Re: Attempts to Refute Cantor's Uncountability Proof?

Hatto von Aquitanien wrote:
Dave L. Renfro wrote:

Cantor's first paper in set theory (1874) was primarily a
proof for the existence of transcendental numbers. The proof
that the set of real numbers is not countable was the tool.

I don't claim expertise in this area, but I thought it had, by that date,
long been established that some numbers were both irrational and
non-algebraic.

I'm not totally sure, but I think that the existence of non-algebraic integers was proved only very shortly before this date - it was shown that sum 10^{-n!} is non-algebraic. I think that pi and e are non-algebraic came later.



Denjoy's extension of the Lebesgue integral, around 1914-16,
was directly based on a transfinite construction process.

There are many more examples, even if one restricts themselves
to before 1915. After the mid 1920's, when the Polish and
Russian analysis and topology schools began to become very
productive, the applications really took off.


I've not encountered these in any applied mathematics. I have a high regard
for what I consider Pure Mathematics - hence my conservativism and pedantry
regarding the use of the term "Axiom". Nonetheless, Kant's warning against
building conclusion upon improper foundations seems apropos here. Since it
appears that many mathematicians are using Cantor's conclusions directly or
indirectly as a foundation upon which to build, the matter of soundness
seems vitally important.

My personal viewpoint is that current mathematics is built upon improper foundations, but that it is pointless and worthless trying to find the flaws at this point in history. For example, I believe that Einstein could never have figured out general relativity had not Riemann and Gauss figured out the mathematics behind curved space, and as such our attempts to figure out or correct the flaws in our current models are just as futile as any attempt to figure out general relativity without the notions of curved space would have been.

While I believe that the current foundations are deeply flawed (and let me reiterate - this belief is not based upon scientific reasoning), I do think that our current foundations are (a) very good and (b) good enough for now. When the flaws are found, I believe that most of mathematics will survive - perhaps only the really weird stuff like inaccessible ordinals will die. Thus it makes very great sense for us to continue our studies in math as if everything is just fine, and it would be an exercise in futility to try to get our foundations completely correct.

From your frequent statements "I am not an expert in XXX" I suggest that you read a good book on mathematical logic. The book from which I learned this stuff is by Mendolsen (spelling?), and it seems to have so many editions that it must be considered by others as well to be a good book. I think that when you know the subject well, you will better appreciate both the strengths and weaknesses of the modern approach.

Stephen
.

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