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Old 19th August 2019, 04:23 AM   #3401
doronshadmi
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Originally Posted by Little 10 Toes View Post
This still doesn't make sense. First set A was a set, then a "formal system", now it's an extension. And it contains things that it already contained? And it must be taken?
From my first post about A, A is a formal system of infinitely many wffs (which is strong enough in order to deal with Arithmetic, exactly because it is an extension of ZF(C)), where all the infinitely many wffs are already included in A, exactly because Infinity is taken in terms of Platonic (or Actual) Infinity (By Platonic (or Actual) Infinity there exists a set of infinitely many things (for example: wffs) as a complete whole).

The maneuvers of jsfisher around A's existence, this is exactly the thing that makes no sense.

Here is jsfisher's last reply, which clearly demonstrates his nonsensical maneuvers around A's existence:
Originally Posted by jsfisher View Post
That's nice, but you still haven't told us how you are actually applying the axiom on ZF(C). Keep in mind, the axiom simply states that there exists a certain set with certain properties. Nothing more.

So, once again, what does "using ZF(C) Axiom Of Infinity on ZF(C) itself" mean?
and he does them in order to avoid the following question:

Originally Posted by doronshadmi View Post
Please explain what do you mean by "The Axiom of Infinity establishes a set in terms of Mathematics." (especially the highlighted part)?

Originally Posted by jsfisher View Post
Keep in mind, the axiom simply states that there exists a certain set with certain properties. Nothing more.
Keep in mind that ZF(C) Axiom Of Infinity is taken in terms of Platonic (or Actual) Infinity, such that there a exists a certain set with infinitely many things as a complete whole.

I take the property of Platonic Infinity from ZF(C) Axiom Of Infinity and relate it to A. Nothing more.

Now, please explain what do you mean by "The Axiom of Infinity establishes a set in terms of Mathematics." (especially the highlighted part)?

(To the other posters:
Quote:
"In the philosophy of mathematics, the abstraction of actual infinity involves the acceptance (if the axiom of infinity is included) of infinite entities, such as the set of all natural numbers or an infinite sequence of rational numbers, as given, actual, completed objects."
https://en.wikipedia.org/wiki/Actual_infinity

Moreover
Quote:
"Historically, logic has been studied in philosophy (since ancient times) and mathematics (since the mid-19th century), and recently logic has been studied in cognitive science (encompasses computer science, linguistics, philosophy and psychology"
https://en.wikipedia.org/wiki/Logic

Also, Infinity is one of the main philosophical subjects, studied by philosophers like Plato, Aristotle and many more philosophers along the years.

The attepmt to define a clear cut distinction between Philosophy and Mathematics in case of Logic and Infinity, is itself some kind of Philosophy, and in this case jsfisher's philosophy about the discussed subject)
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For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video.

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Old 19th August 2019, 06:13 AM   #3402
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Originally Posted by doronshadmi View Post
Keep in mind that ZF(C) Axiom Of Infinity is taken in terms of Platonic (or Actual) Infinity, such that there a exists a certain set with infinitely many things as a complete whole.
Almost. The Axiom is not "taken"; it simply is. A certain set exists; it has certain properties. The von Neumann ordinal is the minimal example of such a set, so we have one example of the set guaranteed to exist. The Axiom alone gives no guidance as to whether there are others.

Your insistence on bringing in philosophic babble is, well, yours.

Quote:
I take the property of Platonic Infinity from ZF(C) Axiom Of Infinity and relate it to A. Nothing more.
Great, you've gone full circular on us. To get A, you "take the property of Platonic Infinity from ZF(C) Axiom Of Infinity and relate it to A."

You probably want to clean that up, and when you do, please explain what "relate it to" means.
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Old 19th August 2019, 07:05 AM   #3403
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Originally Posted by jsfisher View Post
Almost. The Axiom is not "taken"; it simply is.
Let's see: "The Axiom is ... simply is."

jsfisher, maybe this is a very interesting statement. Probably a lot of mathematical work can by done by it and maybe also a profound communication between people can be done by it.

Unfortunately, I do not find such tautology as very useful in our discussion.

Originally Posted by jsfisher View Post
The von Neumann ordinal is the minimal example of such a set, so we have one example of the set guaranteed to exist.
Such set is guaranteed to exist in terms of Platonic (or Actual Infinity) Infinity (which according to it there exists an infinite set as a complete whole).

Originally Posted by jsfisher View Post
Your insistence on bringing in philosophic babble is, well, yours.
Your insistence to establish a clear cut border between Philosophy and Mathematics (and in the discussed case, Logic) is, well, your philosophy.

Originally Posted by jsfisher View Post
please explain what "relate it to" means.
It means that a certain property of x is also a property of y, and in the considered case Infinity is established in terms of Platonic Infinity, both on x and y.

Please explain what do you mean by "The Axiom of Infinity establishes a set in terms of Mathematics." (especially the highlighted part)?
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That is also over the matrix, is aware of the matrix.

That is under the matrix, is unaware of the matrix.

For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video.

Last edited by doronshadmi; 19th August 2019 at 07:08 AM.
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Old 19th August 2019, 07:14 AM   #3404
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Originally Posted by doronshadmi View Post
It means that a certain property of x is also a property of y, and in the considered case Infinity is established in terms of Platonic Infinity.
How would that apply to the case at hand, that being "using ZF(C) Axiom Of Infinity on ZF(C) itself" to get A. What is x; what is y; and what is this certain property?
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Old 19th August 2019, 07:44 AM   #3405
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Originally Posted by jsfisher View Post
How would that apply to the case at hand, that being "using ZF(C) Axiom Of Infinity on ZF(C) itself" to get A. What is x; what is y; and what is this certain property?
The certain property is Platonic (or Actual) Infinity, which is related to ZF(C) Axiom Of Infinity (x) and ZF(C) extension (y).

Quote:
After all natural numbers comes the first infinite ordinal, ω.
https://en.wikipedia.org/wiki/Ordinal_number

ω is not established without the existence of the infinite set of all natural numbers, and the infinite set of all natural numbers is not established (by ZF(C) Axiom Of Infinity) as a complete whole (which enables ω to exist "After all natural numbers") without the "philosophic babble" of Platonic (or Actual) Infinity.

Please explain what do you mean by "The Axiom of Infinity establishes a set in terms of Mathematics." (especially the highlighted part)?
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That is also over the matrix, is aware of the matrix.

That is under the matrix, is unaware of the matrix.

For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video.

Last edited by doronshadmi; 19th August 2019 at 07:51 AM.
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Old 19th August 2019, 09:01 AM   #3406
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Originally Posted by doronshadmi View Post
The certain property is Platonic (or Actual) Infinity
The property would be "infinite" not "infinity", but let's move on.

Quote:
which is related to ZF(C) Axiom Of Infinity (x)
How? The Axiom of Infinity is quite finite.

Quote:
and ZF(C) extension (y).
ZF and ZFC are set theories. What is it you see that is infinite about them? And what is this extension of which you speak?

And assuming there is something infinite about the Axiom and the set theories, how does this relation of a common property between the two give rise to this set, A?


(By the way, for Z' to be an extension of Z where Z' and Z are formal systems like, say, ZF, everything that is decidable in Z must be equally decidable in Z'. There may be additional things decidable in Z', but Z' may not contradict Z.)
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Old 20th August 2019, 02:44 AM   #3407
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Originally Posted by jsfisher View Post
The property would be "infinite" not "infinity", but let's move on.
Let's not move on. The certain property is Infinity in terms of Platonic (or Actual) Infinity.

Originally Posted by jsfisher View Post
How? The Axiom of Infinity is quite finite.
Platonic Infinity is what The Axiom of Infinity establishes, by using finitely many symbols, for example: The infinite set of von Neumann ordinals (https://en.wikipedia.org/wiki/Natura...umann_ordinals).

Originally Posted by jsfisher View Post
ZF and ZFC are set theories. What is it you see that is infinite about them?
Axiom schema (and therefore Infinity) are parts of ZF(C).

Originally Posted by jsfisher View Post
And what is this extension of which you speak?
As done by Godel First Incompleteness Theorem, but in terms of Platonic (or Actual) Infinity (which means that all wffs (whether they are axioms or theorems) are already included in this extension (called formal system A).

Originally Posted by jsfisher View Post
And assuming there is something infinite about the Axiom and the set theories, how does this relation of a common property between the two give rise to this set, A?
Formal system A has infinitely many wffs in terms of Platonic (or Actual) Infinity (formal system A is taken as a complete whole).

Originally Posted by jsfisher View Post
(By the way, for Z' to be an extension of Z where Z' and Z are formal systems like, say, ZF, everything that is decidable in Z must be equally decidable in Z'. There may be additional things decidable in Z', but Z' may not contradict Z.)
In the considered case Z' (which is a complete extension of Z in terms of Platonic (or Actual) Infinity) is indeed strong enough to deal with Arithmetic (as decidable in Z), but unlike Z, all its infinitely many wffs are already included in it (Z' is complete in terms of Platonic Infinity) as follows:

Each wff is encoded by a Gödel number, where at least one of these wffs, called G, states "There is no number m such that m is the Gödel number of a proof in Z', of G" (since G needs a proof, it is not an axiom but a theorem).

Since all wffs are already in Z' and all Gödel numbers are already in Z' (because Infinity is taken in terms of Platonic Infinity) there is a Gödel number of a proof of G in Z', which contradicts G in Z', exactly because Z' is complete (in terms of Platonic Infinity) and therefore inconsistent exactly because Infinity is taken in terms of Platonic (or Actual) Infinity.

Conclusion: Platonic (or Actual) Infinity is the cause of the contradiction (and therefore the inconstancy) of Z' (which is an extension of Z).

---------------------

Please explain what do you mean by "The Axiom of Infinity establishes a set in terms of Mathematics." (especially the highlighted part)?
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That is also over the matrix, is aware of the matrix.

That is under the matrix, is unaware of the matrix.

For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video.

Last edited by doronshadmi; 20th August 2019 at 03:12 AM.
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Old 20th August 2019, 06:28 AM   #3408
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Originally Posted by doronshadmi View Post
...snip...
Many bits of nonsense saved for a later time so as to not further defocus the current thread arc.
Quote:
Originally Posted by jsfisher View Post
And what is this extension of which you speak?
As done by Godel First Incompleteness Theorem, but in terms of Platonic (or Actual) Infinity (which means that all wffs (whether they are axioms or theorems) are already included in this extension (called formal system A).
The proof of Godel's First Incompleteness Theorem is constructive in that it provides one example of an undecidable statement for an incomplete formal system. One. Not infinitely many. Just one.

Quote:
Originally Posted by jsfisher View Post
And assuming there is something infinite about the Axiom and the set theories, how does this relation of a common property between the two give rise to this set, A?
Formal system A has infinitely many wffs in terms of Platonic (or Actual) Infinity (formal system A is taken as a complete whole).
Which infinitely many statements would that be? ZF (or ZFC) can be expressed as a countably infinite number of axioms; Godel can be relied upon for any finite number of additional statements. Which statements did you have mind for your set theory extension?

Quote:
Originally Posted by jsfisher View Post
(By the way, for Z' to be an extension of Z where Z' and Z are formal systems like, say, ZF, everything that is decidable in Z must be equally decidable in Z'. There may be additional things decidable in Z', but Z' may not contradict Z.)
In the considered case Z' (which is a complete extension of Z in terms of Platonic (or Actual) Infinity)...
No such extension exists.

For it to exist, you would need a membership function with respect to your set, A, for the Axiom Schema of Restricted Comprehension. You just need a function that determines whether x is a member of A. You don't have one.

Merely speculating that a set must exist because you want it to does not make it so. Throwing your philosophic baggage at the problem doesn't change that.
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Old 20th August 2019, 08:01 AM   #3409
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Originally Posted by jsfisher View Post
Many bits of nonsense saved for a later time so as to not further defocus the current thread arc.
Another hands waving of yours.

Originally Posted by jsfisher View Post
The proof of Godel's First Incompleteness Theorem is constructive in that it provides one example of an undecidable statement for an incomplete formal system. One. Not infinitely many. Just one.
I agree with you, G is just one statement, as written at the end of my previous post (" Z' " is used instead of "A").

Originally Posted by jsfisher View Post
Which infinitely many statements would that be? ZF (or ZFC) can be expressed as a countably infinite number of axioms;
In order to claim that there are countably infinite number of axioms, you first have to accept that there is an infinite set in terms of a complete whole.

Originally Posted by jsfisher View Post
Godel can be relied upon for any finite number of additional statements.
This is a finite GIT version that can't deduce anything about a set in terms of Platonic (or Actual) Infinity.

By GIT infinite version (in terms of Platonic (or Actual) Infinity, which, as can be seen, was not deduced by you) all the Godel numbers that encode wffs, are already in A, where one of them encodes G wff statement (which is actually a theorem, since it is proven in A).

Originally Posted by jsfisher View Post
No such extension exists.

For it to exist, you would need a membership function with respect to your set, A, for the Axiom Schema of Restricted Comprehension. You just need a function that determines whether x is a member of A. You don't have one.
The Axiom Schema of Restricted Comprehension does not exist, if Infinity is not taken in terms of Platonic (or Actual) Infinity.

If you reject what I wrote about this axiom, you also reject the existence of the infinite set of all natural numbers as a complete whole.

Originally Posted by jsfisher View Post
Throwing your philosophic baggage at the problem doesn't change that.
Your clear cut separation between Philosophy and Mathematics, does not change the fact that it is actually your Philosophy.

---------------------

Please explain what do you mean by "The Axiom of Infinity establishes a set in terms of Mathematics." (especially the highlighted part)?
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That is also over the matrix, is aware of the matrix.

That is under the matrix, is unaware of the matrix.

For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video.

Last edited by doronshadmi; 20th August 2019 at 08:22 AM.
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Old 20th August 2019, 09:15 AM   #3410
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Originally Posted by doronshadmi View Post
This is a finite GIT version... .
By GIT infinite version....
You are making up your own theorems, now, too?

Quote:
...all the Godel numbers that encode wffs, are already in A
If you are trying to tell us that your set, A, contains every possible statement that can be expressed in some formal language sufficient to express the axioms of ZF (or ZFC), then you have a different problem.

The set theory corresponding to your set, A, is not an extension of ZF (or ZFC). There are statements in ZF that would be contradicted in your so-called extended set theory.

As I said before, the set you claim exists does not.

You require it represent an extension to ZF that is Godel complete. No such set exists.
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Old 20th August 2019, 10:55 PM   #3411
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Originally Posted by jsfisher View Post
You are making up your own theorems, now, too?
It is made up exactly as ZF(C) Axiom Of Infinity made up things in terms of Platonic (or Actual Infinity), which enables mathematicians like you to declare that the infinite set if all natural numbers, exists.

Originally Posted by jsfisher View Post
If you are trying to tell us that your set, A, contains every possible statement that can be expressed in some formal language sufficient to express the axioms of ZF (or ZFC), then you have a different problem.

The set theory corresponding to your set, A, is not an extension of ZF (or ZFC). There are statements in ZF that would be contradicted in your so-called extended set theory.
This is my argument right from the beginning of the last discussion, which is:

The very notion of Platonic (or Actual) Infinity necessarily involved with logical contradiction and therefore inconsistency, exactly because a collection of infinitely many things is taken in terms of a complete whole.

Originally Posted by jsfisher View Post
As I said before, the set you claim exists does not.

You require it represent an extension to ZF that is Godel complete. No such set exists.
Until this very moment you are still missing my argument, which is (again, since you are still missing it):

The very notion of Platonic (or Actual) Infinity necessarily involved with logical contradiction and therefore inconsistency, exactly because a collection of infinitely many things is taken in terms of a complete whole.

Because of this logical fallacy also the infinite set of all natural numbers does not exist.

Actually, the very notion of Transfinite System does not exist (in terms of logical consistency), exactly because it is established on the notion of collection of infinitely many things in terms of a complete whole.

---------------------

Again, there is a non-interesting solution about the discussed subject, as follows:

G states: "There is no number m such that m is the Godel number of a proof in A, of G"

If G is already an axiom in A (where A is an infinite set of axioms, such that Infinity is taken in terms of Platonic Infinity) it is actually a wff that is true in A, which does not have any Godel number that is used in order to encode G's proof (since axioms are true wffs that do not need any proof in A).

But then no proof is needed and mathematicians are out of job (therefore it is an unwanted solution).

---------------------

Please explain what do you mean by "The Axiom of Infinity establishes a set in terms of Mathematics." (especially the highlighted part)?
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That is also over the matrix, is aware of the matrix.

That is under the matrix, is unaware of the matrix.

For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video.

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Old 21st August 2019, 04:14 AM   #3412
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Originally Posted by doronshadmi View Post
It is made up....

Have fun with that. Meanwhile, Mathematics not simply made up by you continues unabated by your confusion.



Oh, and the Axiom of Infinity:



...just mathematics.
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Old 21st August 2019, 04:47 AM   #3413
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Originally Posted by jsfisher View Post
Have fun with that. Meanwhile, Mathematics not simply made up by you continues unabated by your confusion.



Oh, and the Axiom of Infinity:

https://wikimedia.org/api/rest_v1/me...e1585b2269bb83 [was replaced by me according to what is written in Wikipedia]

...just mathematics.
Let's see:

"In words, there is a set I (the set which is postulated to be infinite), such that the empty set is in I, and such that whenever any x is a member of I, the set formed by taking the union of x with its singleton {x} is also a member of I." (Please compare it to https://en.wikipedia.org/wiki/Axiom_...rmal_statement, where m is replaced by x).

If Infinity in this axiom is not taken in terms of Platonic (or Actual) Infinity, even the infinite set of all natural numbers does not exist and jsfisher's "...just mathematics" actually does not establish the Transfinite system.

---------------------

jsfisher, please explain what do you mean by "The Axiom of Infinity establishes a set in terms of Mathematics." (especially the highlighted part)?
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That is also over the matrix, is aware of the matrix.

That is under the matrix, is unaware of the matrix.

For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video.

Last edited by doronshadmi; 21st August 2019 at 06:16 AM.
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Old 23rd August 2019, 01:47 AM   #3414
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Originally Posted by jsfisher View Post
You just need a function that determines whether x is a member of A. You don't have one.
Syntactically (by formalism without semantics) There is a set A (the set which is postulated to be infinite), such that the empty set is in A, and such that whenever any x is a member of A, the set formed by taking the union of x with its singleton {x} is also a member of A.

So Syntactically x ---> xU{x} is the bijective membership function of A.

Now we are using also Semantics (adding some meaning) by establish some models about this function, as follows:


Model 1:

Let x be an axiom (wff that is not proven) in A.

Let xU{x} be a theorem (wff that is proven) in A.

Let A be an infinite set of wffs, where Infinity is taken in terms of Platonic (or Actual) Infinity (A is taken as a complete whole).

Each wff (wff that is proven) is encoded by a Gödel number, where one of these wffs, called G, states "There is no number m such that m is the Gödel number of a proof in A, of G" (since G needs a proof, it is not an axiom but a theorem).

Since all wffs (wffs that are proven) are already in A and all Gödel numbers are already in A (because Infinity is taken in terms of Platonic Infinity) there is a Gödel number of a proof of G in A, which contradicts G in A, exactly because A is complete (in terms of Platonic Infinity) and therefore inconsistent.


Model 2:

Let x or xU{x} be axioms (wff that is not proven) in A.

G axiom states: "There is no number m such that m is the Godel number of a proof in A, of G"

Since G is already an axiom in A (where A is an infinite set of axioms, such that Infinity is taken in terms of Platonic Infinity) it is actually a wff that is true in A, which does not have any Godel number that is used in order to encode G's proof (since axioms are true wffs that do not need any proof in A).

But then no proof is needed and mathematicians are out of job (therefore it is an unwanted solution).
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That is also over the matrix, is aware of the matrix.

That is under the matrix, is unaware of the matrix.

For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video.

Last edited by doronshadmi; 23rd August 2019 at 01:52 AM.
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Old 23rd August 2019, 03:59 AM   #3415
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So in both models infinitely in terms of Platonic (or Actual) Infinity (which according to it there exists a collection of things as a complete whole) does not establish an interesting formal system.

An alternative to such non-interesting formal systems, is established such that Platonic (or Actual) Infinity is non-composed (it is not established in terms of collections that are taken as a complete whole) as already given in the following posts:

http://www.internationalskeptics.com...postcount=3302

http://www.internationalskeptics.com...postcount=3303

http://www.internationalskeptics.com...postcount=3304

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That is also over the matrix, is aware of the matrix.

That is under the matrix, is unaware of the matrix.

For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video.

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Old 31st August 2019, 06:30 AM   #3416
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A clearer version of my argument (as firstly was given in http://www.internationalskeptics.com...postcount=3414)

Gödel numbers are used to encode wffs of formal systems that are strong enough in order to deal with Arithmetic.

In my argument, Gödel numbers are used to encode wffs as follows:

Syntactically (by formalism without semantics) there is set A (the set which is postulated to be infinite), such that the empty set is a member of A, and such that whenever any x is a member of A, the set formed by taking the union of x with its singleton {x}, is also a member of A.

So Syntactically xxU{x} is the bijective membership function of A.

Now we are using also Semantics (adding some meaning) by establish some models about this function, as follows:


Model 1:

Let any x be an axiom (wff that is not proven) in A.

Let any xU{x} be a theorem (wff that is proven) in A.

Let A be an infinite set of wffs, where Infinity is taken in terms of Actual Infinity (A is taken as a complete whole).

Each wff (wff that is proven (some xU{x})) is encoded by a Gödel number, where one of these wffs, called G, states: "There is no number m such that m is the Gödel number of a proof in A, of G".

Since all wffs are already in A and therefore all Gödel numbers are already in A (because Infinity is taken in terms of Actual Infinity) there is a Gödel number of an axiom (some x) that proves G (some xU{x}) in A, which is a contradiction in A. Therefore, A is inconsistent.


Model 2:

Let any x or any xU{x} be axioms (wffs that are not proven) in A.

G axiom states: "There is no number m such that m is the Gödel number of a proof in A, of G"

Since G is already an axiom in A (where A is an infinite set of axioms, such that Infinity is taken in terms of Actual Infinity) it is actually a wff that is true in A, which does not have any Gödel number that is used in order to encode G's proof (since axioms are true wffs that do not need any proof in A).

But then no proof is needed and mathematicians are out of job (therefore it is an unwanted solution).

---------------------------

So, in both models infinitely in terms of Actual Infinity (an infinite set that is taken as a complete whole) does not establish an interesting formal system.

---------------------------

Please pay attention to the following remarks, before you reply:

Since A is a set of infinitely many wffs that are taken as a complete whole (this is exactly what Actual Infinity is about) there cannot be a Gödel number that is not already in A, whether whether some wff is an axiom or a theorem in A (see Model 1). So, one can't use G as a wff that is unproven in A, as done in case of GIT, since if one does this, one deduces in terms of Potential Infinity, which is not a part of my argumnt.
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That is also over the matrix, is aware of the matrix.

That is under the matrix, is unaware of the matrix.

For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video.

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Old 1st September 2019, 01:18 PM   #3417
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Originally Posted by jsfisher View Post
You just need a function that determines whether x is a member of A. You don't have one.
Originally Posted by doronshadmi View Post
xxU{x} is the bijective membership function of A.
It is neither bijective nor a membership function. A membership functions are true/false-valued.
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Old 3rd September 2019, 02:34 AM   #3418
doronshadmi
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Originally Posted by jsfisher View Post
It is neither bijective nor a membership function. A membership functions are true/false-valued.
xxU{x} is bijective, for example:

{} → {{}}
{{}} → {{},{{}}}
{{},{{}}} → {{},{{}},{{},{{}}}}
... etc.

and also determine the members of A (exactly as the members of an inductive set are determined by The Axiom Of Infinity).
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That is also over the matrix, is aware of the matrix.

That is under the matrix, is unaware of the matrix.

For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video.

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Old 3rd September 2019, 04:39 PM   #3419
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Originally Posted by doronshadmi View Post
xxU{x} is bijective
Repeating your false claim doesn't make it true. Nothing maps to the empty set.

Be that as it may, it isn't a membership function, either, since it doesn't answer the question, "Is m a member of set A?" F(x) = x U {x} doesn't.


Quote:
and also determine the members of A (exactly as the members of an inductive set are determined by The Axiom Of Infinity).
The Axiom of Infinity doesn't identify any particular set; it doesn't provide a membership function. It merely states two properties the set has (i.e. that it contains the empty set and that every member of the set also has its successor as a member).

The Axiom is silent on whether, for example, {{{{ }}}} is in the set.

The Axiom must to be coupled with other axioms to conclude von Neumann's ordinal is a set in ZF.
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Old 3rd September 2019, 10:27 PM   #3420
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Originally Posted by jsfisher View Post
Repeating your false claim doesn't make it true. Nothing maps to the empty set.
Originally Posted by jsfisher View Post
The Axiom of Infinity doesn't identify any particular set; it doesn't provide a membership function.
Repeating your false claim doesn't make it true. As for the empty set, it is a member of A (as given below) and nothing maps to the empty set since it is a domain object (as seen in http://www.internationalskeptics.com...postcount=3418).

There is set A (the set which is postulated to be infinite), such that the empty set is a member of A, and such that whenever any x is a member of A, the set formed by taking the union of x with its singleton {x}, is also a member of A.

So, both x and xU{x} are members of A.

So, Syntactically xxU{x} is the bijective membership function of A (no x, xU{x} or any gödel number m are missing from A).

Originally Posted by jsfisher View Post
Be that as it may, it isn't a membership function, either, since it doesn't answer the question, "Is m a member of set A?" F(x) = x U {x} doesn't.
m is a Gödel number of a proof of some xU{x} in A (according to model 1 (seen in http://www.internationalskeptics.com...postcount=3416)) exactly because A is determined in terms of Actual Infinity as a complete whole (no m, x or xU{x} are missing from A).

Originally Posted by jsfisher View Post
The Axiom of Infinity doesn't identify any particular set; it doesn't provide a membership function. It merely states two properties the set has (i.e. that it contains the empty set and that every member of the set also has its successor as a member).

The Axiom is silent on whether, for example, {{{{ }}}} is in the set.
I am not talking about ZF, but about the construction of A such that Infinity is taken as a complete whole (no x, xU{x} or any gödel number m are missing from A).

If you don't like xU{x}, it can easily be replaced by {x} as follows:

{} → {{}}
{{}} → {{{}}}
{{{}}} → {{{{}}}}}
... etc.

For example, syntactically (by formalism without semantics) there is set A (the set which is postulated to be infinite), such that the empty set is a member of A, and such that whenever any x is a member of A, the set {x}, is also a member of A.

So, both x and {x} are members of A.

So, Syntactically x → {x} is the bijective membership function of A (no x, {x} or any gödel number m are missing from A).
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That is also over the matrix, is aware of the matrix.

That is under the matrix, is unaware of the matrix.

For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video.

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Old 4th September 2019, 05:17 AM   #3421
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Originally Posted by doronshadmi View Post
...bijective membership function...
You are back to making up your own definitions. It is much easier, I suppose, to spout your nonsense when you are unconstrained by meaning.

Do have fun, but you fail to make any point in Mathematics.
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Old 4th September 2019, 07:27 AM   #3422
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Originally Posted by jsfisher View Post
You are back to making up your own definitions.
The two following equivalent examples are taken from Traditional Mathematics:

Example 1:

There is set A (the set which is postulated to be infinite), such that the empty set is a member of A, and such that whenever any x is a member of A, the set formed by taking the union of x with its singleton {x}, is also a member of A.

So, both x and xU{x} are members of A.

{} → {{}}
{{}} → {{},{{}}}
{{},{{}}} → {{},{{}},{{},{{}}}}
... etc. are all members of A in example 1.

So, Syntactically xxU{x} is the bijective membership function of A (no x, xU{x} or any gödel number m are missing from A).


Example 2:

There is set A (the set which is postulated to be infinite), such that the empty set is a member of A, and such that whenever any x is a member of A, the set formed by {x}, is also a member of A.

So, both x and {x} are members of A.

{} → {{}}
{{}} → {{{}}}
{{{}}} → {{{{}}}}}
... etc. are all members of A in example 2.

So, Syntactically x → {x} is the bijective membership function of A (no x, {x} or any gödel number m are missing from A).

-----------------

Example 2 easily replaces example 1 (which is used in http://www.internationalskeptics.com...postcount=3416) without changing the argument.

-----------------

Originally Posted by jsfisher View Post
Nothing maps to the empty set.
Wrong , {} → {{}} and {{}} → {} are inverses of each other.


---------------------

Originally Posted by jsfisher View Post
It is much easier, I suppose, to spout your nonsense when you are unconstrained by meaning.
jsfisher, please explain what do you mean by "The Axiom of Infinity establishes a set in terms of Mathematics." (especially the highlighted part)?
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That is also over the matrix, is aware of the matrix.

That is under the matrix, is unaware of the matrix.

For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video.

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Old 7th September 2019, 11:28 AM   #3423
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Please observe the following semantic diagram:



As can be seen, there is bijection between X-Axioms and Y-Theorems, as follows:

{}{{}}
{{}}{{},{{}}}
{{},{{}}}{{},{{}},{{},{{}}}}
......


It is a bijection since X-Axioms and Y-Theorems are inverses of each other as follows:

X-axiom proves Y-theorem and Y-theorem is proven by X-axiom.
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That is also over the matrix, is aware of the matrix.

That is under the matrix, is unaware of the matrix.

For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video.

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Old 21st January 2020, 12:50 AM   #3424
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Here is a quote taken from wikipedia: ( https://en.wikipedia.org/wiki/Counta...ithout_details )

Quote:
Formal overview without details

By definition a set S is countable if there exists an injective function f : S → N from S to the natural numbers N = {0, 1, 2, 3, ...}.

It might seem natural to divide the sets into different classes: put all the sets containing one element together; all the sets containing two elements together; ...; finally, put together all infinite sets and consider them as having the same size. This view is not tenable, however, under the natural definition of size.

To elaborate this we need the concept of a bijection. Although a "bijection" seems a more advanced concept than a number, the usual development of mathematics in terms of set theory defines functions before numbers, as they are based on much simpler sets. This is where the concept of a bijection comes in: define the correspondence

a ↔ 1, b ↔ 2, c ↔ 3

Since every element of {a, b, c} is paired with precisely one element of {1, 2, 3}, and vice versa, this defines a bijection.

We now generalize this situation and define two sets as of the same size if (and only if) there is a bijection between them. For all finite sets this gives us the usual definition of "the same size". What does it tell us about the size of infinite sets?

Consider the sets A = {1, 2, 3, ... }, the set of positive integers and B = {2, 4, 6, ... }, the set of even positive integers. We claim that, under our definition, these sets have the same size, and that therefore B is countably infinite. Recall that to prove this we need to exhibit a bijection between them. But this is easy, using n ↔ 2n, so that

1 ↔ 2, 2 ↔ 4, 3 ↔ 6, 4 ↔ 8, ....

As in the earlier example, every element of A has been paired off with precisely one element of B, and vice versa. Hence they have the same size. This is an example of a set of the same size as one of its proper subsets, which is impossible for finite sets.
Please pay attention that |A| is not defined as an accurate value in case of infinite sets (the very notion of size is not well defined in case of infinite sets) so the phrase "Hence they have the same size" has no well defined basis.

For example: writing n ↔ 2n says nothing about an accurate value of |A| > any given n.

As long as this is the case "..." is not some technical problem of writing down infinitely many elements, but it is actually an essential property of literally being an infinity set (which means that no set of infinitely many objects has an accurate amount of objects, as its essential property).

For more details, please look at http://www.internationalskeptics.com...postcount=3422 and http://www.internationalskeptics.com...postcount=3423 .
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That is also over the matrix, is aware of the matrix.

That is under the matrix, is unaware of the matrix.

For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video.

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Old 21st January 2020, 03:16 AM   #3425
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Originally Posted by doronshadmi View Post

Please pay attention that |A| is not defined as an accurate value in case of infinite sets (the very notion of size is not well defined in case of infinite sets) so the phrase "Hence they have the same size" has no well defined basis.
It's based on the very definition of two sets having the same size, namely that there exists a bijection between them. So the basis is rock solid. Try again.
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Old 21st January 2020, 03:43 AM   #3426
doronshadmi
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Originally Posted by Hevneren View Post
It's based on the very definition of two sets having the same size, namely that there exists a bijection between them. So the basis is rock solid. Try again.
"Having the same size" does not actually define the accurate size in case of infinite sets (bijection is simply a 1-to-1 and onto proportion between the given infinite sets), so there is no rock solid basis in your argument. Try again.

Again: writing n ↔ 2n says nothing about an accurate value of |A| > any given n.

If |A| is defined only by bijection, it is not satisfied in terms of actual infinity ("Having the same size" holds only in case of potential infinity where we do not care about any accurate size but only about the 1-to-1 and onto proportion between the given infinite sets) and this is exactly my argument about |A| in case of infinite sets.

In other words, the transfinite mathematical universe (which is based on the notion of actual infinity) is not well defined.
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That is also over the matrix, is aware of the matrix.

That is under the matrix, is unaware of the matrix.

For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video.

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Old 23rd January 2020, 04:23 AM   #3427
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By Traditional Mathematics infinite set N = {1,2,3,...} is in bejection with any infinite subset of it, so according to https://math.stackexchange.com/quest...edirect=1&lq=1 there are no infinite subsets of N with cardinality less than |N|.

By this reasoning it is concluded that |N| is the smallest cardinality > than any given n.

Please pay attention that the accurate value of |N| is undefined and so is the case about the value of the cardinality of any infinite subset of it.

All what traditional mathematicians care is about the bijection between the mapped sets, and yet they claim that even if the accurate value of the cardinality of N or any cardinality of any infinite subset of it is not accurately defined, one can claim that such sets are complete (all of their members are already included, exactly as the members of finite non-empty sets are already included in their sets).

In other words, traditional mathematicians ignore the essential fact that any non-empty finite set has a cardinality with an accurate value (and therefore can be considered as a complete set) where the infinite set N or any infinite subset of it, do not have cardinalities with accurate value (and therefore can't be considered as complete sets).

So, the bijection between infinite set N = {1,2,3,...} and any infinite subset of it, can't be used in order to conclude that they are also complete.

In that case infinite set N = {1,2,3,...} and any infinite subset of it, can't be defined in terms of actual infinity.

So, we have left with potential infinity as the fundamental notion about infinite set N = {1,2,3,...} and any infinite subset of it, where bijection is not the only possible matching between them, for example:

2 ↔ 1
4 ↔ 2
6 ↔ 3
8
...

In this case there are potentially infinite two sets, where the left set always has one more object.

Moreover, the same argument holds for any kind of sets, for example:

# ↔ &
% ↔ ?
@ ↔ !
*
...
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That is also over the matrix, is aware of the matrix.

That is under the matrix, is unaware of the matrix.

For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video.

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Old 28th January 2020, 04:58 AM   #3428
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More about the completeness of the collection of natural numbers.

I use here the word "collection" instead of "set" since set is usually defined as a collection of distinct objects where order is irrelevant.

It is easily understood that by ordering the objects of a given collection, it does not change the number of its objects, even in the case of sets (where order is irrelevant).

So cardinality (the number of objects of a given set) is not influenced by any order.

N = {1,2,3,...} easily enables to match between a given object and a given cardinality, such that every possible object of N is already a member of N and yet the cardinality of N is not accurately defined exactly because the biggest member of N does not exist (N can't be taken as an object of its own rhight exactly because the number of its members (its cardinality) is not accurately defined).

The inability to show some n that is not already a member N can't be used alone in order to conclude that N is a complete set, exactly because the inability to define the biggest member of N also must be considered.

Unfortunately, traditional mathematicians built their mathematical frameworks by using only the inability to show some n that is not already a member N, in order to (wrongly) conclude that N is a complete mathematical object (or in their jargon "an object of its own right" as written in the beginning of https://en.wikipedia.org/wiki/Set_(mathematics)).

Henri Poincare https://en.wikipedia.org/wiki/Henri_...finite_numbers is very clear about the transfinite system (which is a wrong attempt to define the infinite in terms of collection).

There is no wonder that jsfisher (a traditional mathematician) can't answer to the question written at the and of http://www.internationalskeptics.com...postcount=3422 .
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That is also over the matrix, is aware of the matrix.

That is under the matrix, is unaware of the matrix.

For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video.

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Old 29th January 2020, 12:04 AM   #3429
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Why do you think that https://scholar.google.com/scholar?h...mbers%22&btnG= has no results?

If you have other suggestions than "the completeness of the set of natural numbers" in order to provide some results, please write it.

Thank you.
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That is also over the matrix, is aware of the matrix.

That is under the matrix, is unaware of the matrix.

For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video.

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Old 22nd March 2020, 12:43 PM   #3430
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The Axiom Of Infinity (by words, based on Wikipedia): "There is set A (the set which is postulated to be infinite), such that the empty set is a member of A, and such that whenever any x is a member of A, the set formed by taking the union of x with its singleton {x}, is also a member of A."

Please pay attention that given any A successor, it is finite since it is constructed exactly by all the finitely many previous A members, by induction.

But there is nothing in induction that guarantees infinitely many members in A just because it is our wishful thinking, so no axiom that is based on induction guarantees infinitely many members.

For example, let's take N (the set of natural numbers).

The inability to show some n that is not already a member of N, can't be used alone in order to conclude that N is a complete and infinite set, exactly because the inability to define the biggest member of N, also must be considered.

Since the biggest number of N does not exist, |N| accurate value is actually undefined.

It is claimed that order is irrelevant in case of sets, but it is easily understood that order does not change the cardinality (the number of members) of a given set, so the inability to define the biggest member of N has a direct influence about its cardinality and in the considered case |N| accurate value is actually undefined.
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That is also over the matrix, is aware of the matrix.

That is under the matrix, is unaware of the matrix.

For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video.

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Old 23rd March 2020, 01:33 PM   #3431
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Originally Posted by doronshadmi View Post
The Axiom Of Infinity (by words, based on Wikipedia): "There is set A (the set which is postulated to be infinite), such that the empty set is a member of A, and such that whenever any x is a member of A, the set formed by taking the union of x with its singleton {x}, is also a member of A."

Please pay attention that given any A successor, it is finite since it is constructed exactly by all the finitely many previous A members, by induction.

But there is nothing in induction that guarantees infinitely many members in A just because it is our wishful thinking, so no axiom that is based on induction guarantees infinitely many members.

For example, let's take N (the set of natural numbers).

The inability to show some n that is not already a member of N, can't be used alone in order to conclude that N is a complete and infinite set, exactly because the inability to define the biggest member of N, also must be considered.

Since the biggest number of N does not exist, |N| accurate value is actually undefined.

It is claimed that order is irrelevant in case of sets, but it is easily understood that order does not change the cardinality (the number of members) of a given set, so the inability to define the biggest member of N has a direct influence about its cardinality and in the considered case |N| accurate value is actually undefined.
Everyone has stopped responding to your crap cliams.

Have you any idea why that might be?.

In keeping with woo beliefs, you likely believe that people have ceased because your god like beliefs cannot be contested.

In reality, people get bored with a crank eventually. Now to be fair, most of those are not science cranks. There is the telepathy crew.
There are the dowsing crew.
There are the astrology crew.
There are the flat earth crew.
There the creationist crew.
And on and on.


You particular flavour of belief garners no special condideration.

You are just one more wild claim in a sea of wild claims screaming for attention.

Who cares? Why is your claim more special than all of the other thousands?
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Old 23rd March 2020, 05:40 PM   #3432
Little 10 Toes
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Just because doronshadmi doesn't understand infinity .... That's it.
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Old 9th June 2020, 12:30 AM   #3433
doronshadmi
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Originally Posted by abaddon View Post
Everyone has stopped responding to your crap cliams.

Have you any idea why that might be?.

In keeping with woo beliefs, you likely believe that people have ceased because your god like beliefs cannot be contested.

In reality, people get bored with a crank eventually. Now to be fair, most of those are not science cranks. There is the telepathy crew.
There are the dowsing crew.
There are the astrology crew.
There are the flat earth crew.
There the creationist crew.
And on and on.


You particular flavour of belief garners no special condideration.

You are just one more wild claim in a sea of wild claims screaming for attention.

Who cares? Why is your claim more special than all of the other thousands?
Originally Posted by Little 10 Toes View Post
Just because doronshadmi doesn't understand infinity .... That's it.

Comments addressed to the claimant rather than the claim, are fundamentally worthless because they do not deal with the claim.

So, this time please deal with the content of claim http://www.internationalskeptics.com...postcount=3430
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For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video.

Last edited by doronshadmi; 9th June 2020 at 12:38 AM.
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Old 9th June 2020, 11:37 PM   #3434
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Originally Posted by doronshadmi View Post
Since the biggest number of N does not exist, |N| accurate value is actually undefined.
Let's work on this one sentence.

Just because you can't write down something, it does not mean it doesn't exist. You are complaining that infinity does not have a fixed value.

How many numbers are between 1 and 2? An infinite amount.
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Old 11th June 2020, 02:58 AM   #3435
doronshadmi
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Originally Posted by Little 10 Toes View Post
Let's work on this one sentence.

Just because you can't write down something, it does not mean it doesn't exist.
The biggest member of N does not exist since given any N member, there is always at least one member that is bigger than it.

Therefore |N| accurate value is undefined.

This simple fact has nothing to do with the inability to write down something.
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That is also over the matrix, is aware of the matrix.

That is under the matrix, is unaware of the matrix.

For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video.

Last edited by doronshadmi; 11th June 2020 at 03:01 AM.
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Old 11th June 2020, 03:35 AM   #3436
doronshadmi
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Originally Posted by Little 10 Toes View Post
How many numbers are between 1 and 2? An infinite amount.
In case of infinitely many ordered numbers, given any a and b numbers, such that a<b (where a and b are not taken only as integers), there is always at least one c number such that a<c<b.

Since order does not change the amount of the members, the fact that there is always at least one c number such that a<c<b, is equivalent to the fact that N biggest member does not exist.

So, whether there is always at least one next c number between any two given a<b numbers (where a and b are not taken only as integers), or given any N member, there is always at least one next member that is bigger than it, in both cases the accurate infinite amount is undefined.

Again, there is no wonder that jsfisher (a traditional mathematician) can't answer to the question written at the end of http://www.internationalskeptics.com...postcount=3422 exactly because the notion of |N| as an accurate value is no more than an arbitrary agreement based on no more than a belief among group of persons called traditional mathematicians, which force completeness on collections of infinitely many objects.

Originally Posted by abaddon View Post
You particular flavour of belief garners no special condideration.
Infinity as a complete whole in terms of collections, is no more than a belief.

Traditional mathematicians believe that there is value |N|, which is greater than any member of N AND also accurately measures the amount of N members, but this belief is false since order does not change the amount of N members, and since the biggest member of N does not exist (since given any N member, there is always at least one member that is bigger than it), it is impossible to define an accurate value |N|, which is greater than any member of N.
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That is also over the matrix, is aware of the matrix.

That is under the matrix, is unaware of the matrix.

For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video.

Last edited by doronshadmi; 11th June 2020 at 05:34 AM.
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Old 11th June 2020, 06:55 AM   #3437
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Again, infinity is not a number or fixed amount. You just can't subtract 100 from infinity and get a number.
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Old 11th June 2020, 08:11 AM   #3438
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Originally Posted by Little 10 Toes View Post
Let's work on this one sentence.
You can go earlier in Doronshadmi's post than that. The Axiom of Infinity declares the existence of a set with two properties. Almost immediately in the post Doronshadmi is interpreting it as a how-to instead of a there-is.
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Old 12th June 2020, 01:13 AM   #3439
doronshadmi
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Originally Posted by Little 10 Toes View Post
Again, infinity is not a number or fixed amount. You just can't subtract 100 from infinity and get a number.
|N| is defined (by belief) as fixed amount by traditional mathematicians (the accurate cardinality of any inductive set) where N is such set.

What you say actually supports my argument that |N| accurate value is undefined, unlike the agreed belief among traditional mathematicians (for example, jsfisher's belief, which can't deal (yet) with my question to him about the considered subject).

Please observe this:
Originally Posted by jsfisher View Post
It is much easier, I suppose, to spout your nonsense when you are unconstrained by meaning.
jsfisher, please explain what do you mean by "The Axiom of Infinity establishes a set in terms of Mathematics." (especially the highlighted part)?

jsfisher does not give (yet) any answer to that question and I wonder why
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That is also over the matrix, is aware of the matrix.

That is under the matrix, is unaware of the matrix.

For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video.

Last edited by doronshadmi; 12th June 2020 at 02:00 AM.
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Old 12th June 2020, 01:22 AM   #3440
doronshadmi
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Originally Posted by jsfisher View Post
You can go earlier in Doronshadmi's post than that. The Axiom of Infinity declares the existence of a set with two properties. Almost immediately in the post Doronshadmi is interpreting it as a how-to instead of a there-is.
jsfisher ,"there is" does not define A as an infinite complete whole set, as given in http://www.internationalskeptics.com...postcount=3430.

Also you did not reply to http://www.internationalskeptics.com...postcount=3430 and http://www.internationalskeptics.com...postcount=3436.

Moreover:
Originally Posted by jsfisher View Post
It is much easier, I suppose, to spout your nonsense when you are unconstrained by meaning.
jsfisher, please explain what do you mean by "The Axiom of Infinity establishes a set in terms of Mathematics." (especially the highlighted part)?
__________________
That is also over the matrix, is aware of the matrix.

That is under the matrix, is unaware of the matrix.

For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video.

Last edited by doronshadmi; 12th June 2020 at 01:34 AM.
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