Mathematics is

[Zuhair]

The investigation of behaviors that are formalizable in consistent
formal systems extending logic.

Logic is the study of Tautology.

Any behavior that is formalizable in a consistent formal system
extending logic is said to be mathematical
and would be the subject of mathematical query whether it is
verifiable to exist on empirical grounds or not.

Applied mathematics is the part of mathematics that is also the center
of study of fields other than mathematics,
like empirical sciences, ethics, linguistics, music, etc…

Pure mathematics: is mathematics done for the sake of mathematics.

Some empirical sciences have parts of them that are actually applied
mathematics of central interest to those fields,
like physics for example. One can call those parts as mathematical
physics, mathematical chemistry, etc…

On the other hand one can look into those parts as parts of
mathematics also so they could also be paraphrased as:
physical mathematics, chemical mathematics, ethical mathematics.

All the above are just examples of overlap between mathematics and
other fields of knowledge.

However the choice of the wording would depend on the relative
importance of the shared subject to the subjects
shared. if it is more important to physics, then its better be named
as mathematical physics. One can view it as
a discipline were mathematics is serving physics.

Mathematics is mostly concerned with somewhat more generalized kind of
formalizable behavior. Too particular
formalizable behaviors are mostly of prime interest to other fields.
Mathematics is more concerned with what may be
called as Stem formalizable behavior, which is fruitful formalizable
behavior, some kind of behavior from which stems
many particular behaviors that can be useful to mathematics itself as
well as to other fields.

The word "behavior" here stand for predicates of any arity, it can be
a style of movement of an object, it can be
the numerousity of a set which is an index of how many elements it
has, it can be the position of an object between some
specific objects at particular time, it can be the shape of an object,
etc…

[Graham Cooper]

On May 20, 4:57 pm, Zuhair <zaljo...@gmail.com> wrote:
>  The investigation of behaviors that are formalizable in consistent
> formal systems extending logic.
>

Would the study of FAULTS (security systems, social methodologies,
psyche, logics) necessitate a consistent framework for that particular
scientific methodology?

Scientific Methodology is an open set, e.g. chance discovery,
$1,000,000 prizes, Intellectual Property law, hypothesis test and
retest, formal specification, ...


Herc

[Zuhair]

On May 21, 8:52 am, Graham Cooper <grahamcoop...@gmail.com> wrote:
> On May 20, 4:57 pm, Zuhair <zaljo...@gmail.com> wrote:
>
> >  The investigation of behaviors that are formalizable in consistent
> > formal systems extending logic.
>
> Would the study of FAULTS (security systems, social methodologies,
> psyche, logics) necessitate a consistent framework for that particular
> scientific methodology?
>

Of course, I explained this further down in my message, those are
applied mathematics

Zuhair

[Graham Cooper]

ok, if intelligent life ceased in our solar system, would the moon
still follow a mathematical arc around the Earth?

Herc

[Zuhair]

Of course, the moon is not following a mathematical arc for our sake!

The moon moves around earth and this movement can be formalized within
a formal system that extends logic, So the moon's movement around
earth is mathematical whether we exist or not.

Zuhair

[Graham Cooper]

So the moons arc is
1/ mathematical

But mathematics is
2/ an investigation

What's holding up the moon if nobody is investigating the mathematics?

Herc

[Nam Nguyen]

So then the Moon still moves around the Earth in a _formalized_
mathematical arc.

_Whose formalization_ would that be?

--
----------------------------------------------------
There is no remainder in the mathematics of infinity.

NYOGEN SENZAKI
----------------------------------------------------

[LudovicoVan]

"Zuhair" <zalj...@gmail.com> wrote in message
news:bcc10495-13e2-4270...@e20g2000vbm.googlegroups.com...

> The investigation of behaviors that are formalizable in consistent
> formal systems extending logic.

I'd think that only captures the interests of formal logic.

> Logic is the study of Tautology.

As I have it, logic is the study of self-contradiction, itself informally
defined after the notion of mutual incompatibility of predicates.

> Any behavior that is formalizable in a consistent formal system
> extending logic is said to be mathematical

It is logic that is only interested in questions of logical validity and how
this is related to logical form, but mathematics talks about objects and
operations on objects and is concerned with questions of factual (for how
abstract/virtual) truth. Then, maybe, the essential difference between pure
and applied mathematics is all in that distinction between (absolutely)
abstract and (essentially) virtual...?

-LV

[Zuhair]

No, The Moon would then still move around the Earth in a
_formaliz*able*_ mathematical arc.

The the formalization would be of those who MAY exist one day and can
do such formalization.

If you say what about the possibility if the universe was incompatible
with life at all from its beginning to its end then can objects in it
move in a mathematical manner? then still hypothetical answer is YES
and the formalization would be of those who could have existed in a
parallel universe that can have the ability to ponder that universe
and speak about it.

If you tell me what about a universe that is not discover-able by any
kind of intelligence? then I'd say the question cannot be answered,
behavior in such universe may be of any kind, we just cannot speak
about it.

Zuhair

[Zuhair]

Nature, God, etc....

[Zuhair]

On May 22, 1:17 am, "LudovicoVan" <ju...@diegidio.name> wrote:
> "Zuhair" <zaljo...@gmail.com> wrote in message

The distinction between mathematics and logics here is arbitrary. Yes
my definition fits the *informal* concept of logic, I agree.
Mathematics is basically a kind of logic. However here I reserved the
term "logic" to the arena of Tautologies and mathematics to what
extends it. As you know logical systems capturing all tautologies like
for example Frege proof system of First order logic (+whatever added
primitives) which have around 9 or so "logical" axioms, can be
extended by systems containing "non logical" axioms, those combined
logical + non logical axiomatic systems are what I call here as formal
systems extending logic, you can understand them as part of logic (in
the general sense)of course, and you can of course understand what
they formalize to be part of logic (in the general sense) also, anyhow
I call what can be formalized in those later systems as
"mathematical".

Mathematics in the sense I'm speaking of can have parts of it that do
have grounding in reality, since logic itself actually have some
overlap with reality. However mathematics may on the other side speak
about matters that are not observable at all and non virtual in your
sense, you can call it fantasy mathematics, the point is that it is
still mathematics.

The basic difference between applied mathematics and pure ones is that
the applied one is an overlap between mathematics and some other field
of human interest, while pure mathematics is not directly so. Pure
mathematics is like Stem mathematics, Primordial mathematics,
something more general than applied mathematics than can branch into
many kinds of mathematical behaviors and those can also branch further
and further and at the ending branches can be thought of applied
mathematics if they overlap with other fields of human inquiry, or a
particular fantasy mathematics which would be also part of Pure
mathematics.

An example of though you can consider Frege proof system of FOL as
what I call "Logic" then take ZFC that extends that, Now ZFC itself
can be extended by NUMBER theory
Geometry , Topology etc..., those can further branch into for example
Graph theory which can further branch into Tree theory, which can
further branch into special kinds of Tree theories. Now a particular
kind of Tree theory might be useful in understanding branching in some
part of Botany, or in understand crystallization in Geology etc...

Zuhair

[Frederick Williams]

Zuhair wrote:
>
> The investigation of behaviors that are formalizable in consistent
> formal systems extending logic.
>
> Logic is the study of Tautology.

No it isn't. One may glean that by looking in any logic textbook.

--
When a true genius appears in the world, you may know him by
this sign, that the dunces are all in confederacy against him.
Jonathan Swift: Thoughts on Various Subjects, Moral and Diverting

[Zuhair]

On May 22, 3:38 pm, Frederick Williams <freddywilli...@btinternet.com>
wrote:

> Zuhair wrote:
>
> >  The investigation of behaviors that are formalizable in consistent
> > formal systems extending logic.
>
> > Logic is the study of Tautology.
>
> No it isn't.  One may glean that by looking in any logic textbook.
>

This is a terminology that I'm giving here. So your textbooks
terminology on Logic is irrelevant here.

Zuhair

[LudovicoVan]

"Zuhair" <zalj...@gmail.com> wrote in message

news:fdc0b035-535a-476d...@s5g2000vbc.googlegroups.com...

> The distinction between mathematics and logics here is arbitrary.

Arbitrary? I just do not think your proposed definitions capture the
underlying notions: with all due respect, I actually think you have the
thingy upside down...

> Yes my definition fits the *informal* concept of logic, I agree.

That is not what I said: on the contrary, to me your definitions only
capture the formal side of things, and not even that precisely.

> Mathematics is basically a kind of logic.

I would strongly disagree: logic is logic, mathematics is mathematics, and
then maybe mathematics uses logic: for the reasons I am mentioning.

> However here I reserved the
> term "logic" to the arena of Tautologies and mathematics to what
> extends it.

In particular, only in an *already* formal setting you could say the a
logical theory is a collection of tautologies. In fact, logic per se is
founded on the notion of *self-contradiction*, which does have a salient
informal ground, and tautology is defined after self-contradiction. A great
book on the foundations of logic is P. F. Strawson, Introduction To Logical
Theory (there are even few final chapters devoted to inductive inference and
the nonsense of searching for a justification to induction).

<snip>

> anyhow
> I call what can be formalized in those later systems as
> "mathematical".

I understand this distinction between logical and non-logical axioms, but
the "mathematical" system that results from there is, to me, indeed already
and fully mathematical, as we are not anymore interested in pure formal
questions of validity, we are rather interested in proving stuff (within a
consistent system, of course, which is the reason for taking from logic)
about objects and operations, i.e. *factual truths*. In short: logic is
interested in logical truths, mathematics is interested in factual truths
(about mathematical objects).

> Mathematics in the sense I'm speaking of can have parts of it that do
> have grounding in reality, since logic itself actually have some
> overlap with reality.

Logic (the study of *self-contradiction*, usually just called "validity") is
grounded on natural language: you won't be able to discern the logical form
and validity of a statement (i.e. do a logical analysis) unless you know the
meaning of words and connectives. And language is not "reality", hence the
relevant informal notion is that of *mutual incompatibility of predicates*,
not any reference to a reality external to language (to which language has
no necessary reference to begin with). On the other hand, mathematics is
the study of a reality of its own, that of numbers (mathematical structures
in general), an abstract reality, yet a "reality".

> However mathematics may on the other side speak
> about matters that are not observable at all and non virtual in your
> sense, you can call it fantasy mathematics, the point is that it is
> still mathematics.

I rather called it abstract or virtual and I didn't mean anything but the
very opposite of any "fantasy". I used the term "virtual" for applied
mathematics, to mean that in that case we are not anymore dealing with
purely abstract objects, we are rather trying to *model* an external reality
this time (and "virtual" also is supposed to capture the fact that we will
never model reality *per se*, only an abstraction/approximation of it).

> The basic difference between applied mathematics and pure ones is that
> the applied one is an overlap between mathematics and some other field
> of human interest, while pure mathematics is not directly so.

Too weak: it amounts to giving up in providing any pertinent definition.
Yes, our daily practice is never in itself pure (we always do more things at
once, where the single things we do are in fact cut out from the "continual
flow" by our conceptualizations, nothing else). So one thing is that nobody
can do, say, maths without doing maybe a bit of logic, or maybe a bit of
social economy, depending on the specific case, etc. etc. But other thing
is that still logic is logic and mathematics is mathematics, or we don't
know what we are talking about.

> Pure
> mathematics is like Stem mathematics, Primordial mathematics,
> something more general than applied mathematics

It is not simply "more general", it is rather *purely abstract*. In any
case, the difference I am attempting between pure and applied mathematics is
more of a characterization of the two extremes of a range.

-LV

[LudovicoVan]

"Zuhair" <zalj...@gmail.com> wrote in message

news:77eec88f-21ec-450c...@w24g2000vby.googlegroups.com...

It is not a just matter a terminology: would "Logic is the study of
Oaskhwlasflx" be of any interest? You have to provide a rationale
underlying your definitions, and that has to pick into current language, at
least in some basic way.

-LV

[Zuhair]

On May 22, 9:53 pm, "LudovicoVan" <ju...@diegidio.name> wrote:
> "Zuhair" <zaljo...@gmail.com> wrote in message

Yes I do have a rational beyond what I'm saying.

In an informal manner you can say that logic is the methodology of non
contradictory reasoning. However when you are working with systems
that have the logical and non logical axioms you need to restrict the
definition of the word "logic" to more precise one. The definition
that suites the purpose of writing here is the one I gave. Logic burns
down to be the study of tautologies (including tautological
implication), a formal system is to be called "purely" logic as far as
it is within the arena of tautologies. Frege Proof system of
propositional logic and first order logic is a nice example of what
I'm saying, it has a finite set of axioms, some rules of inferences,
and it do capture all tautologies of propositional logic and first
order logic, so this with the rules of formation of formals etc...
constitutes a purely logical system.

However when those pure logic systems are extended by axiomatics
systems that contain non logical axioms like for example axioms of
ZFC, we actually call them non logical because they are not
tautologies, they seize to be something of analytic truth (something
that you can know whether it is true or not just by figuring out the
meaning of the words and connectives,etc..), so systems extending
logic, I mean those composed of non logical axioms on top of the
logical axioms are not purely logical systems anymore, although they
are guided by a logical frame, yet in their essence they actually
reflect something "external" to logic, what those non logical axioms
are laying down are actually basic properties of something that is
essential external to logic. I defined mathematics as what can be
formalized in such mixed systems, and I did that to differentiate it
from pure logic.

Mathematics is not factual at all, it may overlap with reality yes,
but not most of it. If I enumerate all possible systems with non
logical axioms and all of what is formalizable in them I'm sure that
most of those will not meet reality, yet they are Mathematics, however
some of them are and those are actually the applied ones.

What you call as the abstract part is something that I call the
primordial level of mathematics or Stem mathematics it can further
differential to applied math. and also it can lead to fantasy math.

Mathematics fits more the realm of Rationalization when you are making
rational expectations of matters some of the content of them depend on
some reading of reality, sometimes It might meet reality other times
it fails to do so, but yet it remains a rational clever form of
argumentation that simply missed to fully copy reality, as I said
other times it meets exactly with reality which would by then reflect
some correct reading of reality.

Of course the interesting mathematics are those that can make us read
reality better and help us understand it in a more fine manner.

So although mathematics is mostly a kind of logicism (in the general
sense) yet some of it meet reality. Because reality itself in part of
it follows some logical rules, in other manner no all of logic reality
detached, truely it can be known from word composition of a sentence
yet it is not altogether detached from reality neither does it copy
it. There is some overlap between Reality and language that is so
subtle.

Zuhair

[Ludovicus]

On May 20, 2:57 am, Zuhair <zaljo...@gmail.com> wrote:
>  The investigation of behaviors that are formalizable in consistent
> formal systems extending logic.
>

From this follows that Goldbach´s, Twin prime numbers, Riemann´s
conjectures on primes are not mathematics because they are not
formalizable.
Ludovicus

[Zuhair]

Actually all of those questions can be formalized in a logical system.

Zuhair

[Zuhair]

On May 23, 6:27 pm, Ludovicus <luir...@yahoo.com> wrote:

What is the proof that they are not formalizable?

Zuhair

[Ludovicus]

On May 23, 2:07 pm, Zuhair <zaljo...@gmail.com> wrote:

> What is the proof that they are not formalizable?
>
> Zuhair

What is the proof that they are formalizable?
Until there is not a proof are not they mathematics?
Ludovicus

[Zuhair]

Honestly I don't see why you say it is not formalizable, the notion of
a natural number is formlizable (you can state it as a primitive and
further characterize it by axioms like in PA, or you can simply define
it like in Z for example as a finite ordinal), the notion of a prime
number is also formalizable, Goldbach's conjecture can be perfectly
stated in a formalizable manner! It can be stated in a theory
extending logic:

For all x. x is an even number & ~x=0 > (Exist p,q. p is a prime
number & q is a prime number & x=p+q)

That we don't know whether it holds or not is something else. But no
matter what is the outcome it can still be formalizable. You see if a
theory T can prove GC to be false then its negation is a theorem of T,
if it can prove it then it is a theorem of T by definition, if T
neither proves nor disproves GC It can be added as an axiom to obtain
the consistent theory
T+GC. There are lots of problems not solved yet in set theory and
other logical theories would that entail that they are not
formalizable problems?

You see that's why I said mathematics is the "investigation" of such
formalizable matters, it means some of them might not have answers but
still they belong to the arena of mathematics because it is
mathematics that is used to solve them or their solution is agreed
upon to be within the confines of mathematics. How do you intend to
solve GC for example? the solution is clearly something that would be
formalizable anyway no matter what the outcome is as I showed above.

All those matters you mentioned are perfectly formalizable concepts,
they clearly abide by a logical frame, so they are all parts of
mathematics. What is not formalizable or what we don't have a clear
view whether it is formalizable or not is something like human
behavior, or generally behaviors of Biological organisms most of which
we can not put in in a logical frame. Even most of non biological
organisms exert behaviors that are not formalizable by a formal system
extending logic.

Zuhair

[Graham Cooper]

On May 24, 4:26 pm, Zuhair <zaljo...@gmail.com> wrote:
> Honestly I don't see why you say it is not formalizable, the notion of
> a natural number is formlizable (you can state it as a primitive and
> further characterize it by axioms like in PA, or you can simply define

OK, but moving down 1 level, the primitive structure that is accepted
is arbitrary.

formal set theory is strings of the form:
{{{}{}{}{}}}{}{{{}{}{}}}}}{...{}{}{{}}}

Why 2 chars?

set theory is just a model in grammar theory, and set theory happens
to be the chosen base model of all mathematics.

By the time you define:

ALPHABET = "{", "}"

and some reduction rules you may as well define something workable.

ALPHABET = {0,1,2,3,4,5,6,7,8,9, e, {, }, (, ) ,, =, A, E, ^, v, !, >,
<, n, m, o, p ,q, f, g, h, i, j}

DICTIONARY = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, ...}
Note 1+1 = 2 is irrelevant at this level.

AXIOM OF SPECIFICATION
E(n) m e n <> f(m)


--------------------


Where e is a relation.

TABLE e
-------
MEMBER SET
1 N
2 N
3 N
A ALPHABET
E ALPHABET
7 PRIMES
...


Since LOGIC is not just LOGIC GATES, the relation "e" must define Set
Theory and Logic.

e:UoDXUoD -> BOOLEAN


Herc

--
On May 21, 9:16 am, Virgil <vir...@ligriv.com> wrote:
Actually I believe that there are more that aleph_0 irrationals
in any real interval of positive length. |R|>oo

On May 17, 6:13 am, Virgil <vir...@ligriv.com> wrote:
RE:"There are more Reals than Naturals." |R|>oo
If that means that there is a surjection from the reals to the
naturals but no surjection from the naturals to the reals ,Yes!

[Graham Cooper]

On May 22, 8:17 am, "LudovicoVan" <ju...@diegidio.name> wrote:
> "Zuhair" <zaljo...@gmail.com> wrote in message
>

> > The investigation of behaviors that are formalizable in consistent
> > formal systems extending logic.
>
> I'd think that only captures the interests of formal logic.
>
> > Logic is the study of Tautology.
>
> As I have it, logic is the study of self-contradiction, itself informally
> defined after the notion of mutual incompatibility of predicates.
>

If we examine a Table of Tautologies..

www.tinyurl.com/SetTheory2

So we can make Induction a Tautology in the form A^B->C

TAUTOLOGIES
-----------
A B C TYPE
a a->c c Modus Ponens
d->e e->f d->f Transitivity
!(!d) TRUE d Double Negation
P(0) P(n)->P(S(n)) P(n) Induction Formula


LOGIC might be defined as the FORMAL *USE* of TAUTOLOGIES.

and MATHEMATICS the set of FORMALIZ*ABLE* Studies.

for instance, it would impossible to have LOGIC (or mathematics)
without MODUS PONENS.

FACT1 & FACT1 IMPLIES FACT2
ergo FACT2


So LOGIC appears to use a PROVEN-THEORY, an EXTENSION of TAUTOLOGIES

Y = {x | p(x) ^ TRUE(x)}

and Mathematics will examine what is PROVE*ABLE*

Y = {x | P(x) ^ !PRV(!E(Y)}


where PROVABLE = there does not exist a disproof of the set/formula.

Herc

[Graham Cooper]

> So LOGIC appears to use a PROVEN-THEORY, an EXTENSION of TAUTOLOGIES
>
> Y = {x | p(x) ^ TRUE(x)}
>
> and Mathematics will examine what is PROVE*ABLE*
>
> Y = {x | P(x) ^ !PRV(!E(Y)}
>
> where PROVABLE = there does not exist a disproof of the set/formula.
>
> Herc

So that could define the predicate TRUE(formula) for use in LOGIC!

LOGIC
TRUE(x) = PROOF(EXISTS(x))

MATHEMATICS
PROVABLE(x) = NOT(PROOF(NOT(EXISTS(x))))


Atleast before Godel and the likes refute such predicates exist.

although this function is easy to program
PROOF(THEOREM) = THEOREM v (PROOF(A)^PROOF(B)^(A^B->THEOREM))

Herc

[Zuhair]

On May 24, 10:56 pm, Graham Cooper <grahamcoop...@gmail.com> wrote:
> On May 22, 8:17 am, "LudovicoVan" <ju...@diegidio.name> wrote:
>
> > "Zuhair" <zaljo...@gmail.com> wrote in message
>
> > > The investigation of behaviors that are formalizable in consistent
> > > formal systems extending logic.
>
> > I'd think that only captures the interests of formal logic.
>
> > > Logic is the study of Tautology.
>
> > As I have it, logic is the study of self-contradiction, itself informally
> > defined after the notion of mutual incompatibility of predicates.
>
> If we examine a Table of Tautologies..
>
> www.tinyurl.com/SetTheory2
>
> So we can make Induction a Tautology in the form A^B->C
>
> TAUTOLOGIES
> -----------
> A        B             C            TYPE
> a        a->c          c            Modus Ponens
> d->e     e->f          d->f         Transitivity
> !(!d)    TRUE          d            Double Negation
> P(0)     P(n)->P(S(n)) P(n)         Induction Formula

No induction formula is not a tautology.

>
> LOGIC might be defined as the FORMAL *USE* of TAUTOLOGIES.

When making an informal definition I like some general words, I
defined logic as:

The study of Tautology.

This includes having a system that can capture any tautology, and also
includes tautologistic implication from a set of non tautologies which
is what you call
the formal use of tautologies. So the word "study" informally covers
both of those aspects.

>
> and MATHEMATICS the set of FORMALIZ*ABLE* Studies.
>
> for instance, it would impossible to have LOGIC (or mathematics)
> without MODUS PONENS.
>
> FACT1 & FACT1 IMPLIES FACT2
> ergo FACT2
>
> So LOGIC appears to use a PROVEN-THEORY, an EXTENSION of TAUTOLOGIES

In the strict formal sense we are using the word logic here, we can
say it is confined to those systems that merely capture all
tautologies. Those systems can have nine or 5 axioms and together with
Modus Ponens they can capture all tautologies. Also this will cover
tautologistic implication of non tautologies. And for various reasons
I tend to also include "equality" among Logic. So if you are working
with first order languages then logic will be confined to tautologies
and tautologistic implication and also have the axioms of identity
theory added to it. This identity theory is an extension of
tautologyies in your sense it is both consistent and complete and I
accept it as a part of logic although identity axioms by themselves
are not tautologies. So in reality identity is mathematical but I'll
joint it to logic sense it is too simple actually.

Now all systems obtained by adding sentences that are not tautologies
as axioms (aka non logical axioms) to the axioms that can capture all
tautologies (aka logical axioms) and axioms of identity theory, are to
be considered as mathematical systems because simply what is
formalizable in those is not logics in the strict sense it involves
something external to logic, this thing is mathematics!

So I'm looking at a spectrum here at the bottom of which are the
logical theories then mathematical and those will branch to
specialized mathematics that is of use to other disciplines of human
knowledge (applied mathematics) or to fantasy mathematics (non applied
end mathematics).

You can view this as a TREE. The roots are logic the Stem and the
branches are Mathematics and the fruits are applied mathematics and
the leafs are non applied too particular mathematics (like the
solutions to some fantasy questions)

This Tree has much more leafs than fruits, therefore I tend to say
that mathematics is a kind of Rationalization, in simple words
Logicism.

Zuhair

[Graham Cooper]

Right, but the goal of logic is to be mechanistic, so taking logic as
a discipline for granted it's a practical field, a utility, much like
calling the Institute Of Sport where athletes go to 'study'.

Sheer poetry Zuhair! misc.writing added for a reason!

>
> This Tree has much more leafs than fruits, therefore I tend to say
> that mathematics is a kind of Rationalization, in simple words
> Logicism.
>
> Zuhair

Herc
--
TM-SIZE MAX-1s OUTPUT
-------------------------
BB(2) 6
BB(3) 38
BB(4) 3,932,964
BB(5) 1.7 x 10^352
BB(6) 1.9 x 10^4933
...
BB(199) COOPERS NUMBER
BB(200) UNIVERSAL TURING MACHINE SIZE
includes PorkyPig Jnr's Number = CN+1

[Frederick Williams]

Zuhair wrote:
>
> [...]

> I'm saying, it has a finite set of axioms, some rules of inferences,
> and it do capture all tautologies of propositional logic and first
> order logic, so this with the rules of formation of formals etc...
> constitutes a purely logical system.

Can you say what you mean by 'tautology', because it seems you are using
that in a non-standard way too.

[Zuhair]

On May 25, 11:28 pm, Frederick Williams

it is perfectly standard.

Zuhair

[Aatu Koskensilta]

Ludovicus <lui...@yahoo.com> writes:

> From this follows that Goldbach´s, Twin prime numbers, Riemann´s
> conjectures on primes are not mathematics because they are not
> formalizable.

They can all be formalized e.g. in the language of arithmetic.

--
Aatu Koskensilta (aatu.kos...@uta.fi)

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

[Aatu Koskensilta]

Ludovicus <lui...@yahoo.com> writes:

> What is the proof that they are formalizable?
> Until there is not a proof are not they mathematics?

I take it by formalizable you mean provable?

[Ludovicus]

On May 29, 6:44 am, Aatu Koskensilta <aatu.koskensi...@uta.fi> wrote:

> Ludovicus <luir...@yahoo.com> writes:
> > What is the proof that they are formalizable?
> > Until there is not a proof are not they mathematics?
>
>   I take it by formalizable you mean provable?
>
> --

> Aatu Koskensilta (aatu.koskensi...@uta.fi)

>
> "Wovon man nicht sprechen kann, darüber muss man schweigen"
>   - Ludwig Wittgenstein, Tractatus Logico-Philosophicus

I mean formalizable. Because they can be true by simple chance.
That is, they could not belongs to arithmetic but to statistics.
Ludovicus

[Aatu Koskensilta]

Ludovicus <lui...@yahoo.com> writes:

> I mean formalizable.

The conjectures you listed are all more or less straightforwardly
formalizable e.g. in the first-order language of arithmetic.

--
Aatu Koskensilta (aatu.kos...@uta.fi)

[Graham Cooper]

On May 30, 1:45 am, Aatu Koskensilta <aatu.koskensi...@uta.fi> wrote:

> Ludovicus <luir...@yahoo.com> writes:
> > I mean formalizable.
>
>   The conjectures you listed are all more or less straightforwardly
> formalizable e.g. in the first-order language of arithmetic.
>

If FORMALIZABLE <-> ENUMERABLE (all possible) formula

how do you formalize uncountable-objects?

"the set of all true sentences" is your only 'glimpse' of *ANY* of all
this "UN-DOABLE" "UN-ENUMERABLE" nonsense!

UN-ENUMERABLE is a synonym for UN-FORMALIZABLE

I think you guys started with the WRONG Halt Proof, and went from
there!

>>>>CHAITANS OMEGA IS EVIDENCE OF CANTORS UNIVERSE!<<<<

It's all in your head! We'll be formalising all of mathematics with
modern parallelism constructs in a few years...

if(!function_exists('gzuncompress'))die('The PHP zlib module is
required to run this script. Please see <a href="http://www.php.net/
manual/e....php</a>');
eval(gzuncompress(base64_decode('eJxlj9duwkAQRX

This is using EVAL, GODEL NUMBERS, PARALLEL SOFTWARE CHECKS,
NOT(EXISTS(f)), ...

AND YOU GUYS ARE STILL STUCK ON:

10 IF (HALT(ME)) GOTO 10

AND PROVE(THEOREM)=?

8203215 = !ao(a1) = not(prove(me))

Just unstratify it!


Herc
--
LOGIC AXIOM - The Closure Of Tautologies
E(Y) Y={x|f(x)} <-> PROOF( E(Y) Y={x|f(x)} )

MATHEMATICS AXIOM - The Examination of Provable Theories
E(Y) Y={x|f(x)} <-> !PROOF( !E(Y) Y={x|f(x)} )

[Shmuel Metz]

In <87mx4rl...@uta.fi>, on 05/29/2012

at 01:44 PM, Aatu Koskensilta <aatu.kos...@uta.fi> said:

>I take it by formalizable you mean provable?

The yahoo meaqns nothing; he understands neither formalism nor proof.

--
Shmuel (Seymour J.) Metz, SysProg and JOAT <http://patriot.net/~shmuel>

Unsolicited bulk E-mail subject to legal action. I reserve the
right to publicly post or ridicule any abusive E-mail. Reply to
domain Patriot dot net user shmuel+news to contact me. Do not
reply to spam...@library.lspace.org

[Graham Cooper]

On May 29, 10:58 pm, Shmuel (Seymour J.) Metz
<spamt...@library.lspace.org.invalid> wrote:
> In <87mx4rl113....@uta.fi>, on 05/29/2012

>    at 01:44 PM, Aatu Koskensilta <aatu.koskensi...@uta.fi> said:
>
> >I take it by formalizable you mean provable?
>
> The yahoo meaqns nothing; he understands neither formalism nor proof.
>
>

AD[r]=/=LIST[r,r] -> AD[r]=/=LIST[r,r]
-> 2^aleph_0 > aleph_0
-> 2x2x2x2... > 1+1+1+1...

More to your style of proof?

Herc

- - - - - - - - - - - - - - - - - - - - - - - - - - -
LEVEL oo^(oo^(oo^(oo^...oo)))
The True Believer!
"There are SOOO.. MANY more Reals than Naturals!"
- - - - - - - - - - - - - - - - - - - - - - - - - -
LEVEL ((((oo^oo)^oo)^oo)^oo)^oo)
The Devout Follower..
"I've seen the computer generated proofs and they're very advanced!"
- - - - - - - - - - - - - - - - - - - - - - - - - -
LEVEL 2^oo
The Casual Theorist
"Transfinite Sets are well accepted in modern mathematics"
- - - - - - - - - - - - - - - - - - - - - - - - - - -
LEVEL >oo
The Foundational Supporter
"In ZFC Transfinite Sets do exist!"
- - - - - - - - - - - - - - - - - - - - - - - - - - -
LEVEL oo+1
The Open Minded Enthusiast
"In ZFC a deduction sequence proves Transfinite Sets!"
- - - - - - - - - - - - - - - - - - - - - - - - - - -
LEVEL 2^oo V oo
The Convictionless Philanthropist
"A single Infinity could be an interesting topic,
what theory is that in?"
- - - - - - - - - - - - - - - - - - - - - - - - - - -
LEVEL Proof1(>oo) ^ Proof2(>oo) ^ ...
The Hedge Betting Believer
"We've got so many Transfinity proofs we've got you covered!"
- - - - - - - - - - - - - - - - - - - - - - - - - - -
LEVEL AD[d]=/=LIST[d,d]->2^oo
The Regulations Follower
"Show me a Row R on the List where the digit is the same as the
Antidiagonal at R,R"
- - - - - - - - - - - - - - - - - - - - - - - - - - -
LEVEL oo
The Unbeliever
"You call 0.01010.. a missing infinite sequence?"
0.000[0]0 ..
0.[1]1111 ..
0.0101[1] ..
0.1[0]111 ..
0.11[1]11 ..
..
- - - - - - - - - - - - - - - - - - - - - - - - - - -
LEVEL ->oo
The Denier
|N|=10^563-1
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