Discussion:
Worldwide activities concerning dark numbers
Add Reply
WM
2020-11-21 09:43:17 UTC
Reply
Permalink
A natural number is "individually defined" or "instantiated" if it can be communicated such that sender and receiver understand the same and can link it by a finite initial segment to the origin 0.

Individual definition can occur
by direct description in the unary system like |||||||
or by the finite initial segment {1, 2, 3, 4, 5, 6, 7}
or in binary representation 111 or decimal representation 7
or by indirect description like "the number of colours of the rainbow"
or by other words known to sender and receiver like "seven".

Here are some articles about dark numbers:

Russia: Yaroslav D. Sergeyev: A new applied approach for executing computations with infinite and infinitesimal quantities, Informatica, 19 (2008), no. 4, 567–596.

Italy: Gabriele Lolli: Metamathematical investigations on the theory of Grossone, Applied Mathematics and Computation 255 (2015) 3–14

Russia: Yaroslav D. Sergeyev: Numerical infinities and infinitesimals: Methodology, applications, and repercussions on two Hilbert problems, EMS Surv. Math. Sci. 4 (2017), 219–320

Italy: Lorenzo Fiaschi, Marco Cococcioni: Numerical Asymptotic Results in Game Theory
Using Sergeyev’s Infinity Computing, Int. Journ. of Unconventional Computing, Vol. 14 (2018) pp. 1–25 Old City Publishing, Inc.

Norway: Davide Rizza: Numerical Methods for Infinite Decision-making Processes, Int. Journ. of Unconventional Computing, Vol. 14 (2019) pp. 139–158

Germany: Wolfgang Mückenheim: Dark natural numbers in set theory (2019)
https://www.researchgate.net/publication/336220780_Dark_natural_numbers_in_set_theory

Brazil: Walter Gomide: Dark Numbers Academia.edu (2020)
https://www.academia.edu/44462367/Dark_Numbers_academia_edu

Mew Zealand / Romania: Christian S. Calude, Monica Dumitrescu: Infinitesimal Probabilities Based on Grossone, SN Computer Science (2020)

Germany: Wolfgang Mückenheim: Transfinity - A Source Book (monthly updated), https://www.hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf

A lecture about this topic will be given on Saturday, 21 Nov, 6 PM c.t. here:
https://hs-augsburg.zoom.us/j/92096767073?pwd=aG1DUzJOdnFqTzNTYk1FU0RFQUZ0QT09

Regards, WM
Mostowski Collapse
2020-11-21 10:42:34 UTC
Reply
Permalink
Every Crank has a Cutler Beckett moment:

N* = { 1 , 2 , 3 , . . . , ① − 2 , ① − 1 , ① }
Thus, grossone ① is the biggest natural number . . .

from this Masterpiece:

A trivial formalization of the theory of grossone
A. E. Gutman, S. S. Kutateladze
https://www.researchgate.net/profile/Alexander_Gutman/publication/226342669

Only WM is able to confuse N and N*

Beckett's Death Scene


LoL
Post by WM
A natural number is "individually defined" or "instantiated" if it can be communicated such that sender and receiver understand the same and can link it by a finite initial segment to the origin 0.
Individual definition can occur
by direct description in the unary system like |||||||
or by the finite initial segment {1, 2, 3, 4, 5, 6, 7}
or in binary representation 111 or decimal representation 7
or by indirect description like "the number of colours of the rainbow"
or by other words known to sender and receiver like "seven".
Russia: Yaroslav D. Sergeyev: A new applied approach for executing computations with infinite and infinitesimal quantities, Informatica, 19 (2008), no. 4, 567–596.
Italy: Gabriele Lolli: Metamathematical investigations on the theory of Grossone, Applied Mathematics and Computation 255 (2015) 3–14
Russia: Yaroslav D. Sergeyev: Numerical infinities and infinitesimals: Methodology, applications, and repercussions on two Hilbert problems, EMS Surv. Math. Sci. 4 (2017), 219–320
Italy: Lorenzo Fiaschi, Marco Cococcioni: Numerical Asymptotic Results in Game Theory
Using Sergeyev’s Infinity Computing, Int. Journ. of Unconventional Computing, Vol. 14 (2018) pp. 1–25 Old City Publishing, Inc.
Norway: Davide Rizza: Numerical Methods for Infinite Decision-making Processes, Int. Journ. of Unconventional Computing, Vol. 14 (2019) pp. 139–158
Germany: Wolfgang Mückenheim: Dark natural numbers in set theory (2019)
https://www.researchgate.net/publication/336220780_Dark_natural_numbers_in_set_theory
Brazil: Walter Gomide: Dark Numbers Academia.edu (2020)
https://www.academia.edu/44462367/Dark_Numbers_academia_edu
Mew Zealand / Romania: Christian S. Calude, Monica Dumitrescu: Infinitesimal Probabilities Based on Grossone, SN Computer Science (2020)
Germany: Wolfgang Mückenheim: Transfinity - A Source Book (monthly updated), https://www.hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf
https://hs-augsburg.zoom.us/j/92096767073?pwd=aG1DUzJOdnFqTzNTYk1FU0RFQUZ0QT09
Regards, WM
Sergio
2020-11-21 14:14:11 UTC
Reply
Permalink
Post by WM
A natural number is "individually defined" or "instantiated" if it can be communicated such that sender and receiver understand the same and can link it by a finite initial segment to the origin 0.
but there are no witnesses.
Post by WM
Individual definition can occur
by direct description in the unary system like |||||||
or by the finite initial segment {1, 2, 3, 4, 5, 6, 7}
or in binary representation 111 or decimal representation 7
or by indirect description like "the number of colours of the rainbow"
or by other words known to sender and receiver like "seven".
so some human has to make a dark number visible some how, like congering
up a ghost.

again you insert "requirement of an external observer" into qualities of
numbers, that is wrong thinking.

again, what an "external observer" sees or doesn't see is totally
irrelevant.
Post by WM
Russia: Yaroslav D. Sergeyev: A new applied approach for executing computations with infinite and infinitesimal quantities, Informatica, 19 (2008), no. 4, 567–596.
Italy: Gabriele Lolli: Metamathematical investigations on the theory of Grossone, Applied Mathematics and Computation 255 (2015) 3–14
Russia: Yaroslav D. Sergeyev: Numerical infinities and infinitesimals: Methodology, applications, and repercussions on two Hilbert problems, EMS Surv. Math. Sci. 4 (2017), 219–320
Italy: Lorenzo Fiaschi, Marco Cococcioni: Numerical Asymptotic Results in Game Theory
Using Sergeyev’s Infinity Computing, Int. Journ. of Unconventional Computing, Vol. 14 (2018) pp. 1–25 Old City Publishing, Inc.
Norway: Davide Rizza: Numerical Methods for Infinite Decision-making Processes, Int. Journ. of Unconventional Computing, Vol. 14 (2019) pp. 139–158
Germany: Wolfgang Mückenheim: Dark natural numbers in set theory (2019)
https://www.researchgate.net/publication/336220780_Dark_natural_numbers_in_set_theory
Brazil: Walter Gomide: Dark Numbers Academia.edu (2020)
https://www.academia.edu/44462367/Dark_Numbers_academia_edu
Mew Zealand / Romania: Christian S. Calude, Monica Dumitrescu: Infinitesimal Probabilities Based on Grossone, SN Computer Science (2020)
Germany: Wolfgang Mückenheim: Transfinity - A Source Book (monthly updated), https://www.hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf
https://hs-augsburg.zoom.us/j/92096767073?pwd=aG1DUzJOdnFqTzNTYk1FU0RFQUZ0QT09
Regards, WM
WM
2020-11-22 18:10:42 UTC
Reply
Permalink
Post by Sergio
again you insert "requirement of an external observer" into qualities of
numbers, that is wrong thinking.
again, what an "external observer" sees or doesn't see is totally
irrelevant.
Wrong. Mathematics is as observer-dependent as physics. Only this idea leads Sergeyev to claim that his measure of sets does not contradict Cantor's: Cantor has not seen far enough. Cantor has seen the first few items of the bijection and has extrapolated it with the result that there are as many fractions as naturals which is simply ridiculous. Further it depends on "clever" organizing the set of fractions and therefore the "proof" is depending on the ordering of the rationals. This is by far less tolerable than the observer-dependence.

Regards, WM
Sergio
2020-11-22 19:10:50 UTC
Reply
Permalink
Post by WM
Post by Sergio
again you insert "requirement of an external observer" into qualities of
numbers, that is wrong thinking.
again, what an "external observer" sees or doesn't see is totally
irrelevant.
Wrong. Mathematics is as observer-dependent as physics.
Totally Wrong, Math is a language, (actually several). it is not
"observer dependent" at all, this is why it is so important.

Physics uses math to describe sets of observations.
Post by WM
Only this idea leads Sergeyev to claim that his measure of sets does not contradict Cantor's: Cantor has not seen far enough. Cantor has seen the first few items of the bijection and has extrapolated it with the result that there are as many fractions as naturals which is simply ridiculous. Further it depends on "clever" organizing the set of fractions and therefore the "proof" is depending on the ordering of the rationals. This is by far less tolerable than the observer-dependence.
Cantor's list of fractions is not observer dependent at all.

But your Dark Numbers are entirely observer dependent,
Post by WM
Regards, WM
Dan Christensen
2020-11-21 16:25:05 UTC
Reply
Permalink
Post by WM
A natural number is "individually defined" or "instantiated" if it can be communicated such that sender and receiver understand the same and can link it by a finite initial segment to the origin 0.
Individual definition can occur
by direct description in the unary system like |||||||
or by the finite initial segment {1, 2, 3, 4, 5, 6, 7}
or in binary representation 111 or decimal representation 7
or by indirect description like "the number of colours of the rainbow"
or by other words known to sender and receiver like "seven".
Have you or any of your fellow cranks, using ONLY the ZFC axioms, finally been able to prove the existence of the subset of N that are "definable" in this sense?

Didn't think so. Looks like you have failed once again, Mucke!


Dan

Download my DC Proof 2.0 freeware at http://www.dcproof.com
Visit my Math Blog at http://www.dcproof.wordpress.com
WM
2020-11-22 18:18:26 UTC
Reply
Permalink
Post by Dan Christensen
prove the existence of the subset of N that are "definable" in this sense?
Here it is: ∩{E(k) | k ∈ ℕ_def } =/= { }.

Regards, WM
Dan Christensen
2020-11-22 19:04:26 UTC
Reply
Permalink
Post by Dan Christensen
prove the existence of the subset of N that are "definable" in this sense?
Here it is: ∩{E(k) | k ∈ ℕ_def } =/= { }.
Sorry, that won't do. You need is something like:

ℕ_def = {x ∈ ℕ | P(x)}

where P(x) is some wff in whatever formal system you are using. I believe you have ruled out the ZFC axioms. Apparently they don't seem to require these silly "dark" numbers of yours after all. (Hee, hee!)

How about it, Mucke?

Dan

Download my DC Proof 2.0 freeware at http://www.dcproof.com
Visit my Math Blog at http://www.dcproof.wordpress.com
Mostowski Collapse
2020-11-22 19:18:02 UTC
Reply
Permalink
You would never get a job in a vegetable
market. You confuse again apples and
oranges. This here:

/* correct */
∀k ∈ ℕ: ∩{E(1), E(2), ..., E(k)} =/= { }

Ist not the same as:

/* not correct */
∩{E(1), E(2), ...} =/= { }

LoL
Post by WM
Post by Dan Christensen
prove the existence of the subset of N that are "definable" in this sense?
Here it is: ∩{E(k) | k ∈ ℕ_def } =/= { }.
Regards, WM
Me
2020-11-23 01:57:21 UTC
Reply
Permalink
Here it is: ∩{E(k) | k ∈ ℕ_def } =/= { }.
Here is WHAT, you silly idiot?!

Hint: From ∩{E(k) | k ∈ ℕ_def } =/= { } and ℕ_def c IN we can DERIVE that your ℕ_def is finite.

So what?
WM
2020-11-23 14:53:57 UTC
Reply
Permalink
Post by Me
Here it is: ∩{E(k) | k ∈ ℕ_def } =/= { }.
Hint: From ∩{E(k) | k ∈ ℕ_def } =/= { } and ℕ_def c IN we can DERIVE that your ℕ_def is finite.
Not only mine. Your ℕ_def is also finite.
Post by Me
So what?
Of yourse it is clear, but remember that you vehemently denied dark numbers, i.e., the difference between ℕ and ℕ_def. Now you have unrstood?

Regards, WM
zelos...@gmail.com
2020-11-23 06:41:43 UTC
Reply
Permalink
Post by WM
A natural number is "individually defined" or "instantiated" if it can be communicated such that sender and receiver understand the same and can link it by a finite initial segment to the origin 0.
That is all of the natural numbers.
Post by WM
Individual definition can occur
Here you demonstrate that notation is what matters to you.
WM
2020-11-23 14:56:56 UTC
Reply
Permalink
Post by ***@gmail.com
Post by WM
A natural number is "individually defined" or "instantiated" if it can be communicated such that sender and receiver understand the same and can link it by a finite initial segment to the origin 0.
That is all of the natural numbers.
ℕ_def = ℕ ?

Then { } =/= ∩{E(k) | k ∈ ℕ_def } = ∩{E(k) | k ∈ ℕ } = { } or briefly
{ } =/= { } .

Regards, WM
Gus Gassmann
2020-11-23 15:42:00 UTC
Reply
Permalink
Post by WM
Post by ***@gmail.com
Post by WM
A natural number is "individually defined" or "instantiated" if it can be communicated such that sender and receiver understand the same and can link it by a finite initial segment to the origin 0.
That is all of the natural numbers.
ℕ_def = ℕ ?
Give a coherent definition for ℕ_def and/or prove that it exists as a well-defined set. In other words, PUT UP OR SHUT UP .
WM
2020-11-23 20:34:04 UTC
Reply
Permalink
Post by Gus Gassmann
Post by WM
Post by ***@gmail.com
Post by WM
A natural number is "individually defined" or "instantiated" if it can be communicated such that sender and receiver understand the same and can link it by a finite initial segment to the origin 0.
That is all of the natural numbers.
ℕ_def = ℕ ?
Give a coherent definition for ℕ_def and/or prove that it exists as a well-defined set.
It is not a set but a class. It can be increased infinitely without ever becoming infinite.

Here is an implicit definition: |∩{E(k) : k ∈ ℕ_def}| = ℵo .

The class of dark numbers is ℕ \ ℕ_def.

Regards, WM

Loading...