
Welcome to the International Skeptics Forum, where we discuss skepticism, critical thinking, the paranormal and science in a friendly but lively way. You are currently viewing the forum as a guest, which means you are missing out on discussing matters that are of interest to you. Please consider registering so you can gain full use of the forum features and interact with other Members. Registration is simple, fast and free! Click here to register today. 
19th August 2019, 04:23 AM  #3401 
Penultimate Amazing
Join Date: Mar 2008
Posts: 12,936

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: and he does them in order to avoid the following question: 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:
Moreover
Quote:
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) 
__________________
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. 

19th August 2019, 06:13 AM  #3402 
ETcorngods survivor
Moderator Join Date: Dec 2005
Posts: 22,453

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:
You probably want to clean that up, and when you do, please explain what "relate it to" means. 
__________________
A proud member of the Simpson 15+7, named in the suit, Simpson v. Zwinge, et al., and founder of the ET Corn Gods Survivors Group. "He's the greatest mod that never was!"  Monketey Ghost 

19th August 2019, 07:05 AM  #3403 
Penultimate Amazing
Join Date: Mar 2008
Posts: 12,936

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. 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). Your insistence to establish a clear cut border between Philosophy and Mathematics (and in the discussed case, Logic) is, well, your philosophy. 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)? 
__________________
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. 

19th August 2019, 07:14 AM  #3404 
ETcorngods survivor
Moderator Join Date: Dec 2005
Posts: 22,453


__________________
A proud member of the Simpson 15+7, named in the suit, Simpson v. Zwinge, et al., and founder of the ET Corn Gods Survivors Group. "He's the greatest mod that never was!"  Monketey Ghost 

19th August 2019, 07:44 AM  #3405 
Penultimate Amazing
Join Date: Mar 2008
Posts: 12,936

The certain property is Platonic (or Actual) Infinity, which is related to ZF(C) Axiom Of Infinity (x) and ZF(C) extension (y).
Quote:
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)? 
__________________
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. 

19th August 2019, 09:01 AM  #3406 
ETcorngods survivor
Moderator Join Date: Dec 2005
Posts: 22,453

The property would be "infinite" not "infinity", but let's move on.
Quote:
Quote:
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.) 
__________________
A proud member of the Simpson 15+7, named in the suit, Simpson v. Zwinge, et al., and founder of the ET Corn Gods Survivors Group. "He's the greatest mod that never was!"  Monketey Ghost 

Yesterday, 02:44 AM  #3407 
Penultimate Amazing
Join Date: Mar 2008
Posts: 12,936

Let's not move on. The certain property is Infinity in terms of Platonic (or Actual) Infinity.
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). Axiom schema (and therefore Infinity) are parts of ZF(C). 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). Formal system A has infinitely many wffs in terms of Platonic (or Actual) Infinity (formal system A is taken as a complete whole). 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)? 
__________________
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. 

Yesterday, 06:28 AM  #3408 
ETcorngods survivor
Moderator Join Date: Dec 2005
Posts: 22,453

Many bits of nonsense saved for a later time so as to not further defocus the current thread arc.
Quote:
Quote:
Quote:
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. 
__________________
A proud member of the Simpson 15+7, named in the suit, Simpson v. Zwinge, et al., and founder of the ET Corn Gods Survivors Group. "He's the greatest mod that never was!"  Monketey Ghost 

Yesterday, 08:01 AM  #3409 
Penultimate Amazing
Join Date: Mar 2008
Posts: 12,936

Another hands waving of yours.
I agree with you, G is just one statement, as written at the end of my previous post (" Z' " is used instead of "A"). 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. 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). 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. 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)? 
__________________
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. 

Yesterday, 09:15 AM  #3410 
ETcorngods survivor
Moderator Join Date: Dec 2005
Posts: 22,453

You are making up your own theorems, now, too?
Quote:
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 socalled 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. 
__________________
A proud member of the Simpson 15+7, named in the suit, Simpson v. Zwinge, et al., and founder of the ET Corn Gods Survivors Group. "He's the greatest mod that never was!"  Monketey Ghost 

Yesterday, 10:55 PM  #3411 
Penultimate Amazing
Join Date: Mar 2008
Posts: 12,936

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.
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. 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 noninteresting 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)? 
__________________
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. 

Today, 04:14 AM  #3412 
ETcorngods survivor
Moderator Join Date: Dec 2005
Posts: 22,453


__________________
A proud member of the Simpson 15+7, named in the suit, Simpson v. Zwinge, et al., and founder of the ET Corn Gods Survivors Group. "He's the greatest mod that never was!"  Monketey Ghost 

Today, 04:47 AM  #3413 
Penultimate Amazing
Join Date: Mar 2008
Posts: 12,936

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)? 
__________________
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. 

Bookmarks 
Thread Tools  

