> On Nov 20, 2:50 am, zuhair <zaljo
...@gmail.com> wrote:
> > 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
> >http://groups.google.com.jm/group/sci.math/browse_thread/thread/83c63...
> > 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
> 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.
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