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12th August 2019, 08:13 PM  #3361 
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I always find it a bit funny that people would use Godel numbering, using natural numbers, to encode statements in logic when they don't actually need the machinery to prove Godel's theorem. It would seem much simpler to just encode them as strings of symbols (such as ASCII or Latex something).

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"Ideas are also weapons."  Subcomandante Marcos "We must devastate the avenues where the wealthy live."  Lucy Parsons "Let us therefore trust the eternal Spirit which destroys and annihilates only because it is the unfathomable and eternal source of all life. The passion for destruction is a creative passion, too!"  Mikhail Bakunin 

12th August 2019, 09:17 PM  #3362 
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Well, considering Unicode has 137,994 characters, we can use that as a base, with the understanding when we use words vs. numbers. But since we use base10, we're stuck with that.

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I'm an "intellectual giant, with access to wilkipedia [sic]" "I believe in some ways; communicating with afterlife is easier than communicating with me." Tim4848 who said he would no longer post here, twice in fact, but he did. 

12th August 2019, 09:19 PM  #3363 
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we just happen to use base10. We could use base137,994 (the number of characters currently in use in Unicode)/

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I'm an "intellectual giant, with access to wilkipedia [sic]" "I believe in some ways; communicating with afterlife is easier than communicating with me." Tim4848 who said he would no longer post here, twice in fact, but he did. 

12th August 2019, 09:37 PM  #3364 
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We just happen to use base10. We could use base137,994 (the number of characters currently in use in Unicode).

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I'm an "intellectual giant, with access to wilkipedia [sic]" "I believe in some ways; communicating with afterlife is easier than communicating with me." Tim4848 who said he would no longer post here, twice in fact, but he did. 

12th August 2019, 10:18 PM  #3365 
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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. 

12th August 2019, 10:24 PM  #3366 
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We can use any base we want, there are trivial mappings between the sets of finite strings in various bases. Simplest of all we could just use base2, binary bits, where each set of 8 bits is a byte which maps to some symbol in the ASCII table.

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"Ideas are also weapons."  Subcomandante Marcos "We must devastate the avenues where the wealthy live."  Lucy Parsons "Let us therefore trust the eternal Spirit which destroys and annihilates only because it is the unfathomable and eternal source of all life. The passion for destruction is a creative passion, too!"  Mikhail Bakunin 

12th August 2019, 10:37 PM  #3367 
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Both replies are no more than unsupported statements.
So, dlorde and jsfisher please support them. Just saying them is not the same as also logically support them. So the stage is yours and this time do not share only your opinions about http://www.internationalskeptics.com...postcount=3357 but actually logically support your arguments according to its content. 
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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. 

13th August 2019, 12:26 AM  #3368 
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Simple probability tells us that we should expect coincidences, and simple psychology tells us that we'll remember the ones we notice... 

13th August 2019, 04:11 AM  #3369 
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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. 

13th August 2019, 06:32 AM  #3370 
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I'm an "intellectual giant, with access to wilkipedia [sic]" "I believe in some ways; communicating with afterlife is easier than communicating with me." Tim4848 who said he would no longer post here, twice in fact, but he did. 

13th August 2019, 07:36 AM  #3371 
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Set A is a set in infinitely many axioms (where each axiom is written by finitely many symbols) which is established by using ZF(C) Axiom Of Infinity on ZF(C) itself, such that Infinity is taken in terms of Platonic Infinity (By Platonic Infinity there exists a complete set of infinitely many things as a complete whole (without using any process)).
Some example: The set of all natural numbers is infinite in terms of Platonic 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. 

13th August 2019, 09:42 AM  #3372 
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Some correction of the beginning of my previous post.
It has to be corrected to: "Set A is a set of infinitely many axioms ..." So let's write the previews post here in order to complete here my argument.  Set A is a set of infinitely many axioms (where each axiom is written by finitely many symbols) which is established by using ZF(C) Axiom Of Infinity on ZF(C) itself, such that Infinity is taken in terms of Platonic Infinity (By Platonic Infinity there exists a set of infinitely many things as a complete whole (without using any process)). Some example: The infinite set of all natural numbers is taken in terms of Platonic infinity. Now all we care (as written in http://www.internationalskeptics.com...postcount=3357) is about the set of all infinitely many wffs (in terms of Platonic Infinity) that can be established in A. Each wff has some Godel number, where at least one of these wffs, called G, states "There is no number m such that m is the Godel number of a proof in A, of G" Since all wffs are already in A and all Godel numbers are already in A (because Infinity is taken in terms of Platonic Infinity) there is a Godel number of a proof of G in A, which contradicts G in A, exactly because A is complete and therefore inconsistent, since Infinity is taken in terms of Platonic Infinity. So the problem is actually the notion of a complete set of infinity many things in terms of Platonic Infinity, and in order to save the consistency of A, ZF(C) Axiom Of Infinity is taken in terms of Potential Infinity (process is used, exactly as done in case of GIT). But then ZF(C) Axiom Of Infinity can't be used in order to establish sets in terms of Platonic Infinity (for example: the notion of The infinite set of all natural number is logically inconsistent).  Now, Gödel was a Platonist (he agreed with Actual infinity in terms of Cantor (which is actually Platonic Infinity)) and his main motivation behind his Incompleteness Theorems was to logically demonstrate that formal systems that are strong enough in order to deal with Arithmetic, can't be complete AND consistent and also can't prove their own consistency (which means that many "interesting" formal systems can't deal with Platonic realms). But Gödel's Incompleteness Theorems also prove that the very notion of Actual infinity in terms of Platonism (which is also Actual infinity in terms of Cantor) does not hold logically (at least in the strong sense). 
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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. 

13th August 2019, 10:10 AM  #3373 
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13th August 2019, 11:21 AM  #3374 
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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. 

13th August 2019, 11:48 AM  #3375 
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"Ideas are also weapons."  Subcomandante Marcos "We must devastate the avenues where the wealthy live."  Lucy Parsons "Let us therefore trust the eternal Spirit which destroys and annihilates only because it is the unfathomable and eternal source of all life. The passion for destruction is a creative passion, too!"  Mikhail Bakunin 

13th August 2019, 12:20 PM  #3376 
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Well, sets are not "strong enough" in order to deal with Arithmetic. That would require a formal system. I think you must have meant the set of axioms for some formal system. And since you don't care which one, how about the set of axioms for ZFC?
Conveniently enough, that set is infinite.
Quote:
The real problem, though, is with "using [the] Axiom of Infinity on ZF(C)". That does not mean anything. That particular axiom simply says a certain set with a certain property exists within the formal system. It is not something you can "use on" something; you don't, for example, "use it on" another set to establish yet another set with that same certain property mentioned in the Axiom of Infinity. The Axiom of Infinity tells us a certain set exists. (It guarantees one, but as it turns out there are many such sets.) The von Neumann ordinal is the simplest example of such a set. It is not an operator, though. You can't use it on a set. 
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14th August 2019, 12:22 AM  #3377 
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A is a set of axioms, which are a formal system.
It means a lot. A is a formal system with infinitely many axioms, which is established by using ZF(C) Axiom Of Infinity on ZF(C) itself, such that A is complete in terms of Platonic Infinity. No operator is involved by using ZF(C) Axiom Of Infinity on ZF(C) itself, if Infinity is taken in terms of Platonic Infinity. If you reject the notion that ZF(C) Axiom Of Infinity is not taken in terms of Platonic Infinity, then by this axiom, even the infinite set of all natural numbers can't be established in terms of Cantorian Actual 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. 

14th August 2019, 02:47 AM  #3378 
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A is a set of axioms, which are a formal system, so when I wrote "Any set that is strong enough in order to deal with Arithmetic." it was about a set of axioms (in case that you are missing that the last discussion is about a set of axioms in terms of Platonic Infinity).
Now, 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 wff 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. 

14th August 2019, 04:17 AM  #3379 
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14th August 2019, 05:04 AM  #3380 
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Please stop read partially by also ignore the considered context of the discussed subject.
For example, please read all of http://www.internationalskeptics.com...postcount=3372 before you replay to some part of it. Thank you. I mean that A is a set of infinitely many axioms which is "strong enough" in order to deal with Arithmetic, where Infinity is taken in terms of Platonic Infinity. The considered subject is not about the axioms (that, by definition, are not proven) but about wffs in A that have to be proven in A, where the amount of such wffs is taken in terms of Platonic 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. 

14th August 2019, 08:15 AM  #3381 
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Opss...

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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. 

14th August 2019, 08:18 AM  #3382 
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Do Gödel's First Incompleteness Theorem imply the inconsistency of Actual Infinity?
Ok, it is about time to gather the last discussion into one post.
 According to Modern Mathematics (where the majority of mathematicians agree about the notion of actual infinite sets, as established mostly by George Cantor) an inductive set (as given by ZF(C) Axiom Of Infinity) has an accurate cardinality, which implies that it is complete (no one of its members is missing). In other words, by ZF(C) Axiom Of Infinity there exists at least one infinite AND complete set (if we agree with the notion of actual infinity, as mostly established by Cantor). Now, assume a complete set of infinite axioms (according to the reasoning of actual infinity, as established mostly by Cantor and agreed by the majority of modern mathematicians). But by Gödel's First Incompleteness Theorem such set of axioms must be inconsistent as follows: Set A is a set of infinitely many axioms (where each axiom is written by finitely many symbols) which is established by using ZF(C) Axiom Of Infinity on ZF(C) itself, such that Infinity is taken in terms of Platonic Infinity (By Platonic Infinity there exists a set of infinitely many things as a complete whole (without using any process)). Some example: The infinite set of all natural numbers is taken in terms of Platonic infinity. Now all we care is about the set of all infinitely many wffs (in terms of Platonic Infinity) that are established in A. Each wff has some 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 A, of G" (since G needs a proof, it is not an axiom but a theorem). Since all wffs are already in A and all Godel 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 (as shown) and therefore inconsistent. So the problem is actually the notion of a complete set of infinity many things in terms of Platonic Infinity, and in order to save the consistency of A, ZF(C) Axiom Of Infinity is taken in terms of Potential Infinity (process is used, exactly as done in case of Gödel's First Incompleteness Theorem). But then ZF(C) Axiom Of Infinity can't be used in order to establish sets in terms of Platonic Infinity (for example: the notion of The infinite set of all natural number is logically inconsistent).  Gödel was a Platonist (he agreed with Actual infinity in terms of Cantor (which is actually Platonic Infinity)) and his main motivation behind his Incompleteness Theorems was to logically demonstrate that formal systems that are strong enough in order to deal with Arithmetic, can't be complete AND consistent and also can't prove their own consistency (which means that many "interesting" formal systems can't deal with Platonic realms). But Gödel's First Incompleteness Theorem also proves that the very notion of Actual infinity in terms of Platonism (which is also Actual infinity in terms of Cantor) does not hold logically.  There is a noninteresting solution about the discussed subject, as follows: G states: "There is no number m such that m is the Gödell 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 wff 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).  Also please be aware of the following: 1) If ZF(C) Axiom Of Infinity is not necessarily taken in terms of Platonic Infinity, then ZF(C) Axiom Of Infinity is taken in terms of Platonic Infinity OR Not (useless tautology). 2) If ZF(C) Axiom Of Infinity is not necessarily taken in terms of Platonic Infinity, then it can't be used in order to establish even the set of all natural numbers (which means that N (and N) is not necessarily established by ZF(C)). 
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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. 

14th August 2019, 10:21 AM  #3383 
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How do you get from
There exists a set which contains the empty set and for every member of the set it also contains the member's successor^{*}to A is a set of infinitely many axioms which is "strong enough" in order to deal with Arithmetic, where Infinity is taken in terms of Platonic Infinity.That's a breathtaking leap, even for you. Also, nowhere did you "use" the axiom "on ZF(C) itself." ^{*} 'Successor' has been defined various ways for different uses of the axiom in different set theories. ZF and ZFC share a specific successor function, S, where S(x) = x U {x}. 
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15th August 2019, 03:59 AM  #3384 
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All we care is that such set is taken in terms of Platonic Infinity.
I agree with you, modern mathematicians indeed take such axiom in terms of Platonic Infinity. Infinity is taken in A (which is strong enough to deal with Arithmetic) in terms of Platonic Infinity, such that all the infinite wffs are already in A, exactly as ZF(C) Axiom Of Infinity defines an infinite set as a complete whole (Platonic Infinity). jsfisher, please move on to http://www.internationalskeptics.com...postcount=3382. 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. 

15th August 2019, 07:33 AM  #3385 
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Who is this "we" of which you speak?
All "we" (= the set of not you) care about is what the axiom actually asserts, not this extra baggage and bogus inferences you insist upon. So, again, the Axiom of Infinity is very specific in what it asserts. How do you get from there, in small, logical steps, please, to the rather distant conclusion you allege? 
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16th August 2019, 01:32 AM  #3386 
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By Platonic Infinity there exists a set of infinitely many things as a complete whole.
The Axiom of Infinity establishes a set in terms of Platonic Infinity (if Platonic Infinity is rejected, then The Axiom of Infinity can't establish even the infinite set of all natural numbers (which means that N (and therefore N) is not established even in the abstract sense)). Let A (which is a formal system) be the set of all infinite (in terms if Platonic Infinity) wffs (axioms (that do not need to be proven) OR theorems (that need to be proven)) that are encoded by Gödel numbers in A, such that A is strong enough in order to deal with Arithmetic. In that case what is the cardinality of all the infinite Gödel numbers in A (where infinity is taken in terms of Platonic Infinity)? Before you answer to this question, please be aware of the following: 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 A, of G" (since G needs a proof, it is not an axiom but a theorem). Since all wffs 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. 
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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. 

16th August 2019, 04:23 AM  #3387 
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The Axiom of Infinity establishes a set in terms of Mathematics. The properties the set has are based in Mathematics. How you may relate to that philosophically is of no interest to Mathematics.
You have told us that your set, A, is based on "using the ZF(C) Axiom of Infinity on ZF(C) itself." So, a very reasonable question than is what do you mean by that. So far, you have failed to give it any meaning, yet it is fundamental you this set, A, you banter about. 
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16th August 2019, 05:40 AM  #3388 
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By "using the ZF(C) Axiom of Infinity on ZF(C) itself." I mean that A is a formal system with infinitely many wffs in terms of Platonic Infinity, which is strong enough in order to deal with Arithmetic.
Now please answer to http://www.internationalskeptics.com...postcount=3386 but before that please explain what do you mean by "The Axiom of Infinity establishes a set in terms of Mathematics." (especially the highlighted part)? 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. 

16th August 2019, 06:06 AM  #3389 
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You are telling us something about what you think you get by "using the ZF(C) Axiom of Infinity on ZF(C) itself." You first need to tell us what "using the ZF(C) Axiom of Infinity on ZF(C) itself" all by itself means.
Many sets have the characteristics you ascribe to your set, A. Would any such set do (in which case, all this blather about the Axiom of Infinity is extraneous)? 
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17th August 2019, 03:32 AM  #3390 
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A, by itself, is exactly a formal system with infinitely many wffs in terms of Platonic Infinity (exactly as a set with infinitely things is established by ZF(C) Axiom Of Infinity), which is strong enough in order to deal with Arithmetic.
There can be many sets like A, but A is enough in order to address the discussed subject. 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. 

17th August 2019, 08:36 AM  #3391 
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17th August 2019, 11:20 AM  #3392 
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dlorde is not alone in that opinion. You have done nothing but produce scads of navel gazing crapulence in this entire thread. We have reached, and passed the point where you have for protrtacted periods, happily posted to yourself, even at times arguing with yourself. People dip in now and then just to see how deep the rabbit hole has become. And that's it.
You have, for reasons unknowable, reduced yourself to an internet curiosity. Why? Who can tell. All we can do is observe it happening. 
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17th August 2019, 01:06 PM  #3393 
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ZF(C) is an axiomatic formal system that is strong enough in order to deal with Arithmetic.
By using ZF(C) Axiom Of Infinity on ZF(C) itself, there is an axiomatic formal system A that has infinitely many wffs in terms of Platonic (or Actual) Infinity AND it is strong enough in order to deal with Arithmetic, exactly as ZF(C) enables to do it. Now please reply to my question to you in http://www.internationalskeptics.com...postcount=3390. 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. 

17th August 2019, 01:14 PM  #3394 
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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. 

17th August 2019, 05:13 PM  #3395 
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17th August 2019, 10:19 PM  #3396 
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"using ZF(C) Axiom Of Infinity on ZF(C) itself"
is the same as "There is an extension of ZF(C), called A (which is strong enough in order to deal with Arithmetic (exactly as ZF(C) has this property)) where all of its infinitely many wffs are already in A, since Infinity (according to ZF(C) Axiom Of Infinity) must be taken in terms of Platonic (or Actual) Infinity" (otherwise even the infinite set of all natural numbers can't be established by this ZF(C) axiom). Now 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. 

18th August 2019, 12:06 PM  #3397 
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Moderator Join Date: Dec 2005
Posts: 22,453

There is nothing in the Axiom of Infinity that lets you conclude your set A exists. You also have not explained what "using ZF(C) Axiom Of Infinity on ZF(C) itself" means. Telling us the result you think you have isn't the same as telling us how you got it.
So, once again, what does "using ZF(C) Axiom Of Infinity on ZF(C) itself" mean? 
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18th August 2019, 12:41 PM  #3398 
Master Poster
Join Date: Nov 2006
Posts: 2,002

Let's get rid of the extra wording...
"There is an extension of ZF(C), called A where all of its infinitely many wffs are already in A, since Infinity must be taken in terms of Actual Infinity." 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? 
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18th August 2019, 10:05 PM  #3399 
Penultimate Amazing
Join Date: Mar 2008
Posts: 12,938

Wrong, for example: Without the Axiom of Infinity (where Infinity is taken in terms of Platonic (or Actual) Infinity) no axiom schema (for example: the ZF(C) Axiom Schema Of Replacement) exists.
"Given that the number of possible subformulas or terms that can be inserted in place of a schematic variable is countably infinite, an axiom schema stands for a countably infinite set of axioms." https://en.wikipedia.org/wiki/Axiom_...axiomatization In other words, A (exits, exactly as given in http://www.internationalskeptics.com...postcount=3396) or the ZF(C) axiom schema of replacement are both established by taking Infinity in terms of Platonic Infinity, since countably infinite means that there is bijection between, for example, the infinite set of all natural numbers (in terms of Platonic (or Actual) Infinity) and some given set. 
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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. 

19th August 2019, 04:15 AM  #3400 
ETcorngods survivor
Moderator Join Date: Dec 2005
Posts: 22,453

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? 
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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 

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