ABSTRACT:
According to the current dogma, Aleph-0 is less than Aleph-1, but is there evidence to the contrary? Is it really true that something with an unlimited upside is less than another thing with an unlimited upside? This paper offers proof to the contrary.
Equation 1 below is the formula used to build Cantor's cardinal infinities. The next higher-order of infinity is equal to two to the power of a lower-order infinity, assuming we are working with binary numbers. Take note that if we express the higher-order infinity as a binary number, we have a one followed by an infinite number of zeros. We might dare to ask, what kind of infinity is that infinity of zeros? The zeros are discrete and distinct and can be counted. Thus we can infer that the infinity of zeros is the same infinity associated with the set of natural numbers: Aleph-0. There is a one-to-one mapping between the zeros and natural numbers (see mapping below):
Therefore, when we express any order of infinity in binary form, the infinite digits (zeros) following the one must amount to Aleph-0. That means equation 1 is wrong. The correct equation is equation 2:
All cardinal infinites ranging from Aleph-1 to Aleph-infinity are equal to Aleph-1. What about Aleph-0? If we assume it is structured like the other cardinal infinities (2^n) and express it as a binary number, once again we have a one followed by infinite zeros. Since those digits can be counted, we are forced to conclude that their number also amounts to Aleph-0. So Aleph-0 also equals Aleph-1:
The conclusion at equation 5 shows all cardinal infinities are equal to Aleph-1. Of course we are told over and over that this can't be! So let's look at a complete list of whole numbers ranging from 000... to 111... and numbered or paired with natural numbers ranging from 1 to 1000... To have a complete list, we need 2^Aleph-0 rows and Aleph-0 digits or columns when we express the list in matrix form:
If we apply the diagonal method, the matrix needs to be square, i.e., the number of rows must equal the number of columns, so it is physically impossible to do the diagonal method with a complete list of whole numbers unless 2^Aleph-0 rows equals Aleph-0 columns. We could imagine the matrix being square if the number of rows is 2^Aleph-(-1), where Aleph-(-1) is a lower order of infinity than Aleph-0. That would be consistent with equation 1 above. Unfortunately, Aleph-0 is the lowest order of infinity possible--thus Aleph-0 must equal 2^Aleph-0 or Aleph-1. This implies, however, that the diagonal method should fail to produce a new number not on the list:
In the left diagram above, we flip each digit along the diagonal. The created number (in red) must have a leading digit of 1 and a last digit of 0. A complete list, however, has a range of 000...0 to 111...1. The created whole number is greater than 000...0 and less than 111...1, so it already exists on a complete list of whole numbers. This makes perfect sense if you stop and think about it. However, this implies that Cantor did not use a complete list for his diagonal-argument demonstration. He used a n X n matrix format when a complete list requires a 2^n X n format.
Notwithstanding the forging arguments and evidence in favor of all cardinal infinities being equal, it is counter-intuitive to claim that there is a bijective mapping between the set of natural numbers (N) and the set of real numbers (R). If we try to identify and count the next real number above or below zero, for instance, we get stuck. So how do we know there are Aleph-1 members in set R? Take a line of real numbers and divide it in two, then divide each new line segment in two and repeat this exercise an infinite number of times:
Notice that the final result is 2^Aleph-0 or Aleph-1 infinitesimal line segments. Also notice that line segments are discrete and can be counted:
Imagine each line segment with a length of dx shrinking towards zero. The total number of line segments grows to infinity as they continue to shrink, but because they are discrete, they are countable. They only stop being countable when they all have zero length, but by the time they all reach zero length, we have counted all the members of R.
Therefore there is a bijective mapping between sets N and R and Aleph-0 equals Aleph-1. Additionally, we have shown above the that following is true:
References:
1. Cantor, Georg. Grundlagen einer allgemeinen Mannigfaltigkeitslehre (Foundations of a General Theory of Sets). jamesrmeyer.com.
2. Cantor, Georg. Uber eine elemtare Frage de Mannigfaltigketslehre (On an Elementary Question of Set Theory). jamesmeyer.com. 3. Cantor's Theorem. Wikipedia
4. 2020. SP20:Lecture 9 Diagonalization. courses.cs.cornell.edu
5. Cantor's Diagonal Argument. Wikipedia
6. Cardinality of the Continuum. Wikipedia
7. Continuum Hypothesis. Wikipedia
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