From Newsgroup: sci.math


Meanwhile I know three mathematicians [1, 2, 3] who deny that the Binary
Tree can produce the paths belonging to single real numbers. There
remain sheaves or bunches of paths, each one containing uncountably many
paths which are not further distinguishable in the infinite Binary Tree.

/\
/\/\
...

In my opinion this forbids the complete digit sequence of any real
number because a path in the Binary Tree is nothing else than a sequence
of bits. On the other hand Cantor's diagonal argument produces a
complete digit sequence (in the original version [4] a complete bit
sequence, using the symbols W M) of a real number, namely the famous
diagonal number.

How can this contradiction be resolved?

References
[1] Alarming-Smoke1467 in How can the basic element of the Binary Tree
be overcome? https://www.reddit.com/r/learnmath/comments/1sf7jht/how_can_the_basic_element_of_the_binary_tree_be/
(unfortunately deleted meanwhile) said on 7 April 2026 "There are
countably many sheafs, but note that each is uncountable."
[2] Moebius aka Franz Fitsche in Wie kann man die Elemente des Bin|nren
Baums |+berlisten? in the newsgroup de.sci.mathematik said on 17 April
2026 " Ja, es gibt in der Tat nur abz|nhlbar unendlich viele Pfadb|+ndel
'im Baum', aber |+berabz|nhlbar viele Pfade."
[3] Mikko in AI understands where 99 % of mathematicians fail in the
newsgroup sci.logic said on 30 April 2026 "Nodes further down separate
further [paths] but only to infinite subsets."
[4] G. Cantor: |Lber eine elementare Frage der Mannigfaltigkeitslehre. Jahresbericht der Deutsch. Math. Vereing. Bd. I, S. 75-78 (1890-91)

Regards, WM
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From Newsgroup: sci.math

On 07/05/2026 23:48, WM wrote:

Meanwhile I know three mathematicians [1, 2, 3] who deny that the Binary Tree can produce the paths belonging to single real numbers.

Mathematical objecs like a binary tree don't produce. They just are.
There are paths in a binary tree but they don't go anywhere other
tnan to nodes of the tree. Unless at least some nodes are real numbers
the paths don't go to any real number and in any case they don't go
to any other real number.

There
remain sheaves or bunches of paths, each one containing uncountably many paths which are not further distinguishable in the infinite Binary Tree.

Every path is distinguished from every other path by any one of the
nodes that one of them contains and the other does not.

-a/\
/\/\
...

In my opinion this forbids the complete digit sequence of any real
number because a path in the Binary Tree is nothing else than a sequence
of bits. On the other hand Cantor's diagonal argument produces a
complete digit sequence (in the original version [4] a complete bit sequence, using the symbols W M) of a real number, namely the famous diagonal number.

A bit sequence is useful for proving that the power set of a
countable set is not countable. For uncountablility of reals there is
the problem that bit sequences with only finitely many zeros are
different from bit sequences with only finitely many ones but denote
the same real numbers. This problem is avoided with base 3 or higher.

How can this contradiction be resolved?

The most effective way is to stick to formal proofs that are verified
with a good simple proof checker.
--
Mikko
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From Newsgroup: sci.math

On 05/08/2026 05:57 AM, Moebius wrote:

Am 08.05.2026 um 09:58 schrieb Mikko:

On 07/05/2026 23:48, WM wrote:

[...] On the other hand Cantor's diagonal argument produces a
complete digit sequence (in the original version [4] a complete bit
sequence, using the symbols W M) of a real number, namely the famous
diagonal number.

Note: Cantor considered just the set of all sequences of symbols w m,
not "real numbers" (in, say, [0, 1]).

Indeed:

A bit sequence is useful for proving that the power set of a
countable set is not countable. For uncountablility of reals there is
the problem that bit sequences with only finitely many zeros are
different from bit sequences with only finitely many ones but denote
the same real numbers. This problem is avoided with base 3 or higher.

How can this contradiction be resolved?

A good psychiatrist might be helpful.

"Disambiguating quantifiers".

The idea that in combinatorics that the constant "2" is fundamentally
different than the constant "3", yet in asymptotics they run out on
the same orders naively, is for aspects of what's called "Ramsey theory".

Also there's base one and base infinity to consider, when for
an integer is just tally-marks of increment, and a real numbers
is just +- (integer-part) (radix) (non-integer part), that also
the word "radix" fills at least two roles one the idea of the
base of the exponent, the other the divider between integer and non-integer.

A good psychiatrist is not necessarily a "conscientious logician"
of the competent and thorough sort. The "help" may help, yet,
the conscientious logician has a bit of a bigger brain to satisfy.

So, disambiguating quantifiers is a usual account of de-craze-ifying,
since the crazing leads to the cracking, and the failure.


Having an account of "paradox-free" reason may help.


Anyways that it actually matters for the infinitary reasoning
why the binary representation of numbers and trinary/ternary
representations of numbers have different theorems about them, for
example where the binary anti-diagonal has only and exactly _one_ rule
for making the anti-diagonalization and that to avoid "dual
representation" that the usual account is to make the list in a higher
base and say there are "anti-semi-tri-diagonals", instead of an "anti-diagonal", here is that there are accounts like the "Equivalency Function" that only makes for one rule for an anti-diagonal.


So, that "the problem" isn't solve instead just put off.


Then, "Ramsey theory" is a usual umbrella for independence results
of the non-standard, yet these days it's often reduced to talking
about graph-coloring and arithmetic progressions and Szmeredi's
conjectures of all one kind.


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From Newsgroup: sci.math

On 05/08/2026 10:16 AM, Ross Finlayson wrote:

On 05/08/2026 05:57 AM, Moebius wrote:

Am 08.05.2026 um 09:58 schrieb Mikko:

On 07/05/2026 23:48, WM wrote:

[...] On the other hand Cantor's diagonal argument produces a
complete digit sequence (in the original version [4] a complete bit
sequence, using the symbols W M) of a real number, namely the famous
diagonal number.

Note: Cantor considered just the set of all sequences of symbols w m,
not "real numbers" (in, say, [0, 1]).

Indeed:

A bit sequence is useful for proving that the power set of a
countable set is not countable. For uncountablility of reals there is
the problem that bit sequences with only finitely many zeros are
different from bit sequences with only finitely many ones but denote
the same real numbers. This problem is avoided with base 3 or higher.

How can this contradiction be resolved?

A good psychiatrist might be helpful.

"Disambiguating quantifiers".

The idea that in combinatorics that the constant "2" is fundamentally different than the constant "3", yet in asymptotics they run out on
the same orders naively, is for aspects of what's called "Ramsey theory".

Also there's base one and base infinity to consider, when for
an integer is just tally-marks of increment, and a real numbers
is just +- (integer-part) (radix) (non-integer part), that also
the word "radix" fills at least two roles one the idea of the
base of the exponent, the other the divider between integer and
non-integer.

A good psychiatrist is not necessarily a "conscientious logician"
of the competent and thorough sort. The "help" may help, yet,
the conscientious logician has a bit of a bigger brain to satisfy.

So, disambiguating quantifiers is a usual account of de-craze-ifying,
since the crazing leads to the cracking, and the failure.

Having an account of "paradox-free" reason may help.

Anyways that it actually matters for the infinitary reasoning
why the binary representation of numbers and trinary/ternary
representations of numbers have different theorems about them, for
example where the binary anti-diagonal has only and exactly _one_ rule
for making the anti-diagonalization and that to avoid "dual
representation" that the usual account is to make the list in a higher
base and say there are "anti-semi-tri-diagonals", instead of an "anti-diagonal", here is that there are accounts like the "Equivalency Function" that only makes for one rule for an anti-diagonal.

So, that "the problem" isn't solve instead just put off.

Then, "Ramsey theory" is a usual umbrella for independence results
of the non-standard, yet these days it's often reduced to talking
about graph-coloring and arithmetic progressions and Szmeredi's
conjectures of all one kind.

So, relating this also to accounts that when
proof-by-contradiction is the only available template for derivation,
that restriction-of-comprehension always makes accounts
for that the _incomplete_ implies _independence_, that this is what
defines "super-classical" when deductive results are necessary to
establish the completions after the closures, then for the particular
sorts of accounts like "Borel vs. Combinatorics" and "the referee
of the almost-all or almost-none the must-be-middle",
may help explain why usual accounts of "Ramsey theory" as
blind/ignorant to "Erdos' Giant Monster of Mathematical Independence",
as to then a "Great Atlas of Mathematical Independence",
for where 2 < 3 in the asymptotic
(of which "Big O, little o, and theta" are yet of only one account.

So, the constructivism getting involved bring the repleteness into account.




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