15. ÖMG-Kongress
Jahrestagung der Deutschen Mathematikervereinigung

16. bis 22. September 2001 in Wien

Sektion 4 - Mathematische Logik, Theoretische Informatik
Donnerstag, 20. September 2001, 17.00, Hörsaal 46

 

The Puzzle of Transfinite Integers

Endre Kövesi, Wien

Consider the base ten, arabic numerals: $3.10^0 +3.10^1 + 3.10^2 + \dots +3.10^n + \dots$ also written as 333 ...or $\bar 3$. According to set theory the members of the above series are each finite ones, and constitute a least infinite set (of the equinumerical class $\aleph_0$) together. So do nonterminating decimals (like: $0.\bar 3$). Obviously ``$\bar 3$'' is a finite symbol, representing a class $\aleph_0$ set of digits of the numeral ``$3$''. We can say:``$\bar 3$ is $\aleph_0$ long - digit wise.'' In this $\aleph_0$ long chain of three-s, each digit has one immediate neighbour. Since $\bar 3$ actually exists according to theory, it has exactly one first and exactly one last member (digit) at the ``end of infinity''. Clearly, as an individual $\bar 3$ has one predecessor (the sum $2.10^0 + 3.10^1 +3.10^2 + ... +3.10^n + ...$) and one successor (the sum $4.10^0 +3.10^1 +3.10^2 +\dots+3.10^n+\dots$). It is a positive whole number ($\bar 3$). (So are: 000..., 111..., 222..., ..., 999..., 14142..., 314159..., 271828...). If infinitely long integers exist, what is the total number of these numerals? And what do they represent: finite, or infinite (or both) values? We give here some of the relevant answers and show, how to to integrate the numerical systems $\mathbb{N},\mathbb{Z},\mathbb{Q},\mathbb{R},\mathbb{C}$ of any base and any class in one, simple universal construction ${\bf K}$. As a result, (while nothing is wrong with the theory of finite sets) Transfinite Set Theory is eliminated.

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