Hello,

I just noticed that you are referring to my blog post "What is Wrong with Cantor's Diagonal Argument?". In that post I point to two flaws in Cantor's argument. A couple of weeks ago I added two new paragraphs to my post at:

https://hesperusisbosphorus.wordpress.com/2016/07/31/what-is-wrong-with-cantors-diagonal-argument/

to show that the diagonal argument actually involves infinitely many such flaws and not just two!

I had to remove my "Contra Cantor" paper, which contains my much more extended counter-arguments against Cantor, from the Academia.edu site, because that paper is now under review with a journal. You can still reach an earlier version of it at:

https://dokupdf.com/download/contra-cantor-how-to-count-the-uncountably-infinite-_5a02f7ddd64ab2b9bde5d337_pdf

Best,

E. Sayan

Thanks for your reply.

As I explain in my paper ("Contra Cantor") and my blog post ("What is Wrong with...?") I use a popular reconstruction of Cantor's diagonal argument found in many modern textbooks: I tabulate the reals in the interval [0, 1), rather than using Cantor's sequence version (in terms of m and w) in his original proof. The prevalent assumption is that Cantor's proof in terms of sequences is equivalent to the table version.

And, for convenience, in my tables (and in the trees in my paper) I use the binary representation of the reals in [0, 1) rather than the ordinary decimal representations of them. That's why you don't find numbers like 0.50000..., 0.33333.... in my tables (and trees).

"No good reviewer could give you an o.k. as far — excuse me. "

I know it is hard to go against a giant like Cantor (for whom I have great respect, by the way). All I need is revievers, commentators and readers who would give my paper a careful, honest and unprejudiced reading. I haven't been very lucky in that regard. If there are fallacies in my paper, I would like to learn what they are. So far I haven't detected any myself.

I couldn't find your criticism of Cantor's argument in the link you give. Unfortunately don't know German.

Best,

E. S.

I am afraid I am unable to follow your argument and I don't see the relevance of your criticisms of my argument (such as your remark "Where did you show that no fraction is an irrational number?"). Sorry. But let me make a few points which may answer some of your worries.

The table version of Cantor’s diagonal argument is all over the place. You can find many instances of it in these Google links:

https://www.google.com.tr/search?q=cantor%27s+diagonal+argument&rlz=1C1CHFX_enTR773TR773&oq=cantor%27s+&aqs=chrome.0.0j69i65l2j69i57j0l2.5964j0j4&sourceid=chrome&ie=UTF-8

My table has the same format as these examples of the table version of Cantor's diagonal argument. So in that regard my table is nothing new.

You and I have different approaches to attack Cantor. You think it is circular. I argue, instead, that the diagonal of the table d is indeterminate, hence d is not a number, hence d’s inverse i is not a number. This collapses Cantor’s argument.

My arguments against Cantor are self-contained; I don’t need more concepts to make them understandable or complete. Even a good high-school student can follow them (with very careful reading).

The Cantorian table I use is supposed to contain all the reals, i.e. all the rationals and irrationals in [0, 1). (The reason I leave it an open interval on the right should be clear from Figure 5 in my paper.)

I know about Mr. Mückenheim's list of Cantor's critics. My name is in the list. I also saw your name there.

Best,

ES

Dear Peter,

I have some difficulty understanding the relevance of your comments to my paper, and I even have difficulty following your English, I am sorry to say. I give below my replies to some of the points you make, of which I think I have at least partial understanding.

"you don´t give the function f which produces binary values for the fractions in the interval [0, 1]"

Are you asking how I converted the real numbers in [0, 1) into their binary equivalents? I don't find it necessary to include in my paper the procedure of doing that. It is a standard procedure. There are many sites in the internet discussing how it is done. For example, you challenge me to write down an irrational number like π - 3 in binary system. Here it is:

0.0010010000111111…

(See: https://www.quora.com/What-is-pi-in-binary )

This number is in my table (and in my trees) of course. Here is a couple of other examples of irrational numbers expressed in binary system:

0.10110111011110111110...

0.01001100011100001111...

These irrational numbers are also in my table. My table theoretically contains all possible combinations of infinitely many 0's and 1's after the decimal point. Hence it contains all of the rationals and irrationals in [0, 1).

"does your proof collapse if the boundaries are included?"

I don't see why it should.

"Why do you think that the sequences of Cantor´s theorem

(m, m, m, … / w, w, w, … / m, w, m, …)

have to be numbers?"

Again, I am using the table version of Cantor's proof which is popular in modern textbooks, because the table version is much easier to understand than Cantor's own version in terms of sequences.

"The same expression stands for many arguments (x)."

I don't think so.

Hope my replies help.

Best,

ES

Dear Peter,

"good luck with your paper!"

Thanks.

"You don´t understand irrationals if you think one could depict them using rationals."

I know that. That is not a correct criticism of my approach.

"So you don´t count any irrational number!"

Of course I do. But you seem to miss it.

I am glad we don't have take our discussion any further, because it won't be fruitful for either of us. Good luck with your paper too.

Best,

ES