Transcript Implications of Cantorian Transfinite Set Theory

Implications of Cantorian
Transfinite Set Theory on
Creation

A graduate student at Trinity
Computed the 

But the number of digits
Gave him the figits,
So he quit math and took up divinity.
Georg Ferdinand Ludwig
Philipp Cantor
Born: 3 March 1845 in St Petersburg, Russia
Died: 6 Jan 1918 in Halle, Germany
Georg Cantor developed the set theory for transfinite numbers.

2
0
1
3
o ?
Georg a Protestant, the religion of his father. Georg's mother was a Roman Catholic
Cantorian Infinities
David Hilbert described Cantor's
work as:“...the finest product
of mathematical genius and
one of the supreme
achievements of purely
intellectual human activity.”
"I see it but I don't believe it.”
Georg Cantor on his own
theory.
“…the infinite is nowhere to be
found in reality” David
Hilbert.
Hilbert
http://www-gap.dcs.st-and.ac.uk/%7Ehistory/Mathematicians/Cantor.html
Cantorian Infinities…
 is not a number – it is a
process.
 is approached, never achieved.
limn
Cantorian Infinities…
The set of all counting numbers
C={1, 2, 3, 4, …}
has cardinality* 0 .
Other sets have cardinality* 0 if
each element in the set can be
placed on a one-to-one
correspondence to C.
0
* The number of elements in a set.
The Hottentots
• One
• Two
• Three
• Many
(Gamow)
Cantorian Infinities…
The set of even numbers has
cardinality* 0
E={2, 4, 6, 8, …}
Why? Because of the correspondence:
C
1

2

3

E
2
4
6
0
…
n

…
2n
Hilbert’s Hotel…
0 Rooms – all full.
One more person comes.
No Problem!
Send guest in room 1 to room 2, guest 2
to room 3, etc.
This frees room 1 for the new guest.
0
Hilbert’s Hotel…
0 Rooms – all full.
0 more person comes.
No Problem!
Send guest in room 1 to room 2, guest 2
to room 4, 3 to 6, etc.
This frees all the odd rooms for the new
guests.
0
Hilbert’s Hotel…
0 Rooms – all full. 0 guests leave. How
many rooms are left occupied?
1. Guests from all rooms leave.
0 - 0 = 0
2. Guests from rooms 4 and higher leave.
0 - 0 = 4
3. Guests from all the even rooms leave, or,
guest from every tenth room leaves.
0 - 0 = 0
0
Hilbert’s Hotel…
http://www.buzzle.com/editorials/9-9-2002-26002.asp
Hilbert’s Hotel…
http://www.buzzle.com/editorials/9-9-2002-26002.asp
Cantorian Infinities…
Numerator
1
Denominator
A number is
rational
number if it
can be
expressed as
the ratio of two
integers. The
set of rational
numbers, R,
has cardinality
0.
1
1/12/1
3
4
3
4
3/1
4/1
1/4
1/5
 
0
2
2
1/2

2/2

 
1/3 2/3 3/3 4/3

 
1/4 2/4 3/4 4/4
 
Cantorian Infinities…
The set of all
subsets of a
set with 0
elements is of
cardinality
1. This is a
“bigger”
infinity.
Example:
the set of
irrational
numbers
between 0 and 1
(points on a line)
is of cardinality
 1.
1
Cantorian Infinities…
Proof by counterexample: Suppose a mapping exists:
1
0.7568373947578338747575839…
2
0.9585757348938384758439399…
3
0.1938484857657829202938482…
4
0.5000000000000000000000000…
5
0.6549383493904949848484943…
1
Choose any other digit other than the one circled – say the
number after. The number 0.86314… is not in the table.
Contradiction!
Cantorian Infinities…
The number of points on a line, 1, is the same as the
number of points in a square - or in a cube. Consider
a unit interval and a unit square.
(1,1)
For every point, P, in the
square, there is a unique
corresponding point on the
line segment, and visa versa.
y
P
0
z
1
(0,0)
P
1
x
The point on the line is z=0.132456754…. Taking every other
digit, corresponds to x=0.12574… and y=0.346754…
Cantorian Infinities…
n+1, is the set of all subsets of n .
Q: What is an example of 2?
A: All the squiggles that can be drawn on a plane.
2
Cantorian Infinities…
n+1 is the set of all subsets of
n .
Q: What is an example of 3?
A: Like a fifth spatial
dimension, this is beyond
comprehension.
2
Georg Cantor and Pope Leo XIII
• There exists no biggest transfinite
number.
• googol,
• a = o , or a
• The set of all transfinite numbers does
not exist.
• “From me, Christian Philosophy will
be offered for the first time the true
theory of the infinite.” Cantor
William Lane Craig
William Lane Craig
“…since the actual infinite cannot exist
and infinite temporal regress of events
is an actual infinite, we can be sure that
an infinite temporal regress of events
cannot exist, that is to say, the temporal
regress of events is finite. Therefore,
since the temporal regress of events is
finite, the universe began to exist.”
William Lane Craig
Some absurdities of an infinite past:
1. “To try to instantiate an actual infinite
progressively in the real world would be
hopeless, for one could always add one more
element.” (0 versus .)
2. Tristram Shandy Paradox (Russell): If
Tristram Shandry rote his autobiography 365
times as slow as he lived life, he would finish in
an infinite past. (Careful here! Don’t confuse 0
with .) More…
Tristram Shandy Paradox
• Writing from t = 0 onward:
N years
0 N days
t
Bummer
Tristram will fall
further and further
behind.
Tristram Shandy Paradox
• Writing from the past to end at t = 0:
N years
Cool
N days
No matter how big N,
there is always time for
Tristram to finish:
even if N = 0.
t
William Lane Craig
“… an infinite temporal regress is absurd.”
Thus: Time and the universe are finite. The
universe must have been created ex
nihilo.
There are, of course, other
scientific/mathematical arguments that
reach the same conclusion: The Prime
Mover (first cause) & Big Bang
Cosmology.