Physical Set Theory
zuhair
the idea of infinity of sets, and replace with the idea of the
universal finite, so this theory would simply say that
a set is a collection(i.e. class) the size of which is smaller than or
equal to the universal finite.
I will use the theory that I have presented lastly to this Usenet,
present on this link
http://groups.google.com.jm/group/sci.math/browse_thread/thread/83c63f646079edc0?hl=en
But I shall drop axiom 5, to allow for the existence of a maximal set
that is finite.
Now the funny idea is that of a maximal finite, this simply postulates
that the total number of physical objects is finite, and a maximal set
(see the above link) would be one that is equi-numerous to the set of
all physical objects, and thus finite!
"physical object" here is a primitive concept, it stands for
physicians call objects in their theories, something that
have extension is space and time complex.
Let me write the exposition of this theory:
Physical Set Theory : is the set of all sentences entailed ( from
first order logic with identity and membership and the one place
predicate symbol "physical object", and the one place predicate symbol
"class") by the following non logical axioms outlined below
thefollowing definitions:
Define(contained):- x is contained <-> Exist y ( x e y ).
Define(set):- x is a set <-> ( x is a class & x is contained ).
Axioms:
1.Non class-hood:
For all x (x is a physical object -> ~ x is a class)
2.Extensionality: For all classes x,y
for all z ( z e x <-> z e y ) ->x=y
3.Comprehension: if phi is a formula in which x is not free, then all
closures of
Exist a class x for all y ( y e x <-> (y is contained & phi) )
are axioms.
4.Physical sets: For all y (y is physical object -> y is contained)
Define: x=U <-> for all y ( y e x <-> y is a physical object )
so U is the class of all physical objects.
5.Size Limitation:
For all y (y is a set <-> y sub-numerous to U)
y sub-numerous to x <->
(y is a class & x is a class &
Exist f ( f:y-->x , f is injective)).
were "f:y-->x , f is injective" is defined in the standard manner.
6.Finite-hood: U is finite.
were finite is defined in the standard Tarskian way
(i.e. equi-numerous to a natural number).
A natural number x is defined as an ordinal x that is either empty or
one that have an immediate predecessor y, were every non empty member
of x other than y must have an immediate successor in x and an
immediate predecessor in x.
An ordinal is a transitive class of transitive sets
(i.e. Von Neumann ordinals).
y is immediate successor of an ordinal x <->
x Union {x} = y
y is immediate predecessor an ordinal x <->
y U {y} =x.
Theory definition finished/
The craziest thing about this theory is the proper classes that it
has, of course what I mean by proper classes are classes that are not
contained, i.e classes other than sets.
The class of all natural numbers that are sets would be a proper
class, and it is FINITE!
Lets denote the natural number that is equi-numerous to U as "n"
so the class of all natural numbers that are sets "N" would be
N={0,1,2,3,....,n}.
Now according to this theory n is the number of all physical
objects,and it is finite,so N would be finite also!
so we have a finite proper class.
so N would be the maximal ordinal!
Of course we can have Dedekindian infinite proper classes, for example
S={0,{0},{{0}},{{{0}}},....}
This class can be defined from comprehension using the following
formula
For all z ((0 e z & for all u ( u e z -> {u} e z )) -> y e z)
On this set we can define a successor in the following manner
y successor of x <-> y={x}
So one can easily show that the above class is Dedekindian Infinite.
Since S is subclass of V (the class of all contained objects) then
V is Dedekindian infinite.
V should be countable, i.e it should be equi-numerous with S, and thus
well order-able, so global choice must be a theorem of this theory(I
assume).
x is said to be a pure class if and only if it contains no physical
object in its transitive closure.
Cardinality would be better defined in a different manner from the
usual definition.
x is a cardinal <->
(x is pure & x is transitive &
for all y ((y e x & ~y=0) -> y is singleton))
y is singleton <-> Exist z ( for all u ( u e y <-> u=z ) )
So for every class x, cardinality(x) defined as below:
Cardinality(x) = y <-> (y is a cardinal & y equi-numerous to x).
so we'll have Cardinality(V) = S.
However I don't have the full formal prove of the later statement.
Zuhair
zuhair
> This theory would be really a funny one, in this theory I shall omit
> the idea of infinity of sets, and replace with the idea of the
> universal finite, so this theory would simply say that
> a set is a collection(i.e. class) the size of which is smaller than or
> equal to the universal finite.
>
> I will use the theory that I have presented lastly to this Usenet,
> present on this link
>
An Extreme version of this theory that is not equivalent to it, would
be obtained by replacing axiom of size limitation with the following
axiom.
5.Size Limitation:
For all y (y is a set <-> Tc(y) sub-numerous to U)
y sub-numerous to x <->
(y is a class & x is a class &
Exist f ( f:y-->x , f is injective)).
were "f:y-->x , f is injective" is defined in the standard manner.
Tc(y) stands for the transitive closure of y.
Define: x=Tc(y) <-> for all z ( z e x <-> for all u ( u transitional
of y -> z e u ) )
Define: u transitional of y <-> ( y subclass u & u is transitive ).
With this extreme version of finitisim, one would end up with the
class of all contained objects V itself being finite. i.e. we'll have
no infinite classes at all.
Zuhair
zuhair
Although this version of this theory is an extreme finitisim, but yet
wouldn't it be complete,consistent, and Categorical?
Would it escape Godel's incompleteness theorems 1 and 2.
Zuhair
>
> Zuhair
zuhair
> the idea of infinity of sets, and replace with the idea of the
> universal finite, so this theory would simply say that
> a set is a collection(i.e. class) the size of which is smaller than or
> equal to the universal finite.
>
> I will use the theory that I have presented lastly to this Usenet,
> present on this link
>
>
> But I shall drop axiom 5, to allow for the existence of a maximal set
> that is finite.
>
> Now the funny idea is that of a maximal finite, this simply postulates
> that the total number of physical objects is finite, and a maximal set
> (see the above link) would be one that is equi-numerous to the set of
> all physical objects, and thus finite!
>
> "physical object" here is a primitive concept, it stands for
> physicians call objects in their theories, something that
> have extension is space and time complex.
One should exclude the symbols we are writing logic and math with,
from being a physical object.
If we have a computer machine that can generate "objects" randomly up
to a very huge fixed number, then every one of these objects can be
considered as "physical object", and thus U (see below) can be taken
to be the set of all these objects.
So I think the Extreme finite version of this theory (see below) can
be put into a computer program.