The Attacks against Gödel's Incompleteness
1. Attack
The issue under "Limitations of Gödel's theorems" isn't whether "Gödel could use logics too", but whether
1. y is the Gödel number of a formula and x is the Gödel number of a proof of the formula encoded by y
2. y is the Gödel number of a formula THEN
3. Bew(y) = ∃x (x is the Gödel number of a proof of the formula encoded by y)
AND therefore DEFEATS
Gödel's two theorems because the above completely describes the disposition of the field given, i.e., the disposition of logics, the UoD, by Everything (in Mathematics), entities and so on, all the way up to "the whole of mathematics and so on", best seen by
"Bew(y) = ∃ x ( y is the Gödel number of a formula and x is the Gödel number of a proof of the formula encoded by y)",
being the defeated part, under "Construction of a statement about "provability". So the war is on between Gödel/Gödel followers and Gödel critics, whereof I am one. This is the reason that Gödel stands against Tarski in the intellectual World today! But criticism has to be met and we'll see.
2. Attack
The Probable Solution to All Set Theory
'What is it to know? I have absolutely no idea! To "know" has been assigned to me!'
I think the set theory that breaks the Principia Mathematica can be solved by S = Ø (set of solution is empty)
The description:
One should remember that one object/member lower down the hypothetical chain of sets (by categories) triggers necessary objects/members all the way up to the "first natural level where one would otherwise see an empty set right below it". "The first natural level" can also be seen as "the deepest level" before, if any at all, the empty set can occur." "You can add all the (meaningless) categories/set containers you want under a natural set/one set that contains members, but where do you get when the bottom container is empty?
Clearly, it's just rubbish and thus it's not a serious argument against the project that Principia Mathematica represents."
That is, by this explanation, that the maximal number of empty sets under the natural chain of sets, can only be 1, one, but usually is 0, zero, by the usual descriptions of commoners and non-mathematicians. This, thus, represents the final solution to set-theory for all time to come. Good?
(Corroborative for knowledge: Out of 'I know nothing and my set is empty! Can you call illusions knowledge? I don't think so!')
3. Attack
One "unexplainable" smacker for Gödel
One last smacker for Godel: All axioms are needed to establish a (logical) system - Premise
All axioms - Premise
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Logical System - Cond. Elim. - Conclusion by Modus Ponens
You can add the extra reiteration for classical premises, deduction and conclusion to obtain yourself.
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Former Text
The former text follows below:
On the Liar Paradox and more
Generally the liar paradox is shown to be meaningless (now). Next, Tarski and others hold powerful arguments against Gödel's incompleteness theorems. And there is a set theory that you may want to note, http://whatiswritten777.blogspot.no/2011/08/philosophical-notes-of-intellectual.html (a bit down on the page), that has this text: "For the time being, I have this to write. Out of 'I know nothing and my set is empty! Can you call illusions knowledge? I don't think so! What is it to know? I have absolutely no idea! To "know" has been assigned to me! I think the set theory that breaks the Principia Mathematica can be solved by S = ∅ (set of solution is empty). In case of protest, one should remember that one object/member lower down the hypothetical chain of sets (by categories) triggers necessary objects/members all the way up to the "first natural level where one would otherwise see an empty set right below it". "The first natural level" can also be seen as "the deepest level" before, if any at all, the empty set can occur."
"You can add all the (meaningless) categories/set containers you want under a natural set/one set that contains members, but where do you get when the bottom container is empty? Clearly, it's just rubbish and thus it's not a serious argument against the project that Principia Mathematica represents." That is, by this explanation, that the maximal number of empty sets under the natural chain of sets, can only be 1, one, but usually is 0, zero, by the usual descriptions of commoners and non-mathematicians. This, thus, represents the final solution to set-theory for all time to come. Good?
Relationship with computability
Given the below, it must be clear that the halting problem occurs when non-meaningful input has been programmed or that the computer is running an inifinite set, one issue that should be calculated by the machine itself before running/processing of the input happens!
Remember that most testing of these things happen on "scientific" computer, the big mainframes, Tevafloppies and more, i.e., the supercomputers, and as such, qualifying the input by looking for inifinite input should be no problem!
Because there is a significant difference in running input directly vs. checking for infinity input before running the input, i.e., the programming in the loose sense.
From under the "Construction of a statement about "provability"
From 3 sentences,
1. y is the Gödel number of a formula and x is the Gödel number of a proof of the formula encoded by y
2. y is the Gödel number of a formula THEN
3. Bew(y) = ∃x (x is the Gödel number of a proof of the formula encoded by y)
Under "Discussion and implications" by the header of this note, I get:"The incompleteness results affect the philosophy of mathematics, particularly versions of formalism, which use a single system formal logic to define their principles. One can paraphrase the first theorem as saying the following: An all-encompassing axiomatic system can never be found that is able to prove all mathematical truths, but no falsehoods.""On the other hand, from a strict formalist perspective this paraphrase would be considered meaningless because it presupposes that mathematical "truth" and "falsehood" are well-defined in an absolute sense, rather than relative to each formal system.""The following rephrasing of the second theorem is even more unsettling to the foundations of mathematics: If an axiomatic system can be proven to be consistent from within itself, then it is inconsistent. Therefore, to establish the consistency of a system S, one needs to use some other system T, but a proof in T is not completely convincing unless T's consistency has already been established without using S.""Theories such as Peano arithmetic, for which any computably enumerable consistent extension is incomplete, are called essentially undecidable or essentially incomplete."
To this I now answer and generally hold:I question "An all-encompassing axiomatic system can never be found that is able to prove all mathematical truths, but no falsehoods." on grounds of making an axiomatic system that covers all disciplines of mathematics, yet in several parts and "adjacent-"/"contegious-sectors" if you will!There is NO chance that the two incompleteness theorems will survive into the next decade, starting immediately 2020! Call it sci-fi for now, if you want!
For people who think that to make a title "This is not a title" on a book (Raymond Smullyan, fx.) matters, you do not do much other than positing a Austin statement, that is, you commit a speech act, NOT logics!
To say that the total of field isn't provable, isn't good enough, because the field always remain contestable (until one can begin to look on the results consider what "in the World" that can possibly remain in the field to discover!
So criticism toward Gödel still starts with "Everything"!!!
That said, nobody has ever said that any system could be proven by setting up axioms for it!!! Given the axioms themselves, one still doesn't know whether they as a group are enough to cover the field they are designed as seen in geometry with Riemann geometry, under the assumption that the Euclidean geometric planes have been intended to be straight/flat all along!
Some people may think that I've been "after" Gödel, but this is wrong! I've just been saying that I've wanted complete systems and that looking for something /else/ than Gödel's claim over the axioms is, looking at an undeveloped system with very few results, almost impossible, but I don't want to go into this just yet. It may turn out to be a ghost that haunts us, given that advances in logics can very well occur more than "expected", or beyond one's negative taste in case I would give a verdict on it! Let's see what happens!
That is, the current standing on the Gödel's Incompleteness Theorems isn't UP TO DATE!!!
(18 June at 21:41 CEST)
I am of the opinion that criticism should be presented on the same page under the header "Criticism of the Gödel's incompleteness theorems" because this is about presenting the truth. That is, you can't leave out the fact that his incompleteness theorems may be untrue!
(18 June at 22:40 CEST)
One last smacker for Godel: All axioms are needed to establish a (logical) system (P)
All axioms (P)
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Logical System (Cond. Elim. and Concl.)
(19 June at 15:07 CEST)
y is the Gödel number of a formula and x is the Gödel number of a proof of the formula encoded by y
y is the Gödel number of a formula -> (conditionally) (x is the Gödel number of a proof of the formula encoded by y) ∃ x = Bew(y), but Gödel forgets about the premise and gets it WRONG! Good?
(20 June at 18:46 CEST)
Conventionally, the other then: Bew(y) = ∃x (x is the Gödel number of a proof of the formula encoded by y) from under the /premise/: y is the Gödel number of a formula /then/ the former sentence to this premise.
(20 June at 18:46 CEST)
See above in the main body, the very note, for the premise set /before/ the Bew(y)!
(20 June at 18:47 CEST)
Neatly made by this, for time-stamp: From under the "Construction of a statement about "provability"
From 3 sentences,
1. y is the Gödel number of a formula and x is the Gödel number of a proof of the formula encoded by y
2. y is the Gödel number of a formula THEN
3. Bew(y) = ∃x (x is the Gödel number of a proof of the formula encoded by y)
(20 June at 18:53 CEST)
Under "Discussion and implications" by the header of this note, I get:"The incompleteness results affect the philosophy of mathematics, particularly versions of formalism, which use a single system formal logic to define their principles. (more...)
(Friday at 16:11 CEST)
Funny stuff from the Wikip. article: The section of "Limitations of Gödel's theorems" used to be an idiot place even though they've corrected it now to what it should be, I've had a comment to it earlier: "Lastly, for inducing some discipline here: Under "Limitations of Gödel's theorems", I assume ''the theorems still need to hold a point''! Don't they?"
(18 hours ago CEST)
The issue under "Limitations of Gödel's theorems" isn't whether "Gödel could use logics too", but whether
1. y is the Gödel number of a formula and x is the Gödel number of a proof of the formula encoded by y
2. y is the Gödel number of a (more...)
(15 hours ago CEST)
Last from me: For people who think that to make a title "This is not a title" on a book (Raymond Smullyan, fx.) matters, you do not do much other than positing a Austin statement, that is, you commit a speech act, NOT logics!
(8 hours ago CEST)
To say that the total of field isn't provable, isn't good enough, because the field always remain contestable (until one can begin to look on the results consider what "in the World" that can possibly remain in the field to discover!
(8 hours ago CEST)
So criticism toward Gödel still starts with "Everything"!!!
(8 hours ago CEST)
So even if the system isn't provable from the axioms as such, the system can very well become complete in all other senses, and given special considerations of a given field, you begin to consider the field complete from the results you (more...)
(8 hours ago CEST)
That said, nobody has ever said that any system could be proven by setting up axioms for it!!!
(8 hours ago CEST)
However, Gödel still defeats these other lunatics who say that they have these axioms and that this system therefore has to generate these and other results, so Gödel is a winner in these other respects!
(8 hours ago CEST)
Seconds after, time stamp for the above incomplete group of axioms...
(8 hours ago CEST)
Some people may think that I've been "after" Gödel, but this is wrong! I've just been saying that I've wanted complete systems and that looking for something /else/ than Gödel's claim over the axioms is, looking at an undeveloped system (more...)
(8 hours ago CEST)
"Straight" in the Euclidean sense is to be interpreted as "flat", only!!!
(8 hours ago CEST)
For now, I just want to note that I look to the group of axioms and the group of results from the system on two levels and that future investigations in logics to "notions of completeness" start here!
(8 hours ago CEST)
(Note on time: 18 June, 2012.)
(Note on time: 20 June, 2012.)
(Note on time: 22 June, 2012.)
(Note on time: 24 June, 2012.)
Note5: Some of the time-stamps are only "minutes-accurate"! Good?
Note6: The rather coarse and "strange" part from above has been left out and placed here instead, "Thanks, Russell, for pointing out the danger of having a single proposition of knowledge!' TL (I think this quote has been made around 20.11.2009 or a little bit later, but at least in 2009. 23rd Nov. 2009 is by record of Twitter.)"
