NFU as a logical base for Cosmology.
Introduction:
Bertrand Russell advocated the use of logic in discussing
cosmological issues.
Here I will present an axiomatic system that is based on
NFU ( New foundation with Ur-elements), with additional
axioms related to the subject of cosmology.
NFU is chosen here because it allows for the existence
of a universal set of all sets ,without being involved in
Russell's paradox nor Cantor's set paradox,
nor Burali-Forti paradox.
I advocate using a Type set theory with ur-elements
like NFU, as the base set theory of any consistent logical
model of cosmology.
Traditional set theories like ZFC and NBG and MK
cannot serve as a base set theory for logical models
of cosmology , because they do not allow for the existence of
a universal set of all sets, and this will impose a limitation
of the cosmological model by the design of these theories
which is something we don't want in any physical law.
Since cosmology is about the universe, then the set base of any
cosmological model should be a set theory that allow for the existence
of universal set of all sets, in a consistent manner, without being
involved with the paradoxes mentioned above.
Terminology: I will use the terminology used in first order logic
language.
A Universal quantifier = Any = every
E Existential quantifier = at least one exist
e epislon member****p = is a member of = in
~ Negation = Not.
& conjunction= And
v disjunction = Or
-> implication=uniconditional
<-> biconditional
= identity so x=y mean x is identical to y.
SIMPLIFIED VERSION OF NFU BASED COSMOLOGY.
First, a set is a collection of elements that meet a certain
property.
so for example the collection of men, the collection of animals,
etc..
all of these collections are sets.
We can symbolize this as y={x|P} , to mean that y is the collection
of
all x that has property P.
here P is called the property of member****p of y, this mean that if
any x has property P then x is a member of y, and if x is a member
of y then x has the property P.
Example y= { x|Man }, this mean that y is the collection of all men.
Similarly y={x| animal}, this mean that y is the collection of all
animals.
When a set become a member of itself?
Any set become a member of itself only if it fulfils the property of
its member****p.
What I mean is the following:
If y={x|P} and y is P, then we say that y is a member of y.
So y is a member of y , mean that y fulfills the property of its
member****p.
Not all set theories permit such sets that can be members of
themselves,
However the model that I am presenting is based on NFU ( New
foundations
with Ur-elements).
which allows sets to be members of themselves if they fulfill the
property of their member****p.
An example in NFU of a set that is a member of itself ,
is the set of all sets V
V={x|x=x} here V is the set of anything that has the property of
being
equal to itself.
Since everything is equal to itself. Then V is the set of everything.
Now V itself also is equal to itself, i.e V=V, and by then V fulfills
the property of its member****p, therefore we say that V is a member
of
V, so instead of saying that x is a member of y , we simply symbolize
this
as xey. So 'e' mean ' is a member of '.
That was preliminary set.
Now I will introduce my model.
The base set theory of this model is NFU.
First I begin by defining the property ' objective' as below:
x is objective if and only if x exist (in whole or part) irrespective
of intelligent observation.
So for example the Moon do exist even if all intelligent beings in
the
world vanished. so the Moon exist irrespective of intelligent
observation.
The pyramids of Egypt although their start depended on intelligent
observation made by human beings, but their continuing to exit has
nothing to do with intelligent beings observing them, so that even if
all intelligent being vanish, still these pyramids can be imagined to
exist.
Now we start this Model with Assumption 1.
Assumption 1. There exist a thing that is objective.
In symbols we Wright that as Ex( x is objective ).
This assumption can be proved easily, The Moon can be that x, since
it
exists objectively .
Assumption 2. Any collection of objective things IS itself objective.
so if you have a collection of the Moon, Sun, Sebastian , and lets
call this collection y
so y ={ Moon, Sebastian, Sun }
since each member of y is objective, then according to assumption 2,
y
is objective, i.e y exist irrespective of intelligent observation.
Now Lets proceed and define the collection of all objective things.
and lets call this collection J.
So in symbols J={ x| Objective }.
Now according to assumption 2. then J is objective.
So J fulfills the property of its member****p.
So we say that J is a member of itself.
we symbolize this as JeJ.
Now we reached the point were we should define the relation ' cause '
we say that x cause y to mean that: x is the cause of y.
Now: x cause y means that: x is different from y and if x doesn't
exist in J then y doesn't exist in J.
So this mean that if x doesn't exist irrespective of intelligent
observation then this leads to y doesn't exist irrespective of
intelligent observation, and
if x is different from y; then: x cause y.
To make it more simple, we say x cause y if x doesn't exist in the
physical world J then y doesn't exist in the physical world J.
Provided that x is different from y.
Example. IF your father didn't exist in J, then you wouldn't come to
this world J.
so Father cause Son
The formal definition of 'cause' is
x cause y <-> ( ~x=y & (Ey(yeJ)->Ex(xeJ) ).
Now the most im****tant thing is to know the 'cause between an element
and a set'.
The following is the cornerstone of this model, so please
concentrate on it to understand it.
IF x cause the set y , then x cause every member in y.
so if y={x|P} and m cause y , then y cause every x that is in y.
Example: y={ a, b, c }
m cause y leads to m cause a and m cause b and m cause c.
The easiest way to imagine that is to imagine the Painter and its
paintings.
we have y={ x|Painted by the painter z}
so y is the collection of all paintings painted by z.
So z here is the cause of y, since if z doesn't exist in J , then non
of its paintings would have existed in J.
so we have z cause y leads to z cause every painting in y.
we right that in symbols in the following manner
x cause y <-> Az( zey <-> x cause z)
A means 'every'
this formula mean , if x cause y then x cause every member z in y,
and if x cause every member in y then x cause y.
I think this point is clear enough.
>From that it should be understood that if x cause y
then for any subset k of y, x cause k.
Example the painter who painted paintings p1,p2 and p3
is the cause of x={p1,p2,p3}, and also the cause of y={x1,x2}
and the cause of z={x1} , etc...
However what I want everybody to understand here is that
if m is a member of y, then m cannot be the cause of y.**Very
im****tant**.
Because if we have m cause y and mey, then this will lead to
m cause m, which is contradictive, because the definition of 'cause'
require that the cause be different from the caused.
This point should be firmly understood. Because most of what is
present in this model depends on it. And if this point is not
understood, then non of the rest of this thread would be understood.
Note: in the rest of this post whenever '(see above)' is mentioned it
mean this point.
Now we start the model.
We start by asking ourselves Does every member of J has a cause?
The answer is NO.
Why?
Because if we assume that every member of J has a cause, then since J
is a member of itself, then J should have a cause, Now this cause
cannot be a member in J since this will lead to this member causing
itself ( see above), which is contradictive; Nor the cause of J can
be
something outside J, since what is outside J doesn't exist
irrespective of intelligent observation, and so it is not objective,
and what is not objective cannot cause something that is objective.
Accordingly we reach the result that NOT every member of J has a
cause.
Which mean that there should exist at least one member of J that is
uncaused.
IF we define the supernatural as the uncaused
x is supernatural <-> x is uncaused.
x is natural <-> x is caused.
Then from the above proof, we understand that there should exist
at least one supernatural in J.
Now we proceed to define U, were U mean the universe.
The universe is defined as the collection of all caused objective
matters.
So every member of U should have a cause in J, and any member of J
that has a cause in J should be in U.
so U={ x| xeJ & Ey( y cause x) }.
E means 'there exist'.
Now we ask ourselves is U identical to J ?
The answer is NO.
Proof: Either U is uncaused, and by then U is not fulfilling the
property for its member****p, and by then U is not a member of itself.
Now since U has objective existence, because everything in it is
objective, and from assumption 2 this leads to U being objective,
then
UeJ , i.e U is a member of J.
So J has a member in it that is not present in U. According to
Extensionality then J is different from U. Because according to
Extensionality two sets are different if one of them contain a member
that the other doesn't. And here J contain U as a member in it, while
U doesn't contain U as a member in it, So J is different from U.
Now the second possibility is that 'U is caused by a member in J', if
so then this member in J cannot be in U, since this would lead to
this
member causing itself ( see above ), so this member in J which is
causing U should be outside U, and thus
J contain a member that is not a member of U, then J is different
from
U.
So from the above it is proved that ~J=U. ( read as NOT J identical
to
U).
So this mean that there should exist a member in J outside U, and
since it is outside U, then it is supernatural i.e uncaused.
Of course this supernatural can be U itself, since U can be imagined
not to be a member of itself, i.e U is uncaused.
Or this supernatural can be something other than U, and U uncaused,
i.e two supernaturals.
Or this supernatural can be something other than U , that is outside
U
that caused U, and here U would be in itself, since it would fulfill
the property of it member****p.
Now IF we say that U is the supernatural in J.
Then for every member x in U, such as the moon, Sun, etc...
there should be another member in U that caused it, and by then what
we will have is a chain of causes for every member x in U, we call
that chain
X, which is the set of ALL causes of x in U.
so X = { x, x1, x2,x3,............}
were x1 cause x, and x2 cause x1, and x3 cause x2, .................
Now we know that X is a subset of U, but IS X caused or uncaused.
IF we say that x is caused by m for example, then m is either in X or
outside x.
IF m is a member of X, then this will mean that m cause X and meX
which leads to m cause m. which is contradictive ( see above ).
So m should be outside X i.e m should not be a member of X, BUT by
then X would not be the set of ALL causes of x in U, A contradiction.
So X is uncaused, and by then X is supernatural.
What does this mean?
It means that for every member n of U, there will be a cause chain
that is a subset of U, that is uncaused, i.e a supernatural.
Thus if we assume that U is uncaused, then not only U will be the
supernatural, but every cause chain of every element in U would be
supernatural also.
So the number of supernaturals would be at least the same as the
number of elements in U.
Which is an undesirable result.
Because in any logical model of existence, we desire to reduce the
number of supernaturals to minimum.
Why?
because these supernaturals are unpredictable, since they are not
caused, so we cannot put them in a causative law, which makes
interpretating existence by this model a very difficult task if not
impossible.
So from all the above , it seem more plausible to reject the
assumption of 'U is supernatural' , and to adopt the other assumption
which is U is caused.
And this mean that there exist a member of J that is outside U, and
that is not U, that caused U.
This model would be easier, because it only has ONE supernatural in
it.
This model has no proof that the supernatural is ONE.
However it only assume it to be ONE, so that to reduce the number of
supernaturals to minimum, and this can only be made if we assume
that U is caused by something outside it and something that is not U.
While we cannot do that reduction of supernaturals in the model which
has U as a supernatural.
And that's why The model with the one supernatural causing U is more
plausible.
However, who is that one supernatural?
Obviouselly it is J itself.
This model thus would sup****t indian monisim.
RESULT:
The uncaused being do have an objective existence, as objective as the
existence
of the Sun is. And it is more plausible to have a model were the
universe of
all caused matters is itself caused by one supernatural outside it.
Otherwise
we'll have a universe with many supernaturals in it.
And this one supernatural is J itself.
So this model sup****ts indian monisim.
Zuhair
|
