anne wrote: ↑Thu Feb 28, 2019 4:39 am
Godels 1st and 2nd theorems are meaningless invalid and end in paradox
http://gamahucherpress.yellowgum.com/wp ... GODEL5.pdf
1) Gödel’s 1st theorem is meaningless and invalid
Godels 1st theorem states
a) “Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true, but not provable in the theory (Kleene 1967, p. 250)
"... there is an arithmetical statement that is true..."
In other words there are true mathematical statements which cant be proven
But the fact is Godel cant tell us what makes a mathematical statement true thus his theorem is meaningless
b) Gödel uses a G statement to prove his theorem but Godel's G statement is outlawed by the very axiom of the system he uses to prove his theorem -thus his proof is invalid,
Godel's G statement is impredicative '
Godel uses the axiom of reducibility -ie the logic of the system he is using to make his proof
the axiom of reducibility bans impredicative statements
thus Godel's G statement is banned-but he uses it to prove his theorem
thus Godel's theorem is invalid
2)Godel's 2nd theorem ends in paradox
Godel's 2nd theorem is about
"If an axiomatic system can be proven to be consistent and complete from
within itself, then it is inconsistent.”
But we have a paradox
Gödel is using a mathematical system
his theorem says a system cant be proven consistent
THUS A PARADOX
Godel must prove that a system cannot be proven to be consistent based upon the premise that the logic he uses must be consistent . If the logic he uses is not consistent then he cannot make a proof that is consistent. So he must assume that his logic is consistent so he can make a proof of the impossibility of proving a system to be consistent. But if his proof is true then he has proved that the logic he
uses to make the proof must be consistent, but his proof proves that
this cannot be done
THUS A PARADOX
Welcome to the discussion anne
. You come from the same initial conclusion precisely as I have also and appreciate others who directly read and understood the topic.
Some initial recognition for others not in the know of the history, any discussions of this topic treated mathematics on par with logic and while some had different views of separation, everyone then treated the term, "arithmetic" as a particular subset of logic. Gödel's literal theorems were derived upon Russell's and Whitehead's attempt to prove everything from an initial set of simple primitives through a set of volumes of books under the title, "Principia Mathematica", for its main title in honor of Newton's same effort more normally treated for its physics that most ignore was more broad to include a universal means to prove everything from scratch.
Although this theorem is itself 'paradoxical', he, as well as those like Turing and Church, still showed that when we begin with a 'consistent and complete' logic (the 'first-order' logic Gödel used in his theorem), you can prove that it still leads permanently to a paradox, precisely as you concluded. He was trying to be very particular and had to in order to provide confidence of others who relied on the belief that formal logic itself was sufficient to resolve all problems in the realm of 'totality' (not simply our specific universe but to the total collection of all regardless of what one might believe).
Gödel knew this and was equally disappointed afterwards. His resolution was to adapt an 'intuitionist' stance that was more broad and related to a multi-valued system of reasoning that denied the "Law of Excluded Middle" (LEM by logik's shorthand here). This is where logik here is beginning from but why I thought it would be good to step back in this thread to look at its origins more specifically. You're among similar minds but with potential distinct approaches.
What should be noted to be more clear is that Gödel DID begin with a complete and consistent logic and so was't implying that that level of reasoning was not correct but that if you want to prove the paradox, you have to begin with the minimum premises that others are defaulted to thinking is true about logic, which included a need for binary values regarding 'truth'. Even before the "Principia", Russell wrote his first major book called, "Principles of Mathematics" (the same title but in English), as his first introduction to the problem. He intended THAT to be expanded upon until he realized certain concerns about his own approach and then restarted with Whitehead for the latter "Principia". He only tried to address the problem in the appendix, called "Types", to which he segregated the types of 'classes' that could or could not speak about themselves. (like a set defined as "the set of all sets" or "the sets of all sets not included in themselves").
Zermelo-Frankel Set theory opted to 'intuit' the axioms necessary to prevent this but it felt like a cheat or 'made up' because it had rules for creating new sets from old ones (the 'construction' concept which logik
here may discuss as he has in separate threads.) Russel tried to be even more basic but defaulted to presuming the very underlying assumptions of 'consistency' (including 'non-contradiction' and 'exclusion of the middle' -- ie. exclusion of degrees of truth value).
Since 'degrees' of value in math exist as useful, if they are to mean anything for things like science, they too have to be proven with the same vigor and be inclusive from the ground-level logic that was intended to be discovered through the motivating questions, "can there be a universal logic;....and can that universal logic be extended to reality itself?" [a summary expression of my motive questions above.]
The concern begins in Calculus and the set theoretician, Cantor, who later discussed degrees of infinities which perplexed him. For instance, an "infinitesimal" is a distinct set of infinite Real numbers between any two integers. They are 'bounded' which makes them complete but hard to compare between other types of infinity. (like is there the same number of infinites between say, 0 to 1 as there is between 0 to 2?)
While is seems that we can begin in a 'quanta' like way (using definite integer wholes that we don't permit breaking down further than the whole), this 'discrete' approach cannot necessarily prove the 'continuities' that exist in all of reality. Nature appears to be contradictory this way. But some believed (hoped) this could be resolved and why the mathematician/scientist Hilbert formally announced a 'program' challenge for all logicians/scientists to try to address. This was to find some closed way of expressing all of reality without 'cheating' by requiring the introduction of some 'God of the gaps' to fill in the missing information.
Gödel's theories to me are better treated through the visualization that Turing opted to do that represented both 'logic' abstractly and realistically (because many then as now cannot conceive of math as being anything more than an invented but unreal tool.) That Gödel did so formally through logic, Turing added the feature of a physical factor that conforms to that logic which connects the logic to the practical representations of it physically.
Gödel specifically conditioned the argument with ,"effectively generated formal theory", precisely to indicate that his own logic is sufficiently complete and consistent. So he was meaning that you can't prove ALL systems of logic from only the 'consistent' assumption as a prerequisite for all other systems. The more inclusive reasoning of totality begins without ANY laws and why some, like myself, hold a type of "Logical Nihilism" extended to reality ignoring moral implications. [The label "Logical Nihilism" is used in moral
philosophy but implies reality without any moral essence is equivalent to a reality without 'gods' that dictate consistent presumptions about 'fitness' as meaning something with value, like 'good' or 'healthy' or 'supreme' etc.]
So Gödel still presented a theorem that is 'valid' but that shows that it leads to some systems of logic that lacked 'consistency'. To give a simpler example regarding machines that 'function', ...we expect any machine to have a unique outcome to a specific set of inputs. If it has more than one conclusion, than this system is deemed a "relation" but not a function
and so such a system would be like having a calculator that specifically FLIPPED between expected outcomes unpredictably.
The area of math that deals with this with respect to this indeterminate unique and predictable outcome, is "statistics". As such, while the statistics is math that can have initial assumptions that we know are valid in that system (consistent), it cannot be derived from a "first-order" logic without dealing with multiple 'universals' (or literal Universes in the case of physics). So it is to these kinds of 'arithmetics' (meaning any math systems) that you cannot start ONLY with a first-order CONSISTENT logic. You need one that extends itself with some assumptions that allow alternate outcomes (indetermine uniquely expected ones). These are the kinds of logics that are 'true' to us but not both consistent and complete
The addition of the second, somewhat corollary about any system unable to prove itself, was to indicate precisely that if we find some universal system of "logic", it can't be proven as 'logical' because we expect it to be 'consistent' to some degree. Because a truly universal system of reasoning about nature cannot begin in a consistent manner [see my last note post above to logik
at the end], you can have a 'rationale' of Totality that originates with NO LOGIC at all!
And that 'no logic' at all is rational at that stage is such precisely for NOT having even 'laws' to exist there in the extreme case of an absolute 'origin'. At that 'origin' where multiple realities can manifest themselves, where contradiction occurs, totality SPLITS those two options into discrete worlds that cannot be sensed from one another.