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Re: 2.65  Does bijection between infinite sets mean they have the same number of elements?
Posted:
Aug 20, 2014 3:27 PM


On Thursday, August 21, 2014 1:05:59 AM UTC+10, John Gabriel wrote: > Of course not. Cantor was a deluded mythmatician. > > > > The applet at http://tube.geogebra.org/student/m145489 explains it all. > > A bijection between two infinite sets does not mean they have the same cardinality.
In order for these bad ideas to be true, it must be possible to have an infinite number of functions, but as the applet will show, in order to do this, ALL points in the given real interval must be well defined, that is, distinct and describable by a number.
****** LONG REPLY ON DEFINABLE SET THEORY OF REALS FOLLOWS ******
If you meant describable by a FINITE FORMULA then your assertion would be clearer.
Let's list all reals by listing all computer programs anyway!
UniversalTuringMachine( # , DigitPos ) = 0..9
Now we have a TERMINATING process to inspect any digit in the list.
"GIVEN ANY INFINITE LIST THEN CONSTRUCT A NEW REAL" is a NONTERMINATING PROCESS which is why Cantor's Theory is part of ABSTRACT MATHEMATICS
GIVEN AN INFINITE LIST he has an algorithm that runs forever and makes a new missing real.
Fair enough if you define a real as an infinite string of digits where every digit position in N has a digit value in {0..9}.
Then any infinite string defined by computer must also run forever.
Let UTM( n , pos ) = TM_n(pos)
Assume a Halt Function exists, then
UHM( h , pos ) = UTM( n , pos ) where n is the hth TM where all inputs halt.
DUHM( h , pos ) = UHM( h , pos ) MOD 10
gives the computable reals list (with no missing values).
Each (halting) TM corresponds to 1 Real so
r1 = SUM ( DUHM( 1 , p ) * 1/(10^p) )
A Computable Real is defined as: rn = SUM ( DUHM( n , p ) * 1/(10^p) )

So given any rn, Virgil is right he can calculate any digit of an infinite string in such a way all digits mismatch all(n) rn.
LONG VERSION
Let's forget about Russell's Set for a moment and allow the Naive Set theory definition
ALL(p):WFF
EXIST(S) A(X) XeS <> p(X)

Let the above formula = s.
Since s<>s ~(s<>~s)
its already possible to Logically Deduce that ~E(russell) A(X) X e Russell <> X~eX

So although N.S.T. is inconsistent.
ALL(p):WFF
EXIST(S) A(X) XeS <> p(X)

the specification formula can work if p is stratified not to cause a contradiction.
BETTER:
EXIST(p):WFF
EXIST(S) A(X) XeS <> p(X)

an increasing set of WFF' can be Logically constructed.
In essence
phi e WFF' <> phi XOR ~phi
There is no deterministic transitive closure however since
F > ~G G > ~F
either may be theorems, but not both.
(Well Spotted Jesse F Hughes)
You might have a disjoint set theory with several possible 'Universes'.
So given:
ALL(p):WFF' < notice the dash (wff stratification) EXIST(S) A(X) XeS <> p(X)
and some rudimentary Starting Axioms a large transitive closure over inference can be calculated w.r.t. the base of starting axioms.

Now we have AVOIDED the pitfalls of ZFC Specification Axiom.
ZFC avoids Russell's set in a different way by avoiding biimplication in set specification.
ALL(p):WFF EXIST(S) A(X) XeS < p(X)
this is equivalent to ZFC Specification
XeS <> p(X) & SCZ XeS <> p(X) & XeZ (usual form)

By Avoiding The BiConditional a Class System is introduced.
EXIST(S) A(X) XeS < p(X)
which is where neverending bigger infinities seem to derive!

This is in direct opposition to Naive Set Theory where
EVERY MEMBER OF EVERY SET is DEFINABLE BY A PREDICATE
So: Axiom Of Choice was added :
A CHARACTERISTIC ELEMENT DEFINES A SET.
e.g.
midpoint( L : LINE )
if a midpoint exists, then so does a line!

Or in Cantor's Case:
if an AntiDiagonal Exists, then so does a SET OF THOSE REALS!

AOC: E(X) XeS > E(S)
(Well executed Charlie Boo!)
********************************************
But A.O.C. only complicates matters further.
It seems to define uncountable many elements with a specification of 1 element <=> 1 set.
THE GODEL NUMBER OF A CHOICE FUNCTION
2130415 a1(0,1) MIDPOINT(0,1)
But now you have uncountable many choice functions! One for each SET!
 N  =  GODEL NUMBERS   GODEL NUMBERS  =  FUNCTIONS   FUNCTIONS  =  CHOICE FUNCTIONS   CHOICE FUNCTIONS  =  ELEMENTS   ELEMENTS  =  SETS   SETS  >  N 
Once of these must be incorrect!
The only further resolution for Z.F.C. is Uncountable Many Functions
 GODEL NUMBERS  <  FUNCTIONS 
Which means the WFF include Infinite Length Formula!
Such As:
1 e CS <> 1 ~e SET1 AND 2 e CS <> 2 ~e SET2 AND 3 e CS <> 3 ~e SET3 AND ...
CANTORS MISSING SET FORMULA in its expanded term format.
You probably recognise the above as:
CANTORS SET = { n  n ~e f(n) }

These are INFINITE SEQUENCES of CONDITIONAL STATEMENTS not INFINITE SUMS of ARITHMETIC!
That is why the definition of a Real that each individual digit over N digit positions is specified is insufficient to define the LIMIT EXISTS
BETTER:
A LIMIT for A Computable Real rn = SUM ( DUHM( n , p ) * 1/(10^p) )

Every rn in R is definable by a finite formula
R = ( n  SUM ( DUHM( n , p ) * 1/(10^p) ) }
EVERY ELEMENT is defined.

SUMMARY:
for any constructed predicate Pn
NAIVE Pn(X) <> X e SETn
ZFC Pn(X) < X e SETn
PROLOG Pn(X) > X e SETn
In PROLOG Set Membership is a predicate goal to prove
e( 2 , evens ) ?
Set Specification defines which predicate determines set membership
e( X , evens ) < even(X).
A predicate formalises each numerical function.
even( 0 ). even( s(s(X)) ) < even(X).
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