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Christian Stapfer
science forum beginner

Joined: 05 Jun 2006
Posts: 15

Posted: Thu Jul 20, 2006 6:55 pm    Post subject: Re: The list of all natural numbers don't exist

"Patricia Shanahan" <pats@acm.org> wrote in message
 Quote: albstorz@gmx.de wrote: Virgil schrieb: In article <_eydndK1Kp9-jyLZnZ2dnUVZ_uudnZ2d@comcast.com>, "Russell Easterly" wrote: "Christian Stapfer" wrote in message news:99646\$44befd7b\$54497364\$21350@news.hispeed.ch... Peter Niessen wrote: Am Mon, 17 Jul 2006 15:24:56 -0700 schrieb Russell Easterly: If N is the set of all natural numbers, the number of subsets of N is the powerset of N. Does this mean we need an uncountable number of axioms to define all the subsets of N? Um, it depends on what you mean by "define"... \forall m P(m) is a Set. Short enought? That axiom does not give you *all* the subsets of NI. It just gives you the "existence" of the set of all subsets of N. Which is rather an empty shell of a mere tag, in and of itself. If your set theory is consistent at all, it admits of having a merely countable interpretation (Skolem's paradox). So there is something illusory about your suggestion that the powerset axiom does the job of *really* capturing an "uncountable" P(NI) in its entirety. Nearly all the elements of P(N) can't be defined. Nearly all real numbers are inaccessible, too. But not the ones we need. Who needs the ones which are inaccesible? AS Anyone who wants a system simple enough to allow general statements?

Maybe, who knows. But where is your *proof* of that
assertion? (Besides: you only need to be able to
talk *as*if* you had those inaccessible ones, you
to not actually *need*, i.e. "access" them...)

Regards,
Christian
Patricia Shanahan
science forum Guru Wannabe

Joined: 13 May 2005
Posts: 214

Posted: Thu Jul 20, 2006 7:13 pm    Post subject: Re: The list of all natural numbers don't exist

Christian Stapfer wrote:
 Quote: "Patricia Shanahan" wrote in message news:OwKvg.2282\$157.260@newsread3.news.pas.earthlink.net... albstorz@gmx.de wrote: Virgil schrieb: In article <_eydndK1Kp9-jyLZnZ2dnUVZ_uudnZ2d@comcast.com>, "Russell Easterly" wrote: "Christian Stapfer" wrote in message news:99646\$44befd7b\$54497364\$21350@news.hispeed.ch... Peter Niessen wrote: Am Mon, 17 Jul 2006 15:24:56 -0700 schrieb Russell Easterly: If N is the set of all natural numbers, the number of subsets of N is the powerset of N. Does this mean we need an uncountable number of axioms to define all the subsets of N? Um, it depends on what you mean by "define"... \forall m P(m) is a Set. Short enought? That axiom does not give you *all* the subsets of NI. It just gives you the "existence" of the set of all subsets of N. Which is rather an empty shell of a mere tag, in and of itself. If your set theory is consistent at all, it admits of having a merely countable interpretation (Skolem's paradox). So there is something illusory about your suggestion that the powerset axiom does the job of *really* capturing an "uncountable" P(NI) in its entirety. Nearly all the elements of P(N) can't be defined. Nearly all real numbers are inaccessible, too. But not the ones we need. Who needs the ones which are inaccesible? AS Anyone who wants a system simple enough to allow general statements? Maybe, who knows. But where is your *proof* of that assertion? (Besides: you only need to be able to talk *as*if* you had those inaccessible ones, you to not actually *need*, i.e. "access" them...)

If I'm allowed to talk as if they existed, what consequences, if any,
does their non-existence have?

Patricia
zuhair
science forum Guru

Joined: 06 Feb 2005
Posts: 533

Posted: Thu Jul 20, 2006 7:46 pm    Post subject: Re: The list of all natural numbers don't exist

Tez wrote:

Ok , let us stop talking about Cantor's cardinality since this was not
the original subject.

Definitions:

Definitions:

For sets A and B.

1) f:A->B is called "one valued" if for every x in A there is one and
only one y=f(x) , in B.
so if n e A and m e A, then if n=m then f(n) must equal f(m). But if
n<>m then is doesn't imply that f(n) should be inequal to f(m).
Similarly if f(n) <> f(m) then n <> m is a must.
While if f(n) = f(m) then m=n is not a must.

2) f:A->B is called "Bijective" if it is one valued and its inverse
f^(-1):B->A is one valued.

3) f:A->B is called "converjective " if it is one valued and
surjective.

4) f:A->B is called "Generational" if it is converjective and
mentioned explicitely in the definition of B from A.

So if B is explicitely defined as B = { y: y=f(x) , f:A->B }
and if f:A->B is converjective then it is generational of B from A.

and the set B is expressed as B_f , it means set B as generated by
function f.

5) z-card A_h = z-card B_f , if there exist f:A-B that is generational
of B
from A_h. and that has a one valued inverse f^(-1):B->A .

6) z-card A_h> z-card B_f, if there exist f:A->B that is generational
of B
from A _h and that has an inverse f^(-1):B->A that is not one valued.

7) z-card A_f < z-card B_g, if there exist f:B->A that is generational
of A
from B_g and that has an inverse f^(-1): A->B that is not one valued.

8)Axiom 1) for sets A and B, among all the converjective functions
between them, one should be generational.

9) Assumption 1) If the set of all natural numbers N is not defined
explicitely by a generational function f, then it is to be considered
as the set which is primiarly derived from the Peano's five premisses
and it is to be symbolized as N' which means Primary natural numbers
set.

10) Assumption 2) for sets A and B , if the generating fucntions of A
and B, or at least the generating function of one of them from the
other, Is not defined, and there are converjective functions from one
of them to the other that differe in the existance of a one valued
inverse function ( for example g:A->B were g^(-1):B->A is not one
valued, and f:A->B were f^(-1): B->A is one valued ). Then It is the
bijective function that will be considered as the generational
function.

11) If A_f is not identical to B_f , then either A is a proper subset
of B or B is a proper subset of A. Provided of course that f = f. Then
sets A and B are called different equigenerational sets.

Example, the even numbers in the primary natural numbers set N',
denoted as E'( the primary even numbers set ).

12) if the generating funciton of set A is f(x)=x , f:A_g->A, then A_f
= A_g

and A_f is called the copy of A_f , this means that g=f.

Any suggestions.

Zuhair
Peter Niessen
science forum beginner

Joined: 05 Jun 2006
Posts: 19

Posted: Thu Jul 20, 2006 8:19 pm    Post subject: Re: The list of all natural numbers don't exist

Am Thu, 20 Jul 2006 05:50:17 +0200 schrieb Christian Stapfer:

 Quote: Peter Niessen wrote: Am Mon, 17 Jul 2006 15:24:56 -0700 schrieb Russell Easterly: If N is the set of all natural numbers, the number of subsets of N is the powerset of N. Does this mean we need an uncountable number of axioms to define all the subsets of N? Um, it depends on what you mean by "define"... \forall m P(m) is a Set. Short enought? That axiom does not give you *all* the subsets of NI.

No Problem. I have only said:
If m a Set than is P(m) also a Set. Thinking about konsquenzes is an other
Story.
--
Mit freundlichen Grüssen
Peter Nießen
Markus2
science forum beginner

Joined: 20 Jul 2006
Posts: 1

Posted: Thu Jul 20, 2006 8:27 pm    Post subject: Re: The list of all natural numbers don't exist

"Albrecht"
 Quote: Assumption: The list of all natural numbers exists.

lol wasn das fürn Mörderthread hier =)
David R Tribble
science forum Guru

Joined: 21 Jul 2005
Posts: 1005

Posted: Thu Jul 20, 2006 11:22 pm    Post subject: Re: The list of all natural numbers don't exist

Zuhair wrote:
 Quote: My objection to Cantor's cardinality is that it doesn't take into consideration how a set is generated, that is my main objection, you thought I objected to Cantor's cardinality because it is counter-intuitive, and this is not may main objection, in reality z-cardinality is as counter-intuitive as Cantor's cardinality, so I will address it again , my main objection is that Cantor's cardinality doesn't take how a set is generated into consideration.

And how is a set "generated"?

I think you mean to say that your Z-cardinality is some kind of
comparison between the enumeration of the members of two sets.
You say the members of set B are "generated" from the members
of set A, and that this "generating" relation defines the relative
Z-cardinalities of both sets.

In actually, this is nothing more than comparing how the members of
two sets are counted as they are paired (mapped) with each other,
giving you some sort of relative "measure" of the sets with respect
to that particular mapping. It's not cardinality, nor set "size", but
something related to denumerating ("generating") the elements.
David R Tribble
science forum Guru

Joined: 21 Jul 2005
Posts: 1005

Posted: Thu Jul 20, 2006 11:24 pm    Post subject: Re: The list of all natural numbers don't exist

Peter Niessen wrote:
 Quote: \forall m P(m) is a Set. Short enought?

Christian Stapfer wrote:
 Quote: That axiom does not give you *all* the subsets of NI. It just gives you the "existence" of the set of all subsets of N. Which is rather an empty shell of a mere tag, in and of itself. If your set theory is consistent at all, it admits of having a merely countable interpretation (Skolem's paradox). So there is something illusory about your suggestion that the powerset axiom does the job of *really* capturing an "uncountable" P(NI) in its entirety.

Russell Easterly wrote:
 Quote: Nearly all the elements of P(N) can't be defined.

We can _define_ all the elements of P(N). (E.g., using a simple
bijection between the binary real fractions in [0,1) and the subsets
of N.)

We can only _compute_ a countable number of those elements
(or reals), though.
Russell Easterly
science forum Guru Wannabe

Joined: 27 Jun 2005
Posts: 199

Posted: Fri Jul 21, 2006 2:59 am    Post subject: Re: The list of all natural numbers don't exist

"David R Tribble" <david@tribble.com> wrote in message
 Quote: Peter Niessen wrote: \forall m P(m) is a Set. Short enought? Christian Stapfer wrote: That axiom does not give you *all* the subsets of NI. It just gives you the "existence" of the set of all subsets of N. Which is rather an empty shell of a mere tag, in and of itself. If your set theory is consistent at all, it admits of having a merely countable interpretation (Skolem's paradox). So there is something illusory about your suggestion that the powerset axiom does the job of *really* capturing an "uncountable" P(NI) in its entirety. Russell Easterly wrote: Nearly all the elements of P(N) can't be defined. David R Tribble wrote: We can _define_ all the elements of P(N). (E.g., using a simple bijection between the binary real fractions in [0,1) and the subsets of N.)

Russell Easterly wrote:

Binary real fractions?
ZF has binary real fractions?
Which axiom says there are real fractions, binary or otherwise.

 Quote: David R Tribble wrote: We can only _compute_ a countable number of those elements (or reals), though.

Russell Easterly wrote:

Actually, you can only compute a finite number of them,
as I proved with my Zeno machine proof.
(More precisely, no single computation can compute
more than a finite number of naturals or more than a
finite number of digits in a real number.)

I really question definitions of computability that
say we can compute any irrational number.
Turing defined computable numbers in his original paper
as binary strings formed by TM's that never halt.
He realized it was impossible to compute a real number
with a finite number of operations.

Unfortunately for Turing, a sequential machine can't
ever compute an infinite string, even if the TM never halts.

It seems to me the Axiom of Separation doesn't
prevent the existence of any set, it only guarantees
that any predicate of set theory defines a
subset of a set.

How does the Axiom of Separation prevent the
existence of contradictory sets like Russell's
paradox or the set of all sets?

It is obvious ZF without the Axiom of Regularity
is inconsistent, even if set theorists claim the statement
"there exists a set that is a member of itself" is
undecidable in ZF-R. I can still form Russell's
paradox in ZF-R not to mention the Ordinal
of all Ordinals.

Can I prove anything in ZF?
It seems ZF maintains consistency by making
every question undecidable.
I find it hard to believe any axiomatic system
with an Axiom of Union and a definition
of "successor" can fail to prove every natural number
is finite. Yet, ZF-I says "every natural is finite" is undecidable.

Russell
- 2 many 2 count
Dave Seaman
science forum Guru

Joined: 24 Mar 2005
Posts: 527

Posted: Fri Jul 21, 2006 11:29 am    Post subject: Re: The list of all natural numbers don't exist

On Thu, 20 Jul 2006 19:59:27 -0700, Russell Easterly wrote:

 Quote: David R Tribble wrote: We can _define_ all the elements of P(N). (E.g., using a simple bijection between the binary real fractions in [0,1) and the subsets of N.) Russell Easterly wrote: Binary real fractions? ZF has binary real fractions? Which axiom says there are real fractions, binary or otherwise.

All of the axioms together. The real numbers are a defined concept, and
they are defined in terms of set theory.

The reals are similar to the ordinals in that respect. Neither is
actually mentioned in the axioms. Both are defined concepts.

 Quote: David R Tribble wrote: We can only _compute_ a countable number of those elements (or reals), though. Russell Easterly wrote: Actually, you can only compute a finite number of them, as I proved with my Zeno machine proof. (More precisely, no single computation can compute more than a finite number of naturals or more than a finite number of digits in a real number.)

Nevertheless, a countable infinity of real numbers are computable. In
fact, there is a Turing machine that computes the identity function. For
each input n, the machine halts and returns the result n. This machine
enumerates the natural numbers, which are an infinite subset of the
reals.

 Quote: I really question definitions of computability that say we can compute any irrational number.

Not every irrational number. Just a denumerable subset.

 Quote: Turing defined computable numbers in his original paper as binary strings formed by TM's that never halt. He realized it was impossible to compute a real number with a finite number of operations. Unfortunately for Turing, a sequential machine can't ever compute an infinite string, even if the TM never halts.

But for any n, the machine can compute the n'th digit in finite time.
That's all the definition requires.

 Quote: It seems to me the Axiom of Separation doesn't prevent the existence of any set, it only guarantees that any predicate of set theory defines a subset of a set.

Yes.

 Quote: How does the Axiom of Separation prevent the existence of contradictory sets like Russell's paradox or the set of all sets?

It doesn't. How could it? It has been explained to you repeatedly that
you cannot avoid inconsistencies by adding axioms.

 Quote: It is obvious ZF without the Axiom of Regularity is inconsistent, even if set theorists claim the statement "there exists a set that is a member of itself" is undecidable in ZF-R. I can still form Russell's paradox in ZF-R not to mention the Ordinal of all Ordinals.

Ok, let's see your proof that the Russell "set" is actually a set in
ZF-R. If you can prove it in ZF-R, then you can also prove it in ZF.
Just use the same proof. It's not required that you use every axiom in
order for a proof to be valid, you know.

If you can prove a contradiction in ZF-R, you can prove the same
contradiction in ZF. Do you see why you can't avoid inconsistency by

 Quote: Can I prove anything in ZF? It seems ZF maintains consistency by making every question undecidable.

Lots of questions are decidable in ZF. More, in fact, than in ZF-R or in
ZF-I.

 Quote: I find it hard to believe any axiomatic system with an Axiom of Union and a definition of "successor" can fail to prove every natural number is finite. Yet, ZF-I says "every natural is finite" is undecidable.

Wrong. By definition, a natural number is a finite ordinal. Hence,
natural numbers are finite.

I think you mean, ZF-I can't prove that every *ordinal* is finite. That
is correct.

--
Dave Seaman
U.S. Court of Appeals to review three issues
concerning case of Mumia Abu-Jamal.
<http://www.mumia2000.org/>
Aatu Koskensilta
science forum Guru Wannabe

Joined: 17 May 2005
Posts: 277

Posted: Fri Jul 21, 2006 11:57 am    Post subject: Re: The list of all natural numbers don't exist

David R Tribble wrote:
 Quote: Russell Easterly wrote: Nearly all the elements of P(N) can't be defined. We can _define_ all the elements of P(N). (E.g., using a simple bijection between the binary real fractions in [0,1) and the subsets of N.)

What notion of defining you have in mind here?

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

"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
albstorz@gmx.de
science forum Guru Wannabe

Joined: 11 Sep 2005
Posts: 241

Posted: Fri Jul 21, 2006 12:52 pm    Post subject: Re: The list of all natural numbers don't exist

Patricia Shanahan schrieb:

 Quote: albstorz@gmx.de wrote: Virgil schrieb: In article <_eydndK1Kp9-jyLZnZ2dnUVZ_uudnZ2d@comcast.com>, "Russell Easterly" wrote: "Christian Stapfer" wrote in message news:99646\$44befd7b\$54497364\$21350@news.hispeed.ch... Peter Niessen wrote: Am Mon, 17 Jul 2006 15:24:56 -0700 schrieb Russell Easterly: If N is the set of all natural numbers, the number of subsets of N is the powerset of N. Does this mean we need an uncountable number of axioms to define all the subsets of N? Um, it depends on what you mean by "define"... \forall m P(m) is a Set. Short enought? That axiom does not give you *all* the subsets of NI. It just gives you the "existence" of the set of all subsets of N. Which is rather an empty shell of a mere tag, in and of itself. If your set theory is consistent at all, it admits of having a merely countable interpretation (Skolem's paradox). So there is something illusory about your suggestion that the powerset axiom does the job of *really* capturing an "uncountable" P(NI) in its entirety. Nearly all the elements of P(N) can't be defined. Nearly all real numbers are inaccessible, too. But not the ones we need. Who needs the ones which are inaccesible? AS Anyone who wants a system simple enough to allow general statements?

Ockham's razor blade seems to be out of the time.

Best regards
Albrecht S. Storz
science forum beginner

Joined: 10 Jul 2006
Posts: 3

Posted: Fri Jul 21, 2006 1:44 pm    Post subject: Re: The list of all natural numbers don't exist

hello,
I dont know but from your logic if the Subject is true
then
logically then the list of all
emails on this thread also does not exist .

``````LM

stephen@nomail.com wrote:
 Quote: Dave Seaman wrote: On Tue, 27 Jun 2006 09:40:48 EDT, phyti wrote: Those who say the list does not exist are correct. By definition (ZF, Peano, etc.), the natural (basic) integers are constucted using an infinite process. The axioms don't say anything about "construction" or "processes". An infinite set is postulated to exist. That is all. The idea of "construction" and "processes" seems to be at the heart of many people's misconceptions about set theory. Or perhaps more accurately, it is at the heart of their view of mathematics and makes it difficult, if not impossible to understand the standard position. Based on the arguments used by people like Russell, Tony and others, their idea of sets is based on computer implementations. Sets are containers to which objects can be added, or from which objects can be removed. Sets are created by set operations. Given the sets A and B, A union B "creates" a new set, or perhaps it modifies A or B. I am wondering if they think the same away about numbers? My guess would be no, but who knows? For example, given the following code x=7; x=x-2; I do not think they would think we changed 7. x had the value of 7, and after the operation x has the value of 5, but 5 and 7 did not change, and they did not need to be "created". But given the code x={ 2, 3, 5, 7, 11 }; // set assignment x=x - { 2 }; // set difference they think the set x has been modified. Instead of seeing x as a variable whose value has changed from one set to another set, they see x as the set, and operations on x change the set. In the standard view the set { 2, 3, 5, 7, 11 } still "exists" after the operation, it is just the case that x no longer has that value. Stephen
albstorz@gmx.de
science forum Guru Wannabe

Joined: 11 Sep 2005
Posts: 241

Posted: Fri Jul 21, 2006 1:45 pm    Post subject: Re: The list of all natural numbers don't exist

David R Tribble schrieb:

 Quote: Albrecht Storz wrote: If you don't talk about infinity = endlessness = impossibility of completion, use another word or symbol. William Hughes schrieb: Why? Do you think this nonsense is the usual definition of infinity? Albrecht Storz wrote: Nonsense? In what kind of world do you live in? Let's see: You claim: "An infinite set is by definition a set that can be bijected to a proper subset." Yes, that's the usual definition. But I think, infinite sets are at first a special kind of sets which have the propertie to be infinite. I'm right? Depends on what the "property of being infinite" means. Do you mean that infinite sets have the property of being infinite? Seems like a pretty vacuous definition to me.

??? Definition? Why?

 Quote: Can we also say that finite sets have the property of being finite, and that round sets have the property of being round, and that red sets have the property of being red? So please tell me your definition of the propertie "infinite". How about: Set S is infinite if it is not finite.

Not good.

 Quote: Set S is finite if there exists a bijection between it and some natural n.

For me, this is a very good definition for sets. If there is no
bijection, it's not a set.

 Quote: Or, more simply: A set is infinite if there does not exist a bijection between it and any natural.

Sounds like: if a unicorn is invisible, you can't see it.

 Quote: There are other, more formal definitions, of course.

I hope so.

Best regards
Albrecht S. Storz

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