After Kurt Godel and his incompleteness theorem, mathematicians realized axioms can be pretty much whatever you want them to be. Making math, in my opinion, almost as much an art as a science. Here’s the time to get creative and introduce something radical (or not so radical).
Note: This is not a page to leave axiomatic systems unused. If that’s all you got, that’s all you got, but hopefully someone at least will come alongside and develop theorems and new math for all these new axiomatic systems that we will be defining in this space.
So feel free to come here and steal.
Definition: A number is an infinite sequence of numbers.
Definition: Minimum number (denoted 0 or zero number)- a number that is less than any other number.
Definition: Maximum number (denoted 1 or one number)- a number that is greater than any other number.
Definition: Place value– where in the sequence a number falls. For example, in number N=a,b,c,…, number b has a place value of 2.
Definition: A number is said to be equal to another number when all numbers in each corresponding place value are equal to the other number’s.
Definition: A number can be compared to another number by comparing the numbers in the least place value where the numbers are unequal. A number is said to be less than (or greater than) the other number when the number in that place value is less than (or greater than) the number in that same place value of the other number.
Example: N=0,0,0,0,0,0,0,1,1… is less than M=0,0,0,1,1,0,0,1… because the place value of 4 is the least place value that results in an unequal number and for M that number is 1 and for N that number is 0. One is by definition greater than 0 as it is the greatest number.
Note: I am not indicating by the ellipses (…) that there is any pattern to the sequence of numbers, but only drawing to light that the sequences continue indefinitely.
Theorem: The zero number consists of an infinite sequence of zero numbers.
Proof:
Assume the number zero contains a number that is greater than 0. Let B be a number that contains a zero number in the place value that corresponds to the number contained in 0 that is greater than 0 => B<0
>=< by definition of the zero number. => 0 does not contain a number greater than the zero number. Since it cannot contain a number less than the zero number as zero is the minimum, it must contain only minimum numbers.
Therefore, 0=0,0,0,… as we set out to prove. (Note: here the ellipses do refer to a pattern)
The one number can also be proven to consist of an infinite sequence of maximum numbers in a very similar way.
Loose definition: Depth – Loosely, the type of numbers that a number consists of. A number of depth 0 consists of only zero and one numbers. A number of depth one consists of only depth zero numbers.
Definition: A number of depth n consists of only depth n-1 numbers, where n>0.
Definition: A number has depth zero when it consists of only zero and one numbers.
Definition: A number has partial depth when it consists of numbers of varying depths.
The cardinality of the set of all depth 0 numbers is $\beth_1$, c, the cardinality of the continuum.
The cardinality of the set of all depth 1 numbers is $\beth_2$.