Dear All

I am stuck with the following problem. Let an explicit
definition be a formula of the form:

     forall x(p(x) <-> A),
     where p does not occur in A
     A is wff of FOL

Not assume I have a theory T, such that p does not
occur in T. I want to show the following:

    If T is consistent, then T+forall x(p(x) <-> A) is consistent

In elementary type theory I can show this very easily.
Assume T+forall x(p(x) <-> A) would be inconsistent,
then we would have:

      T, Ax(p(x) <-> A) |- f    (f=falsum)

Since elementary theory has lambda expressions, we can
easily show comprehension:

     ——             ——
     A |- A             A |- A
     ————               ————
     (lx.A)x |- A       A |- (lx.A)x
     —————     —————-
     |- A -> (lx.A)x     |- (lx.A)x -> A
     ———————————–
               |- ((lx.A)x <-> A)
               ————————–
               |- Ax((lx.A)x <-> A)
               ————————–
               |- Ey Ax(y(x) <-> A)

Since elementary theory has predicate quantifiers, we
can finally derive:

                             T, Ax(p(x) <-> A) |- f
                          ————————–
                            T, Ey Ax(y(x) <-> A) |- f
                           —————————
|- Ey Ax(y(x) <-> A)       T |- ~Ey Ax(y(x) <-> A)
————————————————–
                   T |- f

So the assumption would lead to inconsistency of T.

Now the vexing problem for FOL is the following.
How can I proof the same for FOL?

a) Syntactically I will be not able to express
    comprehension in FOL (no existential quantifier
    for predicates), so is there another proof
    for the result?

b) Even if I were syntactically able to express
    comprehension, maybe by using gSOL, but not
    assuming a comprehension axiom. Would the
    alternative proof imply comprehension?

Bye

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Call for Paper
The International Journal of Computer Science (IJCS) publishes
original papers on all subjects relevant to computer science,
communication network, and information systems. The highest priority
will be given to those contributions concerned with a discussion of
the background of a practical problem, the establishment of an
appropriate model, the determination of a solution, approximate or
exact, analytical or numerical, and a discussion of the relevance of
the results when applied to the real-life problem. Paper submissions
are invited in the area of computer science, in particular the
technological advances and research results in the fields of
theoretical, experimental, applied electronics, computer science,
communication network and Information technology.
Topics of interest include but are not limited to the following:
     Computer Science

    * Parallel Processing and Distributed Computing
    * Foundations of High-performance Computing
    * Graph Theory and Analysis of Algorithms
    * Artificial Intelligences and Pattern/Image Recognitions
    * Neural Networks and Biomedical Simulations
    * Virtual Visions and Virtual Simulations
    * Data Mining, Web Image Mining and Applications
    * Data Base Management & Information Retrievals Adaptive Systems
    * Bifurcation, Biocybernetics & Bioinformatics
    * Blind Systems, Neural Networks &Control Systems
    * Cryptosystems &Data Compression
    * Evolutional Computation &Fuzzy Systems
    * Image Processing and Image Recognition, Modeling & Optimization
    * Speech Processing, Speech Synthesis & Speech Recognition
    * Video Signal Processing, Watermarking & Wavelet Transform
    * All topics related Computer Science

Communication Network

    * Quantum Computing & Coding
    * Error Controls Agent Computing & Multi-Agents Systems
    * Defining Spectrum Rights and Open Spectrum Solutions
    * Quality of Services and Communication Protocols
    * Satellite and Optical Communication Systems
    * 3G/4G Network Evolutions & CDMA/GSM Communication Protocols
    * Mobile Computing, Transmission/Switching/Distribution
technologies
    * Communication Theory & Signal Processing for Communications
    * Wireless Communications, Wireless & Mobile Networking
    * Optical Networks and Systems &Next-Generation Networking and
Internet
    * Communication QoS &Reliability and Modeling
    * Ad-hoc, Sensor & Mesh Networking
    * Multimedia Services, Communication Software & Services
    * Communication and Information System Security
    * System control and network/service management
    * Network and Internet protocols and standards
    * Client-server, distributed & Web-based communication systems
    * Broadband and multimedia systems & applications
    * Trials of advanced systems and services
    * Any topics related Communication Network

Information and Systems

    * Cryptography and Foundation of Computer Security
    * Authentication/Authorization Issues
    * IDS/Firewall, Anti-Spam mail & Anti-virus issues
    * Biometric authentication & algorithms
    * Fingerprint /Hand/Biometrics Recognitions and Technologies
    * IC-card Security, OTP & Key Management Issues
    * E-commerce, Ubiquitous & RFID Applications
    * Metadata, Meta Modeling, XML & Data Management
    * Knowledge Management, Web Security & Privacy
    * Cyber Threats, Web Services & Web Engineering
    * Web Intelligence, Protocols & Standards
    * Proxies and Servers
    * Multimedia Applications
    * Ontology and the Semantic Web
    * B2B, B2C and C2C
    * E-Business System Design and Development, E-Payment
    * Portal Strategies, Social Networks and Information Systems
    * Social and Legal Issues and Digital Ecology
    * E-Governance, E-Learning and Virtual Classrooms
    * E-Entertainment, E-Journalism
    * Any topics related Information systems

   Electronics

    * Circuits & Devices
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Communications
    * Electromagnetics & Microwaves
    * Instrumentation
    * Measurement & Testing
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    * Optics & Optoelectronic Effects
    * Devices, Systems &Semiconductors
    * Systems & Control Engineering
    * Telecommunications
    * Any topics related Electronics

International Journal of Computer Science (IJCS)
ISSN: 1884-9083
Website: https://sites.google.com/site/ijcsorg/
Manuscript submission to: ijcsorgedi…@gmail.com

All submitted papers will be judged based on their quality by the
technical committee and reviewers. Papers that describe research and
experimentation are encouraged. All paper submissions will be handled
electronically and detailed instructions on submission procedure are
available on IJCS web pages. Researchers and authors are invited to
participate in the peer-review process of IJCS papers if your research
interest matches with the themes of Call for Papers. For other
information, please contact IJCS Managing Editor.

Astronauts Gone Wild part 2 Were the moon landings a hoax?
http://www.youtube.com/watch?v=u2MQuH50_Io

Man, this was funny. Van-Allen Belt I’m
curious of, considering the risks involved,
of such a boundary, would need quite the
preparedness in design that has been alleged
not possible? I’m no expert either.. but
if so, I think that clinches it.

Astronauts Gone Wild part 4 Were the Apollo moon landings a hoax?
http://www.youtube.com/watch?v=QwQGBZ6v89Q

Wow.

-

I am a good man in a world going wrong

Like when a Ron Paul or TV pro says, ‘yeh, they,
sorry people, they lost the money, golly shucks.
go away now I’m a loser like all of everyone else
can be ever, cause Government is always bad’

The Zionist kills innocent Jews for money as a war
criminal, a honest to god war criminal. Justice is
what we must defend as freedoms sacrificed by our
voices of fairness not recognized. I am the good
guy here, you are something else to not know my
free to be path. Or to know it, and despise Life’s
true natural talent at willing for the good every
time as best we can. I am loved, not by AmeriaKan
Zombie Inc. okay, but never the less, loved in
life as more than a half honest, good hearted
human being. I am a good man in a world going
wrong.

Johnny the Jew

-

Jews Want ‘Museum of Tolerance’ Built On Muslim Graves

NOT JEWS. Enemies of the Jew, to steal God’s good name.
To profit off our deaths as Satanic enemies of Humanity.
As so writ, in our Bibles, and just common sense. These
Satanic enemies of Life, steal innocent Christian homes
near daily. The Zionist Nazi Government of Israel, does
this with threats of MURDER, paid for by the American
suckers. The Zionist Nazis broke the truce to get
killed Jews for money. THEY ADMIT THIS FREELY. Ugly,
ungodly, evil, and war criminal. It is only censorship
by a demon enemy happy George Noory, that hides these
challenges to remain un-apposed, for the Zionist Nazi
can’t compete when free to speak, open dialog for
Justice winning is allotted. Hence, Americans refusing
to support my wise words for polite open public debate.
They’d rather die victim to slavery for rich and happy
George Noory, than humble themselves to wise men or
women speaking. See the vid, ‘Learn to Speak Tea Bag’ as
the whole truth. We are hated by corporatized America
not because we are bad People, but because we are good.
And, like the true Liberator I am with knowledge of the,
Justice is suffering because of ourselves condition,
hated even that much more.

-

Chris Hayes Attempts to Explain That the Constitution Does Not Only
Apply to Citizens on Morning Joe
http://videocafe.crooksandliars.com/heather/chris-hayes-attempts-expl…

Scarborough: "You cannot blame what’s going on
in Israel on the Bush administration."

Brzezinski: "You know, you have such a stunningly
superficial knowledge of what went on that it’s
almost embarrassing to listen to you."

-

Bob Kerrey (911 comm) admits 911 was Pre-Planned

Grilled by the TSA on "who did 9/11"

/ / Nick George, a student at Pomona College in
California, was grilled by the TSA on "who did 9/11"
and asked by FBI agents whether he was a communist
after airport security officers found Arabic-English
language learning cards in his luggage last summer \ \
..as to what is expected next? Torture maybe?

What the hell man, Bush did 911. Planned for it
and carried it out by closing Police Investigations..
what’s it to you,.. copper? Who am i? Johnny, Johnny
from Anytown, USA. Wanna see my Badge? Certainly,
Honor for God and Country I hold proud as my badge
of Freedom found by Justice earned, what about you.
What are you doing here not speaking up for Justice
on 911? Claiming our rights to deny such wanting
by remaining ignorant of these facts presented to
US all.. Huh? Eh? Speak louder cause your cowardly
murmerings, are lost to the sounds of sadistic
bushite bombers, rapists, and torturers.

(YOU, yes.. you, forward this to everyone you will)

-

‘Jim Cramer on CNBC while defending Bernanke and Geithner
says that the Bavarian Illuminati, Trilateral Commission,
Goldman Sachs and Queen of England are not all bad’

AmeriKan Zombie Inc.

Bushite grunters are a criminal force who’s
direction of aggression is currently pointed at
Amerika for ransom.  The Thirty Trillion Dollar
bailout was a popular fraud in America called,
Extortion.  Poof!  Now, we have The Government of
AmeriKan Zombie Inc..  Whom, it could be assumed,
will to continue on with a murderously blind
military dictatorship to escape the evil Neocon
Peenackers of 911, or as now including, the Wall
Street trillioniare Bankster Shysters.

With Congress and the Presidents murdering so
many good, honest, hard working men and women in
Iraq without a just cause to escape enemy Neocon,
to warring even more for lawless military
dictatorships around our suffering world,
treating US all to torture, false imprisonments,
or drone strikes due a tip some anonymous heroin
dealer alleged is, AmeriKa Zombie Inc..

We are falling victim by ungodly con men
enriching themselves of our funerals.  The
disloyal Bushite Grunters who murder our innocent
families are not Heroes as CNN and FOX claim.
The do not war God and Humanity for Freedom’s
reign.  As a committed vigilant defense, of
Justice demanded always as anyone truly free.
The Wins of a Civilized World bring many joys.
Bringing the absence of Hunger or indiscriminate
bombing runs, with the loves of experience,
beyond the searchings for immediate shelter or
life saving medicines.  We need to get back all
our stolen values of Justice realized.  Banksters
don’t cover our loans, we do.  They have
defrauded our nations TRILLIONS, driving us
further into fraudulent indebtedness as the
consequence of their criminal causes.  To demand
Justice for all, is where Life truly stands.  We
want the money that was stolen through the
massive Wall Street fraud returned.  The Bailout
was extortion.  Bush and Cheney need arresting
for stopping the Police agencies of America to
properly carry out the arrests of the criminal
suspects.  The charge against George Walk Bush is
called obstruction.  We, The Peoples are not all
betrayers to a greater day.  We, The Peoples are
not proud ignorant enemies of the Republic like
Sarah Palin and every single last Republican,
demanding Justice be denied against terror suspects.
This battle is lead by the forces who champion the
rights of all as fairness you cheats.  No evidence
against an accused means the person is innocent.
(DIE BUSHITE DIE) And, I believe it was the Magna
Charta (but don’t quote me on this) that added,
even if the guy confesses straight away, and goes,
‘no, no, don’t you see? I did it all?’ requires
STILL, acknowledging a confirmation of fact to
substantiate as conclusionary.  Then, something
like, ‘no free man will be imprisoned’ ..I swear
this stuff is really simple.  We can achieve back
what we lost somehow by believing in ourselves
winning a greater day, yes we can.  And We start
that day, by insisting that the culprits responsible
for 911 not escape our wrath over the censorship of
a demon enemy of Life like happy George Noory.  
Expecting all honorable police services to will
the co-operations needed, to bring these bad guys
to Justice.  Listen:  we are going to try all
these bastard peenackers of the bushmob, by fair
public trials.  And yes, get all of everybody’s
monies back, as best we can, from the deceitful
war profiteers of lawless mass murder and torture.

People like Amy Goodman were warned to step back
from WTC7, because the pre-planted explosives were
going to be taking it down momentarily.  Then, it
happened just as they planned it.  Remarkable?  So
much so, that international news networks reported
it falling just before it starting falling.  Who on
Earth could possibly dispute such?  but a bushite
liar, a zionist thief, an ungodly war mongering
enemy to Life, warring YOU for the further escape
of the neocon peenacker highjackers.

AmeriKan Zombie Inc.  "Let them rob us all of
house and home, but just whatever you do, don’t
let wise people like Johnny America speak for
justice on all sides, because I’ll be no longer
sounding better than everyone else, dying US
censored victims for remaining quiet."

INTOLERANCE GROWS AT JERUSALEM�S �MUSEUM OF TOLERANCE�
http://desertpeace.wordpress.com/2010/02/11/intolerance-grows-at-jeru…

"Jewish Chronicle editor says extra-judicial murder is kosher"

This is why GOD calls the Zionist Satanic. They
clearly, truly, are evil godless enemies to
every living soul here.

-

Columbia University declares �Obama never attended Columbia
University�
http://americangrandjury.org/investigators-declare-obama-never-attend…

If this is true, then maybe his father never
was alleged to be an American Citizen, by
anyone,. thereby making Obama not fit for
the Amerikan Zombie Inc. Presidency. See
this is the kinda knowledge you don’t get
anywhere but here, with yours truly,

King Johnny America

-

This from Mike over at: http://whatreallyhappened.com/

/ /

‘Barack Obama accuses Iran of attempt to build a nuclear bomb’

Dear Presid…@whitehouse.gov

You really are an idiot.

Under the terms of the Nuclear Non-Proliferation
Treaty, Iran has a right to use nuclear
technology for peaceful purposes.  This includes
power stations and medical isotopes.  Enrichment
to 20% is consistent with medical usage.
Therefore, Iran’s actions are legal under the
terms of the Nuclear Non-Proliferation Treaty
which Iran has signed right alongside the United
States.

In demanding that Iran surrender their legal
rights as specified in the NNPT, the United
States is in violation of that treaty.  Under
Article IV, the United States is obligated to
assist Iran in building their power stations and
medical facilities.  Clearly, if the United
States were in compliance with the NNPT, we would
know for an absolute certainty what Iran was and
…

Super Tasks

I see Zuhair is getting beat up on for
questioning the diagonal argument.
So, its time to pull out a golden oldie:
Why there is no bijection between N and N.

Assume I have a bijection between two set, A and B.

A = (0,1,2,3)
B = (1,2,3,4)

F = (<0,1>, <1,2>, <2,3>, <3,4>)

I can always construct a bijection, G, with
the following properties:

Every element of (intersection) AxB is paired
with itself and every element of A-B is paired
with an element of B-A.

Such a bijection is easily constructed.

Start with some element of AxB, xi.
If xi is already paired with itself go
to another element. Otherwise, find
the two ordered pairs in F with xi
as an element.

<aj,xi>, <xi,bk>

Replace these two ordered pairs with:
<xi,xi>, <aj,bk>

Repeat this process for every element
in AxB in any arbitrary order:

<0,1>, <1,2> -> <1,1>, <0,2>
<0,2>, <2,3> -> <2,2>, <0,3>
<0,3>, <3,4> -> <3,3>, <0,4>

G = (<0,4>, <1,1>, <2,2>, <3,3>)

Assume I have the following two sets and
bijection:

A’ = (0,1,2,3,…)
B’ = (1,2,3,4,…)

F’ = (<0,1>, <1,2>, <2,3>, …)

I use the method described above
to create a new bijection, G’:

G’ = (<0,z>, <1,1>, <2,2>, …)

z must an element of B’-A’.
Which element is it?

I know most people will argue G’
can’t be constructed. They will
claim I can’t do an infinite number
of exchanges or some such.

I find this argument amusing.
We have to assume we can take the intersection
of two infinite sets just to ask this question.
How is performing an infinite number of
intersections any different than performing
an infinite number of exchanges?

Here is the challenge. Prove my bijection
can’t be constructed or tell me which
member of B’ is paired with 0.

Russell
– Mathematics is the only true religion

.
.
___________________________________________________________

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The following is a theory in FOL with identity "=" membership "e" and
small set-hood "V" , with a universal set in it.

Define: x is small <-> x e V

Extensionality: as in Z

Union, Power, Infinity, Replacement (strong version)  All relativised
to V.

Axiom of small set existence: Exist x: x e V

Anti-foundation axiom:

For every z e V Exist x e V For all y ( y e x <-> ( y e z or y=x ) )

Define the i-th union of x by recursion

Ui(y) = y  if i=0
Ui(y) = U( Ui-1(y)) if i is a finite ordinal other than 0.

Axiom schema of universal comprehension:

If Phi is a formula in which x is not free ,then all closures of

~ For all y ( (y e V & Phi(y)) -> Exist i ( ~ y e Ui(y) ) )
->
Exist x for all y ( y e x <-> Phi(y) ).

is an axiom.

Theory definition finished.

Now all the paradoxes of Russell’s, Buralit-Forti, Cantor’s, and
Lesniewski’s (the singletons paradox) all would be avoided in this
theory, Even Holmes paradox would be avoided here.

I don’t know really if this theory is consistent or not, but anyway it
is so limited to provide significant information other than that ZF
provides.

This theory do prove the existence of a universal set of all sets,
also the set of all sets that are in themselves , also it does prove
the existence of Frege-Russell cardinals for every set in it.

All work that can be done in ZF, can be done in this theory too.

Zuhair

Hi all,

SETS:

 The objects that mathematicians call as *sets*, like those objects of
the universe of discourse of ZF, are actually nothing but a special
kind of containers the identity of which is totally dependent on what
they contain.

 Cans commonly seen in our surroundings do not possess this quality
sets possess, we can have two different cans with the same contents,
i.e. we can have two different cans such that whatever each can
contain the other one also contains and vise versa, and of course
these cans we ordinarily see in our surroundings everyday violate
Extensionality clearly.

 However sets are special kind of cans, they differ from commonly
observed cans in that their identity is completely determined by what
they contain in such a manner that no two different sets contain the
same contents.

Since sets are cans, then we can have an empty set, also the singleton
set is not necessarily identical to itself.

But sets can be members of themselves, and it is not easy to imagine a
can that contain itself?

Actually in used to think that this is a big obstacle to the intuitive
understanding of sets as cans, but later I came to understand that
this is not the case.

First lets resolve the question of weather a Can can be in one of it’s
contents.

Lets have two Cans A and B.

Now let’s assume that Can A only allow Can B to be inside it for some
moment of time, in other words Can B is the sole content of Can A, or
what can be translated to B e A.

Now imagine Can B also only allowing Can A to be inside it for some
moment of time, so we have A e B.

So we have A={B} and we have B={A}.

So the above is imaginable, and there is no problem with Cans being
contained by their contents.

However the problem arise when we want to image A={A}
as a Can.

i.e. can we imagine a Can A such that A={A}, i.e. A is a Can that
solely contain itself.

My personal answer is YES, this is imaginable!

If we think of A as a kind of infinite spiral, then every part of A is
surrounded by other parts of A, and we can say that A is contained in
itself.

The containment relation "e" between two Cans can be defined as having
every part of the contained Can completely surrounded by parts of the
containing can, so imagine for example two circles one inside the
other, then every part of the inner circle is completely surrounded by
parts of the outer circle, so we can say that the inner circle is
contained in the outer circle.

So according to this visual explanation of containment one can have a
spiral that is infinite so in this way every part of that spiral would
be surrounded by other parts of itself, and thus we can say that this
spiral is contained in itself.

So all examples of what is refereed to as *sets* are actually
imaginable as Cans, weather these sets are well founded or not.

An infinite descending membership chain can also be imagined as a
chain of Cans were every Can in that chain have another Can inside it.

So from that account as I said we can understand that sets are nothing
by special kind of Cans the identity of which is solely dependent on
what they contain.

COLLECTIONS:

Collections as commonly used in language are actually quite different
from Cans or containers, a collection can be imagined as
some kind of an abstract object that is the WHOLE of objects specified
by a certain condition or property.

The whole of multiple objects is something that is different from
these objects.

However the whole of a single object is that object itself.

On the other hand we cannot have a whole of no objects!

Of course two collections are identical iff they have the same
members.

We can build collections using any Property that holds for
single objects.

All the above rules are quite different from those of sets.

We can actually build a formal system using the relationship
e’ to denote membership of collections, and we can axiomatize all the
above.

This approach can be used to build up proper classes of sets by
stipulating that each container is a single collection.

However it seems that collections as used in common language have no
mathematical significance. While on the other hand
Sets which can be understood as the special kind of Cans as I outlined
above are very important in mathematics.

The superiority of sets over collections is that sets can be
hierarchical, while collections are flat.

We can always go up the hierarchy of sets, and thus having more and
more fruitful relations and build up more complex sets from simpler
ones, while this cannot be done with collections.

The nutshell of all of this is:

Sets are Cans that are identical when having the same contents.

Sets are containers of objects, these objects were wrongly called as
"members", and they would be better refereed to as "contents"

Sets are not collections of members.

Epsilon membership is better referred to as Epsilon containment, so A
e B is better understood as : A is contained in B.

So after all there is an intuitive interpretation of "sets", and there
is an intuitive interpretation of the epsilon membership relation "e".

 However one must emphasize that this epsilon membership e, is a
relation that we do not encounter in common life, since in common life
we don’t see containers the identity of which is solely dependent on
what they contain.

So the concept of sets is actually a fabricated concept, i.e. it is
artificial, of something that belong to human fantasy, but it is a
useful fantasy that help us explain one of our most useful fantasies
which is mathematics.

Zuhair

Those of you who are contributing to my posts here (for which I am
very grateful) may know by now that I am having some trouble to
obtain, let us call it a "unified picture" of the contents of Levi’s
"Basic Set Theory", Kunen’s "Set theory. An introduction to
independence proofs" and Shoenfield’s "Mathematical Logic". There are
so many concerns I have about this that I hardly know what to ask in
order to disentangle my particular web of doubts.

I have often asked for different literature, but I believe the problem
is not in the literature: I must be thinking things are far more
complicated than they really are.

In Shoenfield, models of first-order theories are sets, but generally
models of set theory are given by means of a relative interpretation
of set theory into itself, i. e. by means of two formulas in the first
order language of set theory M(x, v) and E(x, y, v), with no free
variables other than the ones shown. The first one defining a class of
sets {x | M(x,v)} and the second one a relation {<x,y> | E(x,y,v)},
both with parameter v.

So, in what sense does the picture of models used in Shoenfield to
prove theorems about first order theories and their models fits the
picture used to buid up (for example, transitive) models of set
theory? Am I making a mess of things that have no relation to each
other?