On 2008-04-08, in sci.math, Mariano Suárez-Alvarez
wrote:
That was precisely my point. There is no
point in choosing a particular representation,
because all representations are isomorphic
and nothing one wants to do will depend
on the representation chosen. Hence, it
really does not make any sense to ask
`what is a number'.

So you think those who ask "What is a number?" are
asking "What
representation for numbers should we choose?"?

Given that meaning is independent of language,

--
Aatu Koskensilta (aatu.koskensilta@xortec.fi)

"Wovon man nicht sprechen kann, daruber muss man
schweigen"
- Ludwig Wittgenstein, Tractatus
s Logico-Philosophicus

Calan9400 wrote:
On Apr 8, 11:15 am, Julio Di Egidio
ju...@diegidio.name> wrote:
Pardon me, but IMHO you risk here not to be
thinking "out of the box".

I really don't know what you mean here. I'm willing
to entertain any
number of geometries or mathematical theories given
that they meet
some basic logical requirements, eg. consistency of
axioms and etc.
My requirements are pretty low as far as I'm
concerned, given that the
math is logically tight.

Note: When I say arbitrary, I mean there is no
mathematical reason to
be considering one axiom over an another axiom;
excluding logical
considerations.  It's given the real world will
probably intrude into
my considerations as to whether a mathematical
construction has any
useful purpose.

Yes, and the point I am trying to make is: you have there just shown the very criterion to make our logic "tighter" without "arbitrariness": "purpose", that being a gain in applicability, so usefulness, etc. etc.



This definitely wandering off topic now.

On the edge indeed! :)

Julio





Cheers,
Calan- Hide quoted text -

- Show quoted text -- Hide quoted text -

- Show quoted text -

On Apr 9, 10:07 am, Julio Di Egidio
ju...@diegidio.name> wrote:
Calan9400 wrote:
On Apr 8, 11:15 am, Julio Di Egidio
ju...@diegidio.name> wrote:
Pardon me, but IMHO you risk here not to be
thinking "out of the box".

I really don't know what you mean here. I'm
willing
to entertain any
number of geometries or mathematical theories
given
that they meet
some basic logical requirements, eg. consistency
of
axioms and etc.
My requirements are pretty low as far as I'm
concerned, given that the
math is logically tight.

Note: When I say arbitrary, I mean there is no
mathematical reason to
be considering one axiom over an another axiom;
excluding logical
considerations.  It's given the real world will
probably intrude into
my considerations as to whether a mathematical
construction has any
useful purpose.

Yes, and the point I am trying to make is: you have
there just shown the very criterion to make our logic
"tighter" without "arbitrariness": "purpose", that
being a gain in applicability, so usefulness, etc.
etc.

This definitely wandering off topic now.

On the edge indeed! :)

Julio

Cheers,
Calan- Hide quoted text -

- Show quoted text -

I think I basically agree with you on most points.
However, strictly
speaking, within restricted mathematical world, I
believe that
whatever "purpose" that a mathematical structure may
have it doesn't
change its ontological status, however, useful a
branch of math maybe
to us. It seems the flexability of an axiomatic
system is ultimately
what determines the useful of any set of axioms in
the wider world. I
can think of many instances where the "lack of
purpose" of a branch of
math was roundly critized only to be found it had
some application not
foreseen at its time of creation.

Thanks for your posts and for contributions from
other posters on this
thread.
I have enjoyed reading them.

Calan

Is the empty set a number?

jonas.thornv...@hotmail.com schrieb:
Is the empty set a number?

Historical, numbers are counts of objects/unities.
E.g. an engine
consists of 4 cylinders, 325 screws, and so on. An
engine consists of
zero apples.
In a sense the zero stands for the absence of
something and it defines
a restclass of objects, namely the objects which are
not part of a
totality which is considered.

In this elementary view there is always a number and
an unity and we
can say there is a multiplicative relation between
number and unity:
4 times of cylinders
325 times of screws
0 times of apples

Now 0 * something is nothing. So, in the case of zero
the unity
vanishes. The zero indicates absence, nothing.
In difference to that, in set theory there is always
something, at
least the empty set.
In this sense the numbers are more elementar than the
sets, I think.

The zero indicates absence, nothing.

Albrecht wrote:



That is, we cannot *directly* (analogically vs.
digi-logically) _indicate_ "absence" until within the
domain of hard (digi-logical) sciences.

The zero indicates absence, nothing.

It might now be apparent how I would say that this
sentence is incongruent and, strictly speaking,
incorrect.

Sorry, to clarify a bit:

Julio Di Egidio wrote:

Albrecht wrote:



That is, we cannot *directly* (analogically vs.
digi-logically) _indicate_ "absence" until within
the
domain of hard (digi-logical) sciences.

I meant to cover both analogical vs. digital and
analogical vs. strictly logical. I guess I might have
better ended up with something like "metalogical"...


The zero indicates absence, nothing.

It might now be apparent how I would say that this
sentence is incongruent and, strictly speaking,
incorrect.

There cannot be a direct (non metalogical)
construction for "nothing" of _anything_.

On Apr 7, 2:26 pm, jonas.thornv...@hotmail.com wrote:.

Can you tell me if two empty sets equals one empty set i would be
greatful for an answer.

By the Axiom of Extensionality in ZFC, all empty sets are equal.

It appears that the OP has one of two concerns:

1. Is zero really a number?
2. Is the empty set really a set?

And these two questions have distinct answers.

1. Many of the others have already discussed to which objects
we assign the property of numberhood in this thread. But the
bottom line is that any ring must have an additive identity,
and that element is zero. Indeed, every semigroup can be made
into a monoid by simply attaching such a neutral element.

To standard mathematicians, the unqualified word "number"
refers to an element of C, the set of complex numbers. Thus
zero is a number since 0eC.

One might ask what the set of all numbers is to the OP. If
we let J (for Jonas) be the set of all numbers that the OP
accepts, then clearly J is a proper subset of C, since 0eC
but yet ~0eJ.

So what numbers are in J? I doubt that the OP accepts the
existence of negative numbers, since then one would have
to wonder what -1+1 is.

Historically the Greeks and Romans,
just like the OP, denied the existence of zero, but the
Sumerians, Mayans, and Hindus all had symbols for zero.

The largest set one can have without the existence of zero
is the set of unsigned reals R+, which is labeled by a
script P in Metamath. Notice how Metamath develops the
unsigned fractions and unsigned reals via Dedekind cuts
well before developing zero and signed numbers. This
matches the historical development, where Pythagoras
discovered sqrt(2) over 2000 years before Cardano
introduced negative numbers.

Kronecker said that "God created the integers" -- but it's
uncertain whether he meant the positive natural numbers or
the signed integers. But of course, one trick is simply
to let a signed integer simply be an equivalence class of
ordered pairs of natural numbers, as is usually done, so
that 0 = {(1,1),(2,2),(3,3),...}.

2. But if you deny the existence of the empty set, then
you have a deeper problem than if you merely deny the
numberhood of zero. For the Axiom of Foundation (AKA
Regularity) states that every set is based on the empty
set, in that every set has the empty set as an element of
its transitive closure. So if you don't want an empty set
then you must deny Foundation/Regularity.

One sci.logic poster, Zuhair, also wanted to come up with
a set theory once in which there is no empty set. But
unfortunately, he was not able to come up with such a
theory in a way that it would be consistent.

On 9 Apr, 00:05, lwal...@lausd.net wrote:





On Apr 7, 2:26 pm, jonas.thornv...@hotmail.com wrote:.

Can you tell me if two empty sets equals one empty set i would be
greatful for an answer.

By the Axiom of Extensionality in ZFC, all empty sets are equal.

It appears that the OP has one of two concerns:

1. Is zero really a number?
2. Is the empty set really a set?

And these two questions have distinct answers.

1. Many of the others have already discussed to which objects
we assign the property of numberhood in this thread. But the
bottom line is that any ring must have an additive identity,
and that element is zero. Indeed, every semigroup can be made
into a monoid by simply attaching such a neutral element.

To standard mathematicians, the unqualified word "number"
refers to an element of C, the set of complex numbers. Thus
zero is a number since 0eC.

One might ask what the set of all numbers is to the OP. If
we let J (for Jonas) be the set of all numbers that the OP
accepts, then clearly J is a proper subset of C, since 0eC
but yet ~0eJ.

So what numbers are in J? I doubt that the OP accepts the
existence of negative numbers, since then one would have
to wonder what -1+1 is.

Well that would be dent to no accept negative numbers of course i do.
What i say is that the result from the transaction 1-1 is not a
number, and that is quite another issue. If you accept this and build
your numbersystem and architecture supporting this. You do not have to
think about division by zero. And it still will give perfectly valid
result for any calculation.

Historically the Greeks and Romans,
just like the OP, denied the existence of zero, but the
Sumerians, Mayans, and Hindus all had symbols for zero.

Yes i am aware of that, and i think it ís a problem inherited by our
logic to want to put number to nothing. Because nothing really do not
have a *value*.

Representing a number as base dependent postional value sets works
very good for any base.

Here is some expamples in "DECIMAL" base using positional value
representation and as you can guess 0 is missing representation in
this system.

-2000500009=-{[A,2][5,5][1,9]}
9000020005={[A,9][5,2][1.6]}
0,50000000009={[-1,5][-B,9]}
-1.90005=-oooops-{[1,1][-1,9][-5,5]}





The largest set one can have without the existence of zero
is the set of unsigned reals R+, which is labeled by a
script P in Metamath. Notice how Metamath develops the
unsigned fractions and unsigned reals via Dedekind cuts
well before developing zero and signed numbers. This
matches the historical development, where Pythagoras
discovered sqrt(2) over 2000 years before Cardano
introduced negative numbers.

Kronecker said that "God created the integers" -- but it's
uncertain whether he meant the positive natural numbers or
the signed integers. But of course, one trick is simply
to let a signed integer simply be an equivalence class of
ordered pairs of natural numbers, as is usually done, so
that 0 = {(1,1),(2,2),(3,3),...}.

2. But if you deny the existence of the empty set, then
you have a deeper problem than if you merely deny the
numberhood of zero. For the Axiom of Foundation (AKA
Regularity) states that every set is based on the empty
set, in that every set has the empty set as an element of
its transitive closure. So if you don't want an empty set
then you must deny Foundation/Regularity.

One sci.logic poster, Zuhair, also wanted to come up with
a set theory once in which there is no empty set. But
unfortunately, he was not able to come up with such a
theory in a way that it would be consistent.

You try this- Dölj citerad text -

- Visa citerad text -- Dölj citerad text -

- Visa citerad text -

On 9 Apr, 00:05, lwal...@lausd.net wrote:





On Apr 7, 2:26 pm, jonas.thornv...@hotmail.com wrote:.

Can you tell me if two empty sets equals one empty set i would be
greatful for an answer.

By the Axiom of Extensionality in ZFC, all empty sets are equal.

It appears that the OP has one of two concerns:

1. Is zero really a number?
2. Is the empty set really a set?

And these two questions have distinct answers.

1. Many of the others have already discussed to which objects
we assign the property of numberhood in this thread. But the
bottom line is that any ring must have an additive identity,
and that element is zero. Indeed, every semigroup can be made
into a monoid by simply attaching such a neutral element.

To standard mathematicians, the unqualified word "number"
refers to an element of C, the set of complex numbers. Thus
zero is a number since 0eC.

One might ask what the set of all numbers is to the OP. If
we let J (for Jonas) be the set of all numbers that the OP
accepts, then clearly J is a proper subset of C, since 0eC
but yet ~0eJ.

So what numbers are in J? I doubt that the OP accepts the
existence of negative numbers, since then one would have
to wonder what -1+1 is.

Well that would be dent to no accept negative numbers of course i do.
What i say is that the result from the transaction 1-1 is not a
number, and that is quite another issue. If you accept this and build
your numbersystem and architecture supporting this. You do not have to
think about division by zero. And it still will give perfectly valid
result for any calculation.

Historically the Greeks and Romans,
just like the OP, denied the existence of zero, but the
Sumerians, Mayans, and Hindus all had symbols for zero.

Yes i am aware of that, and i think it ís a problem inherited by our
logic to want to put number to nothing. Because nothing really do not
have a *value*.

Representing a number as base dependent postional value sets works
very good for any base.

Here is some expamples in "DECIMAL" base using positional value
representation and as you can guess 0 is missing representation in
this system.

-2000500009=-{[A,2][5,5][1,9]}
9000020005={[A,9][5,2][1.6]}
0,50000000009=ooops again{[-1,5][-B,9]}should of course be A1,9
-1.90005=-{[1,1][-1,9][-5,5]}





The largest set one can have without the existence of zero
is the set of unsigned reals R+, which is labeled by a
script P in Metamath. Notice how Metamath develops the
unsigned fractions and unsigned reals via Dedekind cuts
well before developing zero and signed numbers. This
matches the historical development, where Pythagoras
discovered sqrt(2) over 2000 years before Cardano
introduced negative numbers.

Kronecker said that "God created the integers" -- but it's
uncertain whether he meant the positive natural numbers or
the signed integers. But of course, one trick is simply
to let a signed integer simply be an equivalence class of
ordered pairs of natural numbers, as is usually done, so
that 0 = {(1,1),(2,2),(3,3),...}.

2. But if you deny the existence of the empty set, then
you have a deeper problem than if you merely deny the
numberhood of zero. For the Axiom of Foundation (AKA
Regularity) states that every set is based on the empty
set, in that every set has the empty set as an element of
its transitive closure. So if you don't want an empty set
then you must deny Foundation/Regularity.

One sci.logic poster, Zuhair, also wanted to come up with
a set theory once in which there is no empty set. But
unfortunately, he was not able to come up with such a
theory in a way that it would be consistent.

You try this- Dölj citerad text -

- Visa citerad text -- Dölj citerad text -

- Visa citerad text -

Julio Di Egidio wrote:
Sorry, to clarify a bit:

Julio Di Egidio wrote:

Albrecht wrote:

That is, we cannot *directly* (analogically vs.
digi-logically) _indicate_ "absence" until within
the
domain of hard (digi-logical) sciences.

I meant to cover both analogical vs. digital and
analogical vs. strictly logical. I guess I might have
better ended up with something like "metalogical"...

The zero indicates absence, nothing.

It might now be apparent how I would say that this
sentence is incongruent and, strictly speaking,
incorrect.

There cannot be a direct (non metalogical)
construction for "nothing" of _anything_.

Indeed, we could write _it_ "absence", "zero", or "empty", and even "silence". In any case, the very fact we write (or say) _it_, makes us write (or say) the _metalogical_ Zero.

BTW, here I guess we encounter an implicit definition for "natural". From the above trivial exclusiveness of Zero, _naturally_ follows that _natural_ isn't but the _ultimate trivial_ (never uninteresting).

Julio- Dölj citerad text -

- Visa citerad text -

On 10 Apr, 02:36, Julio Di Egidio <ju...@diegidio.name> wrote:





Julio Di Egidio wrote:
Sorry, to clarify a bit:

Julio Di Egidio wrote:

Albrecht wrote:

That is, we cannot *directly* (analogically vs.
digi-logically) _indicate_ "absence" until within
the
domain of hard (digi-logical) sciences.

I meant to cover both analogical vs. digital and
analogical vs. strictly logical. I guess I might have
better ended up with something like "metalogical"...

The zero indicates absence, nothing.

It might now be apparent how I would say that this
sentence is incongruent and, strictly speaking,
incorrect.

There cannot be a direct (non metalogical)
construction for "nothing" of _anything_.

Indeed, we could write _it_ "absence", "zero", or "empty", and even "silence". In any case, the very fact we write (or say) _it_, makes us write (or say) the _metalogical_ Zero.

BTW, here I guess we encounter an implicit definition for "natural". From the above trivial exclusiveness of Zero, _naturally_ follows that _natural_ isn't but the _ultimate trivial_ (never uninteresting).

Julio- Dölj citerad text -

- Visa citerad text -

Many layman people like me, actually beleive that there is zeroes in a
number like 10,100,1000,10 000
when there actually is none, the zeroes are only misnomers inherited
by the base and the imagined number of zero. If we had only one hand
some stupid man probably come up with the
0,1,2,3,4,10,11,12,13,14,20,21,22,23,24 and so on, so of course the
zeroes only is inherited property sign. It is a bit retarded to begin
wiht to call anything something +0.

It should of course be something +1.

And it really can't be that hard to see.......- Dölj citerad text -

- Visa citerad text -

On 10 Apr, 04:48, jonas.thornv...@hotmail.com wrote:



On 9 Apr, 00:05, lwal...@lausd.net wrote:

On Apr 7, 2:26 pm, jonas.thornv...@hotmail.com wrote:.

Can you tell me if two empty sets equals one empty set i would be
greatful for an answer.

By the Axiom of Extensionality in ZFC, all empty sets are equal.

It appears that the OP has one of two concerns:

1. Is zero really a number?
2. Is the empty set really a set?

And these two questions have distinct answers.

1. Many of the others have already discussed to which objects
we assign the property of numberhood in this thread. But the
bottom line is that any ring must have an additive identity,
and that element is zero. Indeed, every semigroup can be made
into a monoid by simply attaching such a neutral element.

To standard mathematicians, the unqualified word "number"
refers to an element of C, the set of complex numbers. Thus
zero is a number since 0eC.

One might ask what the set of all numbers is to the OP. If
we let J (for Jonas) be the set of all numbers that the OP
accepts, then clearly J is a proper subset of C, since 0eC
but yet ~0eJ.

So what numbers are in J? I doubt that the OP accepts the
existence of negative numbers, since then one would have
to wonder what -1+1 is.

Well that would be dent to no accept negative numbers of course i do.
What i say is that the result from the transaction 1-1 is not a
number, and that is quite another issue. If you accept this and build
your numbersystem and architecture supporting this. You do not have to
think about division by zero. And it still will give perfectly valid
result for any calculation.

Historically the Greeks and Romans,
just like the OP, denied the existence of zero, but the
Sumerians, Mayans, and Hindus all had symbols for zero.

Yes i am aware of that, and i think it ís a problem inherited by our
logic to want to put number to nothing. Because nothing really do not
have a *value*.

Representing a number as base dependent postional value sets works
very good for any base.

Here is some expamples in "DECIMAL" base using positional value
representation and as you can guess 0 is missing representation in
this system.

-2000500009=-{[A,2][5,5][1,9]}
9000020005={[A,9][5,2][1.6]}

 0,50000000009=ooops again{[-1,5][-B,9]}should of course be A1,9



-1.90005=-{[1,1][-1,9][-5,5]}

The largest set one can have without the existence of zero
is the set of unsigned reals R+, which is labeled by a
script P in Metamath. Notice how Metamath develops the
unsigned fractions and unsigned reals via Dedekind cuts
well before developing zero and signed numbers. This
matches the historical development, where Pythagoras
discovered sqrt(2) over 2000 years before Cardano
introduced negative numbers.

Kronecker said that "God created the integers" -- but it's
uncertain whether he meant the positive natural numbers or
the signed integers. But of course, one trick is simply
to let a signed integer simply be an equivalence class of
ordered pairs of natural numbers, as is usually done, so
that 0 = {(1,1),(2,2),(3,3),...}.

2. But if you deny the existence of the empty set, then
you have a deeper problem than if you merely deny the
numberhood of zero. For the Axiom of Foundation (AKA
Regularity) states that every set is based on the empty
set, in that every set has the empty set as an element of
its transitive closure. So if you don't want an empty set
then you must deny Foundation/Regularity.

One sci.logic poster, Zuhair, also wanted to come up with
a set theory once in which there is no empty set. But
unfortunately, he was not able to come up with such a
theory in a way that it would be consistent.

You try this- Dölj citerad text -

- Visa citerad text -- Dölj citerad text -

- Visa citerad text -- Dölj citerad text -

- Visa citerad text -- Dölj citerad text -

- Visa citerad text -

Julio Di Egidio wrote:
Sorry, to clarify a bit:

Julio Di Egidio wrote:

Albrecht wrote:

That is, we cannot *directly* (analogically vs.
digi-logically) _indicate_ "absence" until within
the
domain of hard (digi-logical) sciences.

I meant to cover both analogical vs. digital and
analogical vs. strictly logical. I guess I might have
better ended up with something like "metalogical"...

The zero indicates absence, nothing.

It might now be apparent how I would say that this
sentence is incongruent and, strictly speaking,
incorrect.

There cannot be a direct (non metalogical)
construction for "nothing" of _anything_.

Indeed, we could write _it_ "absence", "zero", or "empty", and even "silence". In any case, the very fact we write (or say) _it_, makes us write (or say) the _metalogical_ Zero.

BTW, here I guess we encounter an implicit definition for "natural". From the above trivial exclusiveness of Zero, _naturally_ follows that _natural_ isn't but the _ultimate trivial_ (never uninteresting).

Julio- Dölj citerad text -

- Visa citerad text -

jonas.thornv...@hotmail.com> wrote in message

news:9fd1146e-a546-4cb9-9cf6-9f5a7d744c6d@u36g2000prf.googlegroups.com...
On 7 Apr, 18:13, Randy Poe <poespam-t...@yahoo.com> wrote:

On Apr 7, 12:08 pm, jonas.thornv...@hotmail.com wrote:

Well i want you to prove that 0 is a number

OK. Let's start with your definition of "number"
and your definition of "0". What do you mean
by "0"? What do you mean by proving that
something is a number?

- Randy

I do not mean anything by 0 i have been taught that 0 is a number of
no "value"
A number is something that is necessary to represent a "value"
For me is absense of a value not a number, if x=0 4x seems to be a
nonsensical expression.

********************
In answer to the question in your subject line, the empty set is not a
number. The empty set is a set, and whilst you haven't defined "number" I
would assume that you do not consider it as a set - this seems incompatible
with your "definition" of a "number" as a "value".

What logicians have done is to show that the structure which commences with
{} and creates other sets using the rule S(x) = x U {x} is isomorphic to the
definition of numbers given by Peano where S(y) = y+1.

Then, by proving results about sets, we can prove results about numbers.

Note that this definition of S(x) = x U {x} is arbitrary; other mappings are
possible,
including S(x) = {x}.

Note that this isomorphism also does not require {} <-> 0.

The same thing would work if we defined the isomorphism as

{} <-> 1 (with the disadvantage we have no direct way of expressing zero)

or

{{}} <-> 0 (with the disadvantage that we have to type additional brace
characters all the time)

So for reasons of simplicity, the most commonly used isomorphism is based
upon

{} <-> 0

So no, {} is not a number. It can be associated with a number (as defined by
Peano) through a straightforward isomorphism, but equally other sets could
be mapped to zero, or {} could be mapped to some other number, and the
isomorphism would still hold. Its a logical but arbitrary choice that we
usually create the isomorphism using {} to represent 0.

Hope this helps

Peter Webb