One of Georg Cantor’s best-known and widely applied suppositions is the foundation for his famous diagonalization argument. And, it makes the assumption that infinity is a fixed and reachable locus on the number-line. A location one might surpass entering the realm of transfinite values.
It should be said that the root of the word, “supposition”, is the term, “suppose”. This is the same intellectual perspective used when constructing a “proof by contradiction”. And, it might be a fully “unsound” premise. If, a convincing counter-example is found.
If, one examines the modern theory of computation. Which does not have the same standard as mathematical “law”, such as the "Rule of Pythagoras". He will see that its canonical proof-work establishes that “subtraction” is not computable, based upon the model of computation used within this theory. And, that result partially depends upon Cantor’s diagonalization argument.
By the definition of a number and the rules for their construction, one might argue that the production of a larger number is
as simple of prefixing a sequence of digits with another one greater than zero or adding a digit as a suffix and sifting
each one already found in the number right one position in the direction of the highest tens position.
So, by definition, infinity is a location that ultimately remains unreachable. And, it must. Otherwise, the fabric of our
numbering system, that is the basis of mathematics, shall be rent irreparably.
And, this is best evidenced by the “proof” that one of the four fundamental operations which is not “commutable” is classifiable as not “computable”.
One might make certain assertions based upon coincidences arising from “time and chance” such as Cantor’s surname, his choice of names for digits beyond infinity: Aleph, Beth, Gimmel, Daleth, ….; plus, his most probable personal-faith. And, one might argue that his “proof-work” on the transfinite focuses the mathematically-minded on the passages of Judeo-Christian Scripture that is numbered in such a fashion and the weightier matters of life. As, one finds in Psalm 119.
Ultimately, any great mathematician is first a philsopher and second an artisan who crafts theorems, corollaries, lemmas, axioms, and such from mathematical-implements.