been following this thread very closely, but let me make a short
remark:
Rather than go about it this way (ask whether abstractions are members of the
"real world") you could try to describe the situation by asking the question in
the opposite way: "Is the real world an *application* of pure mathematics"?
It seems much more reasonable to me to ask the question in *this* direction and
by all appearances it looks the answer is affirmative. As far as I am concerned,
everything in my universe appears to be a creation of the Natural numbers.
Your question (in the direction posed) opens a whole can of worms leading to the
Platonic duality, which traps many mathematicians.
> MoeBlee
--
I.N. Galidakis
Han de Bruijn
> MoeBlee wrote:
> [snip]
>
>>But what is the "real world"? Are abstractions members of its
>>universe?
>
> [snip]
>
> I haven't been following this thread very closely, but let me make a short
> remark:
>
> Rather than go about it this way (ask whether abstractions are members of the
> "real world") you could try to describe the situation by asking the question in
> the opposite way: "Is the real world an *application* of pure mathematics"?
> It seems much more reasonable to me to ask the question in *this* direction and
> by all appearances it looks the answer is affirmative. As far as I am concerned,
> everything in my universe appears to be a creation of the Natural numbers.
No problem with the Natural Numbers. But the question in this thread is
whether everything in our universe appears to be a creation of common
Set Theory.
> Your question (in the direction posed) opens a whole can of worms leading to the
> Platonic duality, which traps many mathematicians.
Han de Bruijn
Hero
>... But the question in this thread is
> whether everything in our universe appears to be a creation of common
> Set Theory.
>
That's to the point!
Now common set theory is static ( no variable set possible). Whereas a
creation is a process, a movement.
With a smile
Hero
I.N. Galidakis
[snip]
> No problem with the Natural Numbers. But the question in this thread
> is whether everything in our universe appears to be a creation of
> common Set Theory.
I would surmize so. The notion of counting (Naturals) is intricately connected
to the notion of the Von Neumann integer and thus also connected to the notion
of sets.
In effect, everything seems to be the creation of the null set {}=0, and "that"
(consciousness) which recognizes the null set, 1={0}.
Whether {} and {0} beget the Naturals first or Set Theory first, is open to
interpretation, because the sets begotten from the Von Neumann construction can
be seen to be *exactly* the Naturals.
I would tend to think that the notion of "set" might be slightly more
"primitive" to consciousness than that of counting (and hence may "predate" the
Naturals on the time axis), because whereas "counting" requires the use of all
Naturals, "collecting/segregating" (objects, stuff), does not require more than
the first two Naturals, 0={} and 1={0} (i.e. the characteristic function of
sets: 1=belong, 0=doesn't belong).
Even if this is the case, binary arithmetic (and therefore logic) immediately
interjects itself in the construct, hence all three are ultimately connected at
a very primitive level (Naturals, Set Theory, Binary Arithmetic)
As far as I am concerned, consciousness and {} beget a mental construct (Set
Theory) which immediately allows the Naturals to manifest first. Once the
Naturals are born, the rest is history.
> Han de Bruijn
--
I.N. Galidakis
Han de Bruijn
That's why the _potential_ infinite (which is dynamic, in contrast
to the static actual infinite) will necessarily deprive set theory
from its predominant place in mathematics.
> With a smile
Han de Bruijn
Han de Bruijn
> Han de Bruijn wrote:
> [snip]
>
>>No problem with the Natural Numbers. But the question in this thread
>>is whether everything in our universe appears to be a creation of
>>common Set Theory.
>
> I would surmize so. The notion of counting (Naturals) is intricately connected
> to the notion of the Von Neumann integer and thus also connected to the notion
> of sets.
>
> In effect, everything seems to be the creation of the null set {}=0, and "that"
> (consciousness) which recognizes the null set, 1={0}.
>
> Whether {} and {0} beget the Naturals first or Set Theory first, is open to
> interpretation, because the sets begotten from the Von Neumann construction can
> be seen to be *exactly* the Naturals.
But there exists at least one other (HdB) construction of the naturals,
which is exposed in the "Bit mapped Set Theory" thread, starting with
the following OP:
http://groups.google.nl/group/sci.math/msg/d0e63882f157ecbe
In _this_ theory all sets (and naturals) are constructed from the empty
set as well. But there is one major difference: the naturals do not form
a part of set theory, but instead: they are equivalent with it.
MoeBlee
> Your question (in the direction posed) opens a whole can of worms leading to the
> Platonic duality, which traps many mathematicians.
I was asking someone, who used the expresssion "the real world", what
he believes it to be and whether abstractions are part of its
universe. Whatever Platonic duality you have in mind, at least I don't
feel trapped by it.
MoeBlee
MoeBlee
> No problem with the Natural Numbers. But the question in this thread is
> whether everything in our universe appears to be a creation of common
> Set Theory.
What? Who ever said anything such as that "everything in the universe
appears to be a creation of common set theory"?
MoeBlee
Hero
The potential infinite was invented to have a restricted space (up to
the crystal sphere of fix-stars) - it's infinite in time. The actual
infinite is static, but infinite in space. So i have my doubts about
Your prediction.
Set theory and logic starts with the sign =.
I would like to see Qi ( Ch'i), the wind of change, enter into the
beginnings. This is like the ideas of Herklit (the dark one) about the
river - but this never made it into math up to today.
It must be like an answer to the question arising, when we are told
>From { } = 0 we go to {{ }} = { 0} = 1, from this to the set of both
and also to the four ordered pairs of these, and so forth.
The question of course is, how can we give to that, which let change 0
to 1, a mathematical expression - the first step from = to =|=?
With friendly greetings
Hero
herbzet
I just wanted to say this is very interesting and unique: at least,
I haven't seen anything like it.
Please see also my following collateral reply to Hero.
--
hz
herbzet
Hero wrote:
> Set theory and logic starts with the sign =.
I don't know what you mean by that. Personally, I'm of the
opinion that everything starts with =/=, the recognition, or
the decision, that everything is not the same thing.
> I would like to see Qi ( Ch'i), the wind of change, enter into the
> beginnings. This is like the ideas of Herklit (the dark one) about the
> river - but this never made it into math up to today.
> It must be like an answer to the question arising, when we are told
> >From { } = 0 we go to {{ }} = { 0} = 1, from this to the set of both
> and also to the four ordered pairs of these, and so forth.
> The question of course is, how can we give to that, which let change 0
> to 1, a mathematical expression - the first step from = to =|=?
I think you might enjoy "Laws of Form" by G. Spencer Brown. There's
a good deal of literature on this book on the Net -- google it.
He creates what he calls "the primary arithmetic" which he maintains
is the arithmetic of the Boolean algebra -- something different
from an interpretation of the Boolean algebra, according to him.
The primary arithmetic is created by the use of only one operator,
which in ascii-speak would be written (). This operator has an
inside and an outside. Expressions in the primary arithmetic are
finite collections of the operator, such as ()() or (()). The
Boolean algebra arises by considering expressions with variables,
which variables stand for the presence or absence of arbitrary
expressions.
The primary injunction in "Laws of Form" is "Draw a distinction".
The primitive operator () is intended to indicate one side or the
other of this primary distinction. The laws of form arise by
considering the ways in which we might represent this first,
primary distinction to ourselves.
The movement from = to =/=, from non-being to being, from
timelessness to time, is considered in connection with the
the notion of drawing, or observing, a distinction.
--
hz
Hero
Daniel wrote:
>
>
>
> > I have run into a dilemma defining the Cartesian product. I would
> > appreciate it if someone could help me out.
>
> > 1. I define the Cartesian product of AxB as
> > AxB = { (a,b) | a in A <and> b in B }
> > where (a,b) is an ordered pair and
> > <and> is the "logical AND" operator
>
> > 2. But this definition of the Cartesian product depends on the binary
> > operator <and>. A binary operator (or a function), is a special case
> > of a relation. So before I can define the Cartesian product, I need to
> > first define a relation.
>
> > 3. But the definition of a relation R is that it is a subset of a
> > Cartesian product.
>
> > 4. So, in the end, my definition of Cartesian product depends on the
> > definition of the Cartesian product --- hence my dilemma.
>
> > Stated another way,
> > AxB = { (a,b) | a in A <and> b in B }
> > but <and> is of the form
> > <and>:{T,F}x{TxF} --> {T,F}
> > ... and so the definition of the Cartesian product depends on an
> > operator defined with the help of a Cartesian product.
>
> > would be especially appreciated.
>
> Well ...hmm , may be not
>
> And ...
What about
AxB := { (a,b) | {a, b} Subset of A union B } ?
With friendly greetings
Hero
Han de Bruijn
At least it's my impression that many mathematicians find set theory
more "fundamental" than the natural numbers.
Han de Bruijn
Hero
>
> What about
> AxB := { (a,b) | {a, b} Subset of A union B } ?
>
No, this one is definitely wrong. Sorry.
Hero
Han de Bruijn
Herklit (the dark one) actually declared the paradox as the primary
source of change. Something remains the same while it isn't: "Panta
rhei, ouden menei". Let me quote something from an old poster:
http://groups.google.nl/group/sci.math/msg/3199c520f5c89bfd
> Uhm, did I say that 0 = 1 has not a sensible model in physics? Take a
> look at the following picture (but first take a non-proportional font):
>
> plus + ___________
> O contact
> :::------------=====-------------O 0
> | spring ______ _ |
> | coil //// | / \ 1
> | \\\\ | | o | bell
> |____________//// | \ _ /
> minus -
>
> This apparatus is called a "buzzer" in English, if I'm informed well.
> It's a two state machine. Let's call the states 1 and 0. If the machine
> is in state 0, an electric current will flow through the coil and will
> make it magnetic. So the contact will be broken and the bell rings: 1.
> But if the contact is broken, the string will pull back the contact: 0.
> And so on, and so forth. So 1 causes 0 and 0 causes 1. Like a paradox,
> which is repeatedly resolved in time. Let x be a boolean with values 1
> and 0, then the process is described by: x := ~ x . Sloppy: x =/= x .
Such an oscillator is the heart of a (digital) counter, proceeding from
0 to 1, to 10, to 11, to 100, and so on and so forth.
Han de Bruijn
Han de Bruijn
while Google displays it correctly]:
look at the following picture (but first take a non-proportional font):
plus + ___________
O contact
:::------------=====-------------O 0
| spring ______ _ |
| coil //// | / \ 1
| \\\\ | | o | bell
|____________//// | \ _ /
minus -
This apparatus is called a "buzzer" in English, if I'm informed well.
Han de Bruijn
MoeBlee
So, I take it that you don't have an instance of anyone saying
anything such as "everything in our universe appears to be a creation
of common set theory".
MoeBlee
Grouchy
Well put.
A linguistic "conjunction", like "And" or "Et" or "Y" does NOT carry
the same logical implications as logical conjunction (and "ordered
pair" has no determinable "truth-value")
What I suspect they are looking for is the Computational Operation
ADD, not AND.. or even better use a topological relations, not
propositonal operators.
Interestingly, in electronics "NOT" or "negation" cannot be determined
without a control V (it's an inverter gate). So Strictly speaking
computational logics and applied logics would NEVER produce a silly
propositional "not" in an existential system. Even MORE
interestingly, the ACTUAL phases a electronic inverter gate goes
through requires A MINIMUM of three distinct "states" and FIVE to
SEVEN changes
Value A
A >= Vcc?
no
A
A>=Vcc?
yes
A<=Vcc?
no
Value ?
A <=Vcc?
yes
value 0
etc. (actally a flywheel inverter works differently, and an accurate
trace of the changes within the circuit represented as a algorithm
would ring and waggle like crazy.
My point is that although you are correct D Greehoe, I suspect that
"truth" for most students of Logic (especially propositional systems)
is a subjective term related to pleasing their TA or professor, rather
than grasping the beauty of what is great about anachronistic systems
and entering the modern era.
Still Thank you for the effort.
Grouchy
On Aug 13, 10:11 am, "Daniel J. Greenhoe" <dgreen...@yahoo.com> wrote:
> On Aug 13, 8:24 pm, David C. Ullrich <ullr...@math.okstate.edu>
>
> Thank you for your reply. I do genuinely appreciate it when people
> take time to respond to my posts.
>
> > This is silly.
> > We don't need to regard "and" as a Cartesian product.
>
> In mathematics, it is not what we regard something to be that is
> significant, but rather what an object is as determined by the utility
> it performs. And in this case, the utility that "and" performs is the
> _operation_ of the <logical and>. And the <logical and> is indeed a
> relation. And a relation is indeed a subset of a Cartesian product.
> And as such, how can we not take care to rigorously define it? Because
> "and" performs a very real mathematical binary operation, of course we
> must define it rigorously.
>
> > "And" is simply an English word with a well-known
> > meaning.
>
> "And" indeed is an English word and its meaning is well-known. But
> this is not surprising, because propositional logic is fundamental to
> the human thought process; and so one would expect that every one of
> the 6000 or so human languages would contain vocabulary for the
> <logical and> and <logical or>. In conversational languages, the
> meaning of logical operators is understood by intuition partly because
> we use them since early childhood. But in mathematics, understanding
> by intuition is not sufficient. At some point every concept must be
> well defined based on whatever axiomatic system we are in.
>
> > I'm not going to attempt to give a definition of
> > "and" - I'd end up with something like
> > "A and B" is true if and only if A is true and B is true
> > which is clearly circular, or with something that just
> > hides the circularity with a synonym.
>
> Actually, your definition here is essentially the same as the standard
> definition of the <logical and> used in classical logic:
> A B | A and B
> --------------
> F F | F
> F T | F
> T F | F
> T T | T
>
> That is, "A and B" is T(rue) if and only if A is true and B is true...
> which is clearly not circular (meaning that the definition of <logical
> and> in no way depends on itself).
>
> > You can't possible define _everything_ in terms of
> > previously defined terms. If the reader does not
> > understand ordinary English then no amount of
> > explanation, in English, is going to make the
> > reader understand.
> > In fact your readers _do_ know what "and" means,
> > so there's no problem.
>
> It would seem that such an argument is based on pedagogical
> considerations rather than considerations regarding mathematical
> construction. That is, such an argument is concerned with what is
> necessary to make someone understand the concept of the Cartesian
> product. And if one was preparing class notes for a high school class,
> this may be a reasonable approach --- most all high school students
> know what "and" means and so that is that. But in mathematics,
> especially in analysis, mathematicians are concerned with
> "constructions" --- how can we construct this or that starting with
> just a few axioms such as the notion of a set?
>
> And it is in the spirit of construction, not pedagogy, that I try to
> define the Cartesian product --- that is, I try to construct it
> starting from the simple notion of the set and not from the notion of
> concepts pre-built into this or that European language that drifts
> about every few hundred years.
>
> And I am certainly not alone in this thinking. Would one also tell all
> the mathematicians who worked on constructing the real number system
> from integers, ring operations, and metric spaces/topology concepts
> that their work is "silly" because after all, everyone knows what a
> real number is? Or would Frege tell Russell that his "Russell's
> Paradox" argument was "silly", because, after all, every one know what
> we mean when we talk about a set?
>
> Kind Regards,
> Dan Greenhoe
>
> On Aug 13, 8:24 pm, David C. Ullrich <ullr...@math.okstate.edu> wrote:
>
> > On Mon, 13 Aug 2007 05:15:14 -0000, "Daniel J. Greenhoe"
>
> > <dgreen...@yahoo.com> wrote:
> > >I have run into a dilemma defining the Cartesian product. I would
> > >appreciate it if someone could help me out.
>
> > >1. I define the Cartesian product of AxB as
> > > AxB = { (a,b) | a in A <and> b in B }
> > > where (a,b) is an ordered pair and
> > > <and> is the "logical AND" operator
>
> > >2. But this definition of the Cartesian product depends on the binary
> > >operator <and>. A binary operator (or a function), is a special case
> > >of a relation. So before I can define the Cartesian product, I need to
> > >first define a relation.
>
> > >3. But the definition of a relation R is that it is a subset of a
> > >Cartesian product.
>
> > >4. So, in the end, my definition of Cartesian product depends on the
> > >definition of the Cartesian product --- hence my dilemma.
>
> > This is silly.
>
> > We don't need to regard "and" as a Cartesian product.
> > "And" is simply an English word with a well-known
> > meaning.
>
> > I'm not going to attempt to give a definition of
> > "and" - I'd end up with something like
>
> > "A and B" is true if and only if A is true and B is true
>
> > which is clearly circular, or with something that just
> > hides the circularity with a synonym.
>
> > You can't possible define _everything_ in terms of
> > previously defined terms. If the reader does not
> > understand ordinary English then no amount of
> > explanation, in English, is going to make the
> > reader understand.
>
> > In fact your readers _do_ know what "and" means,
> > so there's no problem.
>
> > >Stated another way,
> > > AxB = { (a,b) | a in A <and> b in B }
> > >but <and> is of the form
> > > <and>:{T,F}x{TxF} --> {T,F}
> > >... and so the definition of the Cartesian product depends on an
> > >operator defined with the help of a Cartesian product.
>
> > >Is there any standard way out of this dilemma? Any classic references
> > >would be especially appreciated.
>
> > >Many thanks in advance,
> > >Dan Greenhoe
>
> > ************************
>
> > David C. Ullrich
Hero
> Hero wrote:
>
> > Set theory and logic starts with the sign =.
> > I would like to see Qi ( Ch'i), the wind of change, enter into the
> > beginnings. This is like the ideas of Herklit (the dark one) about the
> > river - but this never made it into math up to today.
> > It must be like an answer to the question arising, when we are told
> > From { } = 0 we go to {{ }} = { 0} = 1, from this to the set of both
> > and also to the four ordered pairs of these, and so forth.
> > The question of course is, how can we give to that, which let change 0
> > to 1, a mathematical expression - the first step from = to =|=?
>
> Herklit (the dark one) actually declared the paradox as the primary
> source of change. Something remains the same while it isn't: "Panta
> rhei, ouden menei".
That's Plato (and Simplicius) about Heraklit.
For them, he is dark.
> Let me quote something from an old poster:
>
> http://groups.google.nl/group/sci.math/msg/3199c520f5c89bfd
>
>
>
> > Uhm, did I say that 0 = 1 has not a sensible model in physics? Take a
> > look at the following picture (but first take a non-proportional font):
>
> > plus + ___________
> > O contact
> > :::------------=====-------------O 0
> > | spring ______ _ |
> > | coil //// | / \ 1
> > | \\\\ | | o | bell
> > |____________//// | \ _ /
> > minus -
>
> > This apparatus is called a "buzzer" in English, if I'm informed well.
> > It's a two state machine. Let's call the states 1 and 0. If the machine
> > is in state 0, an electric current will flow through the coil and will
> > make it magnetic. So the contact will be broken and the bell rings: 1.
> > But if the contact is broken, the string will pull back the contact: 0.
> > And so on, and so forth. So 1 causes 0 and 0 causes 1. Like a paradox,
> > which is repeatedly resolved in time. Let x be a boolean with values 1
> > and 0, then the process is described by: x := ~ x . Sloppy: x =/= x .
>
> Such an oscillator is the heart of a (digital) counter, proceeding from
> 0 to 1, to 10, to 11, to 100, and so on and so forth.
Just as in Computer-math one writes
(Let) x = x + 1
(take the value of the register with the name x, add 1 and place the
result back into the memory cell).
The content of this memory cell, with the name x, is a variable
number, in such a way.
Here we have change, math in action.
Han, what is so paradox about change?
One might think the constant is paradox, just like You sit at rest
behind the monitor, but still moving around the axis of earth. Of
course there are eternal constants, like pi - but proofed to be
constant how? By changing the diameter and consequently the length of
the perimeter.
Resolving in time is a nice description.
With friendly greetings
Hero
PS
http://kr.geocities.com/hyun_sinnayo/presoc.htm#HERACLITUS
http://www.wsu.edu/~dee/GREECE/HERAC.HTM
http://www.pinselpark.de/philosophie/h/heraklit/texte/natur.html
http://www.iep.utm.edu/h/heraclit.htm
Hero
One might think to go nuts with only endless strings of one operator.
Think of the other endless stream of 0 and 1 inside the computers.
It's similar to :Now from 0 to 1 , from { } to { { } } needs an
operation, isn't it? With lot's of things its a collecting, but one
can look at it also as a seperation: { car, cookie } is including a
car and a cookie and excluding the rest of the universe. Or from 1 to
zero, it's like taking out of it: like the child answered, why there
is now no cookie,where there was one before - it's eaten up, gone.
So there are attempts to find a base for movement in math. We will
see more of this in future, this is my prediction.
> Hero wrote:
> > Set theory and logic starts with the sign =.
herbzet wrote:
> I don't know what you mean by that. Personally, I'm of the
> opinion that everything starts with =/=, the recognition, or
> the decision, that everything is not the same thing.
I once saw the empty set { } defined by { x| x=|=x}, by a logical
contradiction. So inside is a logical contradiction and excluded from
the space between the brackets is the rest of math.(Logical
contradictions are a powerful tool in math to prove things.)
The sign = does not denote being equal, 3+5 = 8 denotes that there is
equality of quantum, just like on the scales in both scales is
something different, but in equilibrum 3+5 is of the same value as 8.
This = is also needed for the sets being the same, or for ordered
pairs - but in itself is defined as a relation on AxA . So we have
another circle for Daniel.
And in logic we have Aristoteles most basic law, in which he is
reasoning about this 'the same', this =.
Important here is the expression 'simultaneously' resp 'at the same
time'.
That's the point where we need developement in axiomatic area. Growing
and shrinking, creation and extinction, operations and variation of
size, change can be done with math too.
Here's Aristoteles Metaphysics , IV:
" Evidently then such a principle is the most certain of all; which
principle this is, let us proceed to say. It is, that the same
attribute cannot at the same time belong and not belong to the same
subject and in the same respect; we must presuppose, to guard against
dialectical objections, any further qualifications which might be
added. This, then, is the most certain of all principles, since it
answers to the definition given above. For it is impossible for any
one to believe the same thing to be and not to be, as some think
Heraclitus says. For what a man says, he does not necessarily believe;
and if it is impossible that contrary attributes should belong at the
same time to the same subject (the usual qualifications must be
presupposed in this premiss too), and if an opinion which contradicts
another is contrary to it, obviously it is impossible for the same man
at the same time to believe the same thing to be and not to be; for if
a man were mistaken on this point he would have contrary opinions at
the same time. It is for this reason that all who are carrying out a
demonstration reduce it to this as an ultimate belief; for this is
naturally the starting-point even for all the other axioms."
http://classics.mit.edu/Aristotle/metaphysics.4.iv.html
With friendly greetings
Hero
herbzet
Han de Bruijn wrote:
> > Uhm, did I say that 0 = 1 has not a sensible model in physics? Take a
> > look at the following picture (but first take a non-proportional font):
> >
> > plus + ___________
> > O contact
> > :::------------=====-------------O 0
> > | spring ______ _ |
> > | coil //// | / \ 1
> > | \\\\ | | o | bell
> > |____________//// | \ _ /
> > minus -
> >
> > This apparatus is called a "buzzer" in English, if I'm informed well.
> > It's a two state machine. Let's call the states 1 and 0. If the machine
> > is in state 0, an electric current will flow through the coil and will
> > make it magnetic. So the contact will be broken and the bell rings: 1.
> > But if the contact is broken, the string will pull back the contact: 0.
> > And so on, and so forth. So 1 causes 0 and 0 causes 1. Like a paradox,
> > which is repeatedly resolved in time. Let x be a boolean with values 1
> > and 0, then the process is described by: x := ~ x . Sloppy: x =/= x .
>
> Such an oscillator is the heart of a (digital) counter, proceeding from
> 0 to 1, to 10, to 11, to 100, and so on and so forth.
Speaking of oscillators, postulates of the Boolean algebra have always
reminded me of a basic multivibrator circuit:
http://en.wikipedia.org/wiki/Multivibrator#Astable_Multivibrator_circuit
Note the cross-coupling of the two transistors, where the output of
one is fed back into the input of the other. The output is a
square wave, oscillating between logic 1 and logic 0.
Here are postulates for the Boolean algebra (Huntington 1904):
A class of elements B together with two binary operations
(+) and (*) is a Boolean algebra if and only if the following
postulates hold:
P1 The class B is closed under the operations (+) and (*)
P2. The operations are commutative.
P3. Each operation is distributive over the other.
P4. There exists in B distinct identity elements 0 and 1
relative to the operations (+) and (*), respectively.
P5. For every a in B there exists an inverse element a' in B
such that
a + a' = 1 and a * a' = 0
Note the cross-coupling of P4 and P5: The identity element 0
relative to the (+) operator is the element for which an inverse
element is defined relative to the (*) operator, and vice versa.
--
hz
herbzet
Hero wrote:
>
> Thanks for Your lively intro to G.S. Brown. I didn't know him.
There's an edition of "Laws of Form" in german. You might look
for it. Try google.
> One might think to go nuts with only endless strings of one operator.
In ASCII it must be rendered as a string. On paper we're not
confined to one dimension. We can draw a circle on a piece
of paper as the primitive operator. It has an inside and an
outside. We can draw another circle, either inside the first,
or outside the first. This is a compound expression, analogous
to (()) or ()() respectively.
We can draw any finite number of circles, and get a compound
expression. It does not have to be a string. The only restriction
is that the boundaries don't cross each other -- for any two
circles A and B we should be able to determine whether one
is inside the other, or whether they are both outside of each
other.
The primary arithmetic has two axioms:
1) Condensation: ()() = ().
2) Cancellation: (()) = .
Cancellation allows expressions to vanish, or, in the
reverse direction, allows expressions to pop into existence
out of the void. The blank " " along with () are the two
primitive expressions of the primary arithmetic.
By means of the two axioms we can reduce any (finite) compound
expression to one of the two primitive expressions. In the reverse
direction, the axioms allow the generation of any (finite) compound
expression from the primitive expressions.
We can consider arbitrary expressions by the use of variables.
For example consider the expression
1) a ()
where a is any expression. By means of the two axioms a can be
reduced to either () or " ". So expression (1) becomes either
2) ()()
or
3) ()
respectively. By condensation ()() = (), so in either case
we have
4) a () = ()
regardless of what expression a might be.
By similar reasoning, we arrive at two axioms for the
primary algebra (Boolean algebra):
I) ((a)a) = .
II) ((a)(b))c = ((ac)(bc)).
These axioms form a complete basis for the primary algebra.
In the later chapters Brown considers infinitely extended expressions,
which might alternatively be considered as finite expressions which
"feed back" into themselves. The values of such expressions
will in some cases be indeterminate, or may be considered, by the
introduction of time, to oscillate in value.
[...]
> I once saw the empty set { } defined by { x| x=|=x}, by a logical
> contradiction. So inside is a logical contradiction and excluded from
> the space between the brackets is the rest of math.
Nice.
--
hz
herbzet
"I.N. Galidakis" wrote:
> In effect, everything seems to be the creation of the null set {}=0, and "that"
> (consciousness) which recognizes the null set, 1={0}.
[...]
> As far as I am concerned, consciousness and {} beget a mental construct (Set
> Theory) ...
It seems to me that there is a difference between {} and absolute
nothingness.
I think that the null set is already a construction of consciousness.
The null set contains nothing, but it is, arguably, not nothing.
I don't think this really messes up your metaphysics, just pushes it
back a step.
--
hz
"The king is a thing -- of nothing."
Hamlet IV, 2
I.N. Galidakis
[snip]
> It seems to me that there is a difference between {} and absolute
> nothingness.
> I think that the null set is already a construction of consciousness.
If a difference exists, it is very very subtle. To me it seems that {} is the
most primitive "symbolic description" of AN (Absolute Nothingness).
Consciousness is using its own primitive set-theoretic symbols {"{","}"} and
first attempts to define that which it "sees" first. As such, on a theoretical
level (independently of consciousness) it may well be that {}=/=AN, although on
a realistic/conscious level (from the point of view of the consciousness which
surveys the beginning) {}=AN.
In other words, AN is independent of consciousness. {} is consciousness' attempt
to characterize/describe AN and thus is dependent on consciousness.
> The null set contains nothing, but it is, arguably, not nothing.
That's true, but at the level you are describing the two, consciousness hasn't
built its symbols yet to differentiate between AN and {}. It only has the first
2 primitive symbols in its posession: {"{","}"}, hence the first thing it does
is to classify AN as {}. For *that* consciousness, {} IS AN. That's how it
understands it.
> I don't think this really messes up your metaphysics, just pushes it
> back a step.
Not necessarily, although I understand your distinction. Your distinction makes
sense only apart from consciousness, as such I think it is a moot point, because
we are describing the process from the point of view of a single consciousness
itself.
Let me make it more specific: To say that {}=/=AN amounts to introducing a
"second" observer/consciousness (independent of the first consciousness which
perceives AN, and who has alredy built its set-theoretic tools) who can "see"
the difference between AN and {} and "point it out" to the first observer. Such
an observer does not exist. The only observer that exists, immediately
"describes" AN as {}.
When this observer constructs the Naturals, via {}->{0}, etc, ONLY THEN he has
the capacity (and the set-theoretic tools) to go back in time and "examine"
whether AN IS {} (like we do here). Therefore this comparison (whether AN={})
makes sense only a posteriori.
--
I.N. Galidakis
Han.de...@dto.tudelft.nl
[ referring to all of his posters in this thread ]
Thank you for jumping in, Hero. Quite a refreshment!
Han de Bruijn
G. Frege
<morp...@olympus.mons> wrote:
>>
>> It seems to me that there is a difference between {} and absolute
>> nothingness.
>>
Obviously. Since from {} = {} we get
Ex(x = {})
i.e. the is something which is (identical with) the empty set.
While the following might be considered a formal version of the claim
that there is _nothing_:
~Ex(x = x).
(Assuming that for any existing thing /a/ we have /a/ = /a/.)
>
> If a difference exists, it is very, very subtle.
>
No, it's in no way subtle.
Ex(x = {})
can be read as
{} exists.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
"We may indeed take 'Ex(x = a)' as expressing 'a exists'."
(W. V. O. Quine, Existence and Quantification, 1969)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
>
> To me it seems that {} is the most primitive "symbolic description" of AN
> (Absolute Nothingness).
>
You are clearly wrong about this. Since (see above) {} _is_; it cannot
"mean" AN.
The following might be considered to be the expression you are looking
for:
~Ex(x = x).
Though there may be some debate about this (my) claim.
>
> [...] AN is independent of consciousness.
>
Right. Moreover if there WERE (sic!) some consciousness, then this
would be in contradiction with AN.
>>
>> The null set contains nothing, but it is, arguably, not nothing.
>>
Right.
F.
--
E-mail: info<at>simple-line<dot>de
Hero
> On Fri, 17 Aug 2007 20:46:23 +0300, "I.N. Galidakis"
>
>
>
> > To me it seems that {} is the most primitive "symbolic description" of AN
> > (Absolute Nothingness).
>
> You are clearly wrong about this. Since (see above) {} _is_; it cannot
> "mean" AN.
>
> The following might be considered to be the expression you are looking
> for:
>
> ~Ex(x = x).
>
Is there a difference between
nothing
and
Absolute Nothingness?
With friendly greetings
Hero
G. Frege
wrote:
>>>
>>> To me it seems that {} is the most primitive "symbolic description" of AN
>>> (Absolute Nothingness).
>>>
>> You are clearly wrong about this. Since (see above) {} _is_, [hence] it
>> cannot "mean" AN.
>>
>> The following might be considered to be the expression you are looking
>> for:
>>
>> ~Ex(x = x).
>>
> Is there a difference between
>
> "nothing"
>
> and
>
> "Absolute Nothingness?"
>
Sure. The two expressions belong to different grammatical categories.
G. Frege
>>>
>>> It seems to me that there is a difference between {} and absolute
>>> nothingness.
>>>
> Obviously. Since from {} = {} we get
>
> Ex(x = {})
>
> i.e. the is something which is (identical with) the empty set.
>
> While the following might be considered a formal version of the claim
> that there is _nothing_:
>
> ~Ex(x = x).
>
> (Assuming that for any existing thing /a/ we have /a/ = /a/.)
>
>>
>> If a difference exists, it is very, very subtle.
>>
> No, it's in no way subtle.
>
> Ex(x = {})
>
> can be read as
>
> {} exists.
>
> ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
>
> "We may indeed take 'Ex(x = a)' as expressing 'a exists'."
>
> (W. V. O. Quine, Existence and Quantification, 1969)
>
> ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
>
>>
>> To me it seems that {} is the most primitive "symbolic description" of AN
>> (Absolute Nothingness).
>>
> You are clearly wrong about this. Since (see above) {} _is_, [hence] it
> cannot "mean" AN.
>
Moreover that there is an empty set (i.e. {}) does not exclude the
existence of other sets beside {}.
But...
>
> ...the following might be considered to be the expression you are looking
> for:
>
> ~Ex(x = x) ,
>
since this claims that there is nothing.
>
> Though there may be some debate about this (my) claim.
>
F.
Hero
Is there a (logical difference) between
~Ex(x = x)
and
A x ( x =|=x)
Nam D. Nguyen
Yes there is. "Nothing" is formalizably knowable with a symbol, say, "{}".
"Absolute Nothingness", otoh, can't be precisely formalized; nor can one
know for certain what it is. That's is to say, unless one is a Zen Master,
or is dead, which in either case one couldn't utter "Absolute Nothingness"
in a way others would understand it, and know the its difference from "nothing".
In brief, "nothing" as defined as "{}" is a 1st-order nothingness (and there
are different ways to define it), while AN as you've alluded to is not in
the domain of mathematical logic. Imho.
Cheers,
---Nam
>
> With friendly greetings
> Hero
>
Han.de...@dto.tudelft.nl
>
> I think you might enjoy "Laws of Form" by G. Spencer Brown. There's
> a good deal of literature on this book on the Net -- google it.
>
> He creates what he calls "the primary arithmetic" which he maintains
> is the arithmetic of the Boolean algebra -- something different
> from an interpretation of the Boolean algebra, according to him.
>
> The primary arithmetic is created by the use of only one operator,
> which in ascii-speak would be written (). This operator has an
> inside and an outside. Expressions in the primary arithmetic are
> finite collections of the operator, such as ()() or (()). The
> Boolean algebra arises by considering expressions with variables,
> which variables stand for the presence or absence of arbitrary
> expressions.
>
> The primary injunction in "Laws of Form" is "Draw a distinction".
> The primitive operator () is intended to indicate one side or the
> other of this primary distinction. The laws of form arise by
> considering the ways in which we might represent this first,
> primary distinction to ourselves.
>
> The movement from = to =/=, from non-being to being, from
> timelessness to time, is considered in connection with the
> the notion of drawing, or observing, a distinction.
Hmm, Spencer Brown's formula formation rules (()) and ()() bear
great resemblance to the formation rules for my 'Bit mapped Set
Theory', as exposed in:
http://groups.google.nl/group/sci.math/msg/d0e63882f157ecbe
Quote:
> Formation rules:
> 0. {} is a set (the empty set)
> 1. if A is a set and B are sets then AB are sets
> 2. if A are sets then {A} is a set
> 3. There are no other ways to form (sets or) a set
Coincidence or not?
Han de Bruijn
Hero
> Hero wrote:
> > G. Frege wrote:
> >> I.N. Galidakis wrote:
> >> <morph...wrote:
>
> >>> To me it seems that {} is the most primitive "symbolic description" of AN
> >>> (Absolute Nothingness).
> >> You are clearly wrong about this. Since (see above) {} _is_; it cannot
> >> "mean" AN.
>
> >> The following might be considered to be the expression you are looking
> >> for:
>
> >> ~Ex(x = x).
>
> > Is there a difference between
> > nothing
> > and
> > Absolute Nothingness?
>
> Yes there is. "Nothing" is formalizably knowable with a symbol, say, "{}".
> "Absolute Nothingness", otoh, can't be precisely formalized; nor can one
> know for certain what it is. That's is to say, unless one is a Zen Master,
> or is dead, which in either case one couldn't utter "Absolute Nothingness"
> in a way others would understand it, and know the its difference from "nothing".
>
> are different ways to define it), while AN as you've alluded to is not in
> the domain of mathematical logic. Imho.
>
Your thinking is about orders, domains and masters.
A Master in knowledge about nothing?
And he can teach us the way from nothing to AN?
When You ask a Zen Master, what nothing is - he will hit You. It's a
silly question, as You know already, what You will get as an answer,
when You don not ask.
You do not believe me, listen to the sounds of silence.
Hero
PS {} is a 1st-order nothingness - wrong.
{ } is the size, the quantity of nothing, thus naught.
G. Frege
<namduc...@shaw.ca> wrote:
>
> "Nothing" is formalizably knowable with a symbol, say, "{}".
>
Not in mathematics. Here {} is known as empty set, i.e. a set which
doesn't contain any elements.
>
> "Absolute Nothingness", otoh, can't be precisely formalized.
>
One might try to express it the following way:
~Ex(x = x).
"Nothing exists."
(Note that we only can express/say this since it is FALSE that nothing
exists, for example the very claim just expressed above exists.)
herbzet
Hero wrote:
> Is there a (logical difference) between
> ~Ex(x = x)
> and
> A x ( x =|=x)
> ?
Your question is rhetorical?
~Ex(x = x) = Ax~(x = x)
= Ax(x =/= x).
Obviously your question relates to what you wrote before:
"I once saw the empty set { } defined by { x| x=|=x}, by a logical
contradiction. So inside is a logical contradiction and excluded from
the space between the brackets is the rest of math."
So let's compare expressions:
1) Ax(x =/= x)
2) {} =df {x | x =/= x}
Where does this leave us? I don't know.
--
hz
herbzet
"I.N. Galidakis" wrote:
> herbzet wrote:
> [snip]
>
> > It seems to me that there is a difference between {} and absolute
> > nothingness.
> > I think that the null set is already a construction of consciousness.
>
> If a difference exists, it is very very subtle.
Yes.
[...]
> > I don't think this really messes up your metaphysics, just pushes it
> > back a step.
>
> Not necessarily, although I understand your distinction. Your distinction makes
> sense only apart from consciousness, as such I think it is a moot point, because
> we are describing the process from the point of view of a single consciousness
> itself.
If we take consciousness as the fundamental substrate of the world ...
"Outside time, without extension" ...
There is an awakening, conciousness takes cognizance of itself,
takes cognizance of taking cognizance of itself ...
Thus is all the world generated??
Do we all re-enact the primal act of consciousness?
I'm thinking something along these lines. The world as a manifestation
of the Brahman, which remains, nevertheless, outside the world,
outside time, unmanifest.
Something like that.
Still, an interesting essay on your part connecting conciousness to
mathematics.
--
hz
herbzet
Han.de...@DTO.TUDelft.NL wrote:
> Hmm, Spencer Brown's formula formation rules (()) and ()() bear
> great resemblance to the formation rules for my 'Bit mapped Set
> Theory', as exposed in:
>
> http://groups.google.nl/group/sci.math/msg/d0e63882f157ecbe
>
> Quote:
>
> > Formation rules:
> > 0. {} is a set (the empty set)
> > 1. if A is a set and B are sets then AB are sets
> > 2. if A are sets then {A} is a set
> > 3. There are no other ways to form (sets or) a set
>
> Coincidence or not?
Perhaps. Re rule 1: If A is a set and B is a set, what is AB?
--
hz
Nam D. Nguyen
> On Sat, 18 Aug 2007 18:46:21 GMT, "Nam D. Nguyen"
> <namduc...@shaw.ca> wrote:
>
>> "Nothing" is formalizably knowable with a symbol, say, "{}".
>>
> Not in mathematics. Here {} is known as empty set, i.e. a set which
> doesn't contain any elements.
Is there much of a difference between emptiness and nothingness?
It doesn't seem mathematically wrong to think an empty set is
a set that contains "nothing". (Any way, I wasn't trying to rename
an "empty" a "nothing" set; though there exist other names e.g. "null"
set, etc...)
>
>> "Absolute Nothingness", otoh, can't be precisely formalized.
>>
> One might try to express it the following way:
>
> ~Ex(x = x).
>
> "Nothing exists."
>
> (Note that we only can express/say this since it is FALSE that nothing
> exists, for example the very claim just expressed above exists.)
From what I know, ~Ex(x = x) is still a FOL= expression and therefore
is not above a reasoning framework as AN would be.
>
>
> F.
>
Nam D. Nguyen
Correct. That's why he's different from a scientist who'd
claim to have knowledge about "something".
> And he can teach us the way from nothing to AN?
No. But he could point out the way one could teach oneself.
>
> When You ask a Zen Master, what nothing is - he will hit You. It's a
> silly question, as You know already, what You will get as an answer,
> when You don not ask.
So how much have you been hit by asking the question "A Master in
knowledge about nothing?"?
>
> You do not believe me, listen to the sounds of silence.
I believe in the sounds of silence; I just don't listen to some
people.
I.N. Galidakis
[snip]
> If we take consciousness as the fundamental substrate of the world ...
>
> "Outside time, without extension" ...
>
> There is an awakening, conciousness takes cognizance of itself,
> takes cognizance of taking cognizance of itself ...
>
> Thus is all the world generated??
Yes, in my opinion. Consciousness taking cognizance of itself generates the
Naturals from {}.
> Do we all re-enact the primal act of consciousness?
Depends on your view. In effect, what I (and you, understanding) have done is
regress back to the time of the first Mind and see the process of Creation. The
real "Big Bang" is the creation of the Naturals by Consciousness. The material
"Big Bang" is a later consequence of the genesis of the Naturals.
> I'm thinking something along these lines. The world as a
> manifestation of the Brahman, which remains, nevertheless, outside
> the world, outside time, unmanifest.
>
> Something like that.
>
> Still, an interesting essay on your part connecting conciousness to
> mathematics.
What's interesting is having at least one other human who understands it.
That's the pinnacle of communication between Brahman and human mind.
It's that "little something" which makes the difference between {} and AN ;o)
--
I.N. Galidakis
herbzet
"I.N. Galidakis" wrote:
> herbzet wrote:
> > Still, an interesting essay on your part connecting conciousness to
> > mathematics.
>
> What's interesting is having at least one other human who understands it.
Well, "understanding" may be too strong a term.
You might want to have a look at "Laws of Form" by G. Spencer Brown.
For a non-mathematical approach I really like "An Introduction to
Zen" by D.T. Suzuki, with a forward by Carl Jung. Also, "The
Field of Zen" by the same author, if it can be found. Actually,
anything by D.T. Suzuki, Professor of Occidental Philosophy at
Otani University for many years (now deceased) is worth looking
at.
--
hz
Han de Bruijn
Then AB are sets (i.e. not a _single_ set).
Han de Bruijn
schoenf...@gmail.com
[...]
> Is there any standard way out of this dilemma? Any classic references
> would be especially appreciated.
There is no "time" in mathematics. The axioms are all simultaneously
true irrespective of the order of their declarations. Thus a cyclic-
dependency, as you have, is not really a problem.
You can if you want forward declare the existence of a relation before
defining what it is.
For example
1. Declare the existence of a "relation" .
2. Define the Cartesian product using the relation 'and'.
3. Define a relation in terms of the Cartesian product.
4. Define 'and'.
Parsing that in order should avoid problems. This same problem arises
in compiler design. Forward declarations solve it.
Alternatively, you can avoid set-theoretic formulations of mathematics
and use functional ones like the Lambda Calculus. I like lambda
calculus quite a lot but they carry some theoretical problems (but
practically they are superior).
schoenf...@gmail.com
Also, another solution is to embed the definition of a 'relation' and
of 'and' together with that of the 'cartesian product' in their
n'order logic forms (making sure to avoid cyclic-dependencies).
The penalty with this is that you unnecessarily complicate the
definition of a cartesian product and you (redundantly) duplicate part
of that logic when you define a 'relation' and 'and' separately.
herbzet
herbzet wrote:
> "I.N. Galidakis" wrote:
> > herbzet wrote:
>
> > > Still, an interesting essay on your part connecting conciousness to
> > > mathematics.
I meant to put the following in yesterday's post, but I couldn't find
it. Turns out it was in the preface!
In "Infinity and the Mind" (1982) Rudy Rucker writes:
"I started having ideas for [this book] in that most Sixties of
years, 1972. At that time I was ... attending a logic seminar
led by the eminent proof theorist Gaisi Takeuti at the Institute
of Advanced Study in Princeton, New Jersey. The first time I met
Takeuti I asked him what set theory was really about. "We are
trying to get exact description of thoughts of infinite mind,"
he said And then he laughed, as if filled with happiness by
this impossible task."
The idea of "a set is the form of a possible thought" recurs through-
out the book as a motif.
--
hz
herbzet
I think maybe you need to specify that A =/= B.
Also, maybe, that {A} =/= A.
--
hz
Han de Bruijn
I've taken notice. Thanks.
Han de Bruijn
cbr...@cbrownsystems.com
But that is not how his "implementation" works. In fact A, AA, and AAA
are distinct "set/sets" (note that equality is based on some lexical
function to the naturals in his implmentation). "are sets" are
actually (ordered) 'sequences' of either "are sets" or "is a set".
Thus, generally AB is not equal to (as he has defined it) BA (where
both such constructions exist).
This makes the "membership" relation (which I will note as \mem as
opposed to the usual 'in') a bit confusing: if A is a set and B are
sets, then A \mem AB, but not B \mem AB. On the other hand, if A is a
set and B is a set, then B \mem AB. If we use standard set membership,
then A is not in AB, because AB is an ordered set - it contains
"something like" <A,1>, but not A itself. In fact the relation A \mem
B when A are sets implies that B = {A}.
This renders such statements as "A = {x | {} \mem x}" non-starters: A
is not an ordered set, nor a singleton set, so it is not likely to be
an object in this system; and certainly not a /unique/ object in the
system.
Because another related difficulty that arises is that we fail to have
(A = B) iff (for all x, x \mem A iff x \mem B); so if we are /not/
working in a finite world, it's not 100% clear how "=" should be
defined without resorting to some standard set theoretic model.
How complicated this can get depends on how we are to understand rule
(3) "There are no other ways to form (sets or) a set ".
At its simplest, it could mean "this is the last axiom regarding set
formation", which might as well be "this axiom intentionally left
blank".
It could also mean "every set/sets is one that was formed in this
way", or more exactly: "If A is a set, then A = {} or exists a unique
B such that A = {B}. If A are sets, then exists unique C, D such that
C is a set and (D is a set or D are sets) and A = CD".
Neither of these interpretations would rule out the existence of an
infinite "sets" S, for which A \mem S for more than any finite number
of distinct "A"s. Nor would they rule out the existence of A with A =
{A}.
Finally (and this is what I think he /meant/ to say), it could mean
"If A is a set or A are sets, there exists a finite series of
applications of the above rules which produce A". But that's much less
interesting!
Cheers - Chas
Daniel J. Greenhoe
> I have run into a dilemma defining the Cartesian product. I would
> appreciate it if someone could help me out.
>
> 1. I define the Cartesian product of AxB as
> AxB = { (a,b) | a in A <and> b in B }
> where ... <and> is the "logical AND" operator
>
> 2. But this definition of the Cartesian product depends on the binary
> operator <and>. A binary operator (or a function), is a special case
> of a relation. ...
>
> 3. But the definition of a relation R is that it is a subset of a
> Cartesian product.
Here is the solution that I am currently using to my previous dilemma.
It starts with the concept of a set and from there constructs
relations, the Cartesian product, functions, logical operators, and
set operations:
1. Define an ordered pair as
(a,b) = { {a}, {a,b} } (Kuratowski's definition).
2. Define a relation R as any set of ordered pairs:
R = { (x1,y1), (x2,y2), ... }
(note: before I defined a relation as a subset of a Cartesian
product).
3. Define a "relational and"
<rand> = { ({f,f},f), ({t,f},f), ({t,t},t) }.
This is a temporary definition used for the construction of the
Cartesian product. It is not the "logical and" --- this definition
will come after we have defined functions. The relational and is
constructed using the concept of the relation only. Note also the
definition of <rand> does not need the ordered pair ({f,t},f) because
this is equivalent to the ordered pair ({t,f},f).
4. Define the Cartesian product XxY as
XxY = { (x,y) | x in X <rand> y in Y }.
5. Proposition: Any relation R:X-->Y is a subset of the Cartesian
product XxY.
Note: before this was essentially the definition of a relation rather
than a proposition.
6. Define a function f:X-->Y as a relation R:X-->Y that satisfies the
following:
(x,y1), (x,y2) in f ==> y1=y2
7. Define the standard logical operators as functions:
<not> = { (t,f), (f,t) }
<and> = { ({f,f},f), ({t,f},f), ({t,t},t) }.
<or> = { ({f,f},f), ({t,f},t), ({t,t},t) }.
<xor> = { ({f,f},f), ({t,f},t), ({t,t},f) }.
8. Define the standard set operators for subsets of a set X:
A <union> B = { x in X | x in A <and> x in B }
A <intersect> B = { x in X | x in A <or> x in B }
A <symmetric difference> B = { x in X | x in A <xor> x in B }
A <difference> B = {x in X | x in A <and> <not>(x in B) }
<complement> A = {x in X | <not>(x in A) }
And so that is it --- it is a way to "boot up" a reasonably useful
family of mathematics starting from the humble set without the
circular definition I had before.
(
note: this construction is also available in more detail and with
better typesetting (by LaTeX) at http://banyan.cm.nctu.edu.tw/~dgreenhoe/fsd/construct.pdf
)
I am certainly not arguing that this is the only way to construct
functions and set operations. But I will make the following notes:
1. Note the strong similarities between the construction here and the
previous post in this thread by "schoenfeld....@gmail.com" at
http://groups.google.com/group/sci.logic/msg/570926f7642974bc
where he proposed
> 1. Declare the existence of a "relation" .
> 2. Define the Cartesian product using the relation 'and'.
> 3. Define a relation in terms of the Cartesian product.
> 4. Define 'and'.
2. Using Halmos' axiomatic naive set theory, we could invoke the
"axiom of specification" to just make the dilemma go away. This axiom
basically says that if you can _specify_ a set using any intelligible
means, then that set is valid. In other words, we would not need to go
to the trouble of defining the "relational and" <rand>; instead we
could just define the Cartesian product as
XxY = { (x,y) | x in X and y in Y }.
Because English is an intelligible language and the word "and" has a
specific meaning in English, by this axiom, we would be OK to define
XxY as just given.
3. However, as expedient as axiomatic systems are, I think the axioms
need to be chosen in such a way as to avoid, to such an extent as
possible, circular arguments. And for this reason, I would prefer to
not invoke such an axiom to solve this particular dilemma, but would
rather use a construction such as the one given above.
4. Others may argue that the whole issue is not a dilemma at all and
doesn't even really call for clever axioms because, as already
mentioned, "and" is well understood in English. But my objection here
is that using the English "and" is just an _implicit_ use of the
_explicit_ "logical and"; and in using the English "and" in this way,
we would only be attempting to cover up circular reasoning.
Dan Greenhoe (original poster)
On Aug 13, 1:15 pm, "Daniel J. Greenhoe" <dgreen...@yahoo.com> wrote:
> I have run into a dilemma defining the Cartesian product. I would
> appreciate it if someone could help me out.
>
> 1. I define the Cartesian product of AxB as
> AxB = { (a,b) | a in A <and> b in B }
> where (a,b) is an ordered pair and
> <and> is the "logical AND" operator
>
> 2. But this definition of the Cartesian product depends on the binary
> operator <and>. A binary operator (or a function), is a special case
> of a relation. So before I can define the Cartesian product, I need to
> first define a relation.
>
> 3. But the definition of a relation R is that it is a subset of a
> Cartesian product.
>
> 4. So, in the end, my definition of Cartesian product depends on the
> definition of the Cartesian product --- hence my dilemma.
>
> Stated another way,
> AxB = { (a,b) | a in A <and> b in B }
> but <and> is of the form
> <and>:{T,F}x{TxF} --> {T,F}
> ... and so the definition of the Cartesian product depends on an
> operator defined with the help of a Cartesian product.
>
> Is there any standard way out of this dilemma? Any classic references
> would be especially appreciated.
>
> Many thanks in advance,
> DanGreenhoe
Daniel J. Greenhoe
what Robert Israel had to say (2007 August 13) also has strong
similarity to the proposed solution:
> "and" doesn't require the
> definition of relation or cartesian product in general, it
> requires a specific relation on finite sets that can be
> written out explicitly.
Many thanks to Robert Israel for this insightful contribution.
Dan Greenhoe (original poster)
> better typesetting (by LaTeX) athttp://banyan.cm.nctu.edu.tw/~dgreenhoe/fsd/construct.pdf
george
> Here is the solution that I am currently using to my previous dilemma.
Not really. This is not a solution to anything, simply because this
is
preposterous. I don't mean that dismissively: I mean it literally.
It has something that is supposed to come "after" coming "before".
It doesn't respect the right order of the definitions ("pre" meaning
before,
"post" meaning after).
> It starts with the concept of a set and from there constructs
> relations, the Cartesian product, functions, logical operators, and
> set operations:
You CAN'T start with sets and then construct logical operators.
You HAVE to START with logical operators.
Gee, I hope that clears THAT up.
herbzet
"Daniel J. Greenhoe" wrote:
>
>
> 3. Define a "relational and"
> <rand> = { ({f,f},f), ({t,f},f), ({t,t},t) }.
Do you mean
<rand> = { ({f},f), ({t,f},f), ({t},t) } ?
--
hz
I left out sci.math.num-analysis because my news-server
only allows posting to 3 NGs. Hope you see this.