Every natural number n has an endsegment {n+1, n+2, n+3, ...}. The first endsegments are
{2, 3, 4, 5, 6, 7, ...}
{3, 4, 5, 6, 7, ...}
{4, 5, 6, 7, ...}
...
Some defenders of set theory claim that the intersection of all endsegments is empty while no endsegment is empty. I call this statement matheology, a touchstone of irrationality.
Regards, WM
Cantor's measure of infinite sets has been disproved. See https://www.reddit.com/r/AspectsOfTheInfinite/comments/1tc6v1l/proof_of_the_existence_of_dark_numbers/ and https://www.reddit.com/r/AspectsOfTheInfinite/comments/1tdz9dm/classical_mathematics_contradicts_set_theory/
Not all infinite sets can be compared by size, but we can establish some useful rules.
- The rule of subset proves that every proper subset has fewer elements than its superset. So there are more natural numbers than prime numbers, and more complex numbers than real numbers. Even finitely many exceptions from the subset-relation are admitted for infinite subsets. Therefore there are more odd numbers than prime numbers.
- The rule of construction yields the number of integers |Z| = 2|N| + 1 and the number of fractions |Q| = 2|N|^(2) + 1 (there are fewer rational numbers). Since all products of rational numbers with an irrational number are irrational, there are many more irrational numbers than rational numbers.
- The rule of symmetry yields precisely the same number of real geometric points in every interval (n, n+1] and with at most a small error same number of odd numbers and of even numbers in every finite interval and in the whole real line.
This theory makes the number of natural numbers (and of course of other sets too) depending on the numerical representation. The set {1, 11, 111, ...} of natural numbers has only comparatively few elements. Therefore the set of natural numbers in unary or binary notation has fewer, in hexadecimal notation more than |N| elements. The set {10, 20, 30, ...} has |N|/10 elements, but if the zeros are only applied as decoration, this set, like {1', 2', 3', ...}, has |N| elements.
If every rational number were equal to |N| fractions, then only |N| rational numbers would exist. This is clearly wrong. The solution of this paradox lies in the fact that small rationals are equal to more fractions than large rationals. 1 = 1/1 = 2/2 = 3/3 = ... has |N| equal fractions, 100 = 100/1 = 200/2 = 300/3 = ... has only |N|/100 equal fractions. All definable rational numbers have ℵ₀ equal fractions, but almost all numbers are undefinable.
There are fewer real numbers of the form 100.1415... than of the form 0.1415... (see above).
It will be a matter of future research to investigate the effect of different numerical systems in detail.
[W. Mückenheim: "Evidence for Dark Numbers", ELIVA Press, Chisinau (2024) pp. 1-36.] https://www.elivapress.com/en/book/book-5647011244/]
Regards, WM
Example: Take the set ℕ, remove 1 and with n remove also n+1. Does the empty set remain?
Let X(n) = {1, 2, 3, ..., n} be a finite initial segement of ℕ. For every natural number n: ℕ \ X(n) is nonempty. That means it is impossible to insert all n into the template X(n). Almost all remain outside. How can it be explained that all n can completely be inserted into the template (m, n) of a bijection f(n) = m between the sets M and ℕ?
Let X(n) = {1, 2, 3, ..., n} be a finite initial segement of ℕ. For every natural number n: ℕ \ X(n) is nonempty. That means it is impossible to insert all n into the template X(n). Almost all remain outside. How can it be explained that all n can completely be inserted into the template (m, n) of a bijection f(n) = m between the sets M and ℕ?
The product of odd numbers is odd, the product of even numbers is even, but the mixed product is even.
I know two alternatives:
In potential infinity there is nothing next to 1. We can come as close as we like, but we can never close the gap. A gap remains.
In actual infinity, there is a point next to 1. Of course this point cannot be known. It is dark.
Is there a third alternative?
I know two alternatives:
In potential infinity there is nothing next to 1. We can come as close as we like, but we can never close the gap. A gap remains.
In actual infinity, there is a point next to 1. Of course this point cannot be known. It is dark.
Is there a third alternative?
Choose two fractions as close together as you like. Between them there remain infinitely many fractions. Even if you divide the interval by 2 or by 10^(10000000) this will never change.