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Century long battles about Transfinite numbers
Georg Cantor created the transfinite numbers as a consequence of its Diagonal Theorem in 1874.
Georg Cantor’s former mentor, Leopold Kronecker, accepted the proof of the diagonal theorem, but
did not accept the conclusion that the subset of real numbers [0, 1] was uncountable which led to transfinite
numbers. As a world class Mathematician, Leopold Kronecker was very influential, he managed somehow to block any hope for Georg
Cantor to publish on the Crelle’s Journal  p:7
Georg Cantor was fearing Leopold Kronecker’s reaction who once said Cantor’s theory was “of no real significance ” p:7
Georg Cantor’s transfinite numbers remained controversial until the very influencial David Hilbert came.
David Hilbert was sympathetic to Cantor’s actual infinity p:3.
David Hilbert was considered open minded by his peers and was generally fair p:7. However David
David Hilbert, was competing against Kronecker on finiteness  p: 27, and was expressing concerns about Leopold Kronecker’s strong opposite views p:7.
As a consequence, David Hilbert elevated Cantor’s Mathematics as paradise compare to Kronecker’s rigorist ones p:54
Justifying the withdraw of the initial Article
The initial draft article “Was Leopold Kronecker right about Transfinity?” siding with Leopold Kronecker’s and Henri Poincaré’s viewpoints has been withdrawn at it was premature to publish anything fundamental before:
- understanding David Hilbert’s viewpoint about actual infinity
- learning finitism
Pro’s about actual infinity
- Promoted by David Hilbert
- Promoted by Standard Mathematics ZFC Set Theory
- A new model mixing actual infinity and potential infinity is studied by TMT
Con’s about actual infinity
- Extract from professor Alexander A. Zenkin’s article p:36 …”So, according to Aristotle, actual infinity is simply the logical negation of potential infinity and, as such, is a scientific impossibility”…
Pro’s about transfinite numbers
- Promoted by David Hilbert
- Promoted by Standard Mathematics ZFC set theory
Con’s about transfinite numbers
- Significant list of transfinite-sceptic mathematicians kept up to date by Professor Wolfgang Mückenheim’s in his “Transfinity: A Source Book”
- Extract from professor Alexander A. Zenkin’s article p:31 Henri Poincaré: all Cantor’s set theory as well as all modern “nonnaive” axiomatic
set theories are really “built on a sand” [Poincaré 1983].
- Inconsistencies: transfinite numbers are part of axiomatic ZFC set theory while being defined outside ZFC’s own axiomes, the worst being that transfinite numbers and overruling ZFC’s own infinity axiome
- As Cantor’s ordinal ω is defined as the largest integer, is ω an even of an odd integer?
- After a century of “existence”: no useful scientific applications
- Einbu’s bijection breaks Cantor’s diagonal theorem outcome: R is the same size as ZxN:
In 2014, John Einbu, a Norwegian hobby mathematician, has made one of the most incredible discovery of the 21 st century in Mathematics with just 3 pages [here]:
A bijection between between [0 , 1] and N: Here is how it is built in decimal base (radix):
1 ↔ 0.1
2 ↔ 0.2
9 ↔ 0.9
10 ↔ 0.01
20 ↔ 0.02
99 ↔ 0.99
100 ↔ 0.001
200 ↔ 0.002
999 ↔ 0.999
1000 ↔ 0.0001
2000 ↔ 0.0002
↔ meaning one to one correspondence
Nevertheless, 6 years later, it remains completely unnoticed apart in W. Mückenheim’s “Transfinity: A Source Book” p:147
Relevance of actual infinity?
Actual infinity created by Georg Cantor and promoted by David Hilbert is relevant if and only if we can figure out a way to make it logically compatible with Aristotelian potential infinity.
Was Leopold Kronecker right about Transfinite numbers?
He was the first to strongly reject the existence of transfinite numbers: the Einbu’s bijection is proving him right 140 years later.