So you're telling me if I construct a system from axioms which are, by definition, true (like 1 + 1 = 2) and I use the laws of that system to create other assertions (like 2 - 1 = 1), those assertions could be unprovable or contradictory?
And there's no way around this?
So first Wittgenstein ends philosophy with "whereof one cannot speak, thereof one must be silent" then this guy tells that even in mathematics we can't speak with any certainty.
Where does that leave us?
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Postmodernity
In a comfy place free from theorical wankery and now back to poetics as the main guide for the human experience as in the times of Homer before Plato infested western civilization with his autism
hmmm would you guys kiss a boy even if he looked like a sparrow?
1+1=2 has a clear, demonstrated and logical proof that a 10-second google search would have gotten you.
Henceforth, neck yourself.
I don't think you can read.
I think you don't know the definition of axiom.
Take a vow of silence if you can't stand it so much
I wanna punch you in the throat
Certain simple systems are complete. Look up Presburger Arithmetic.
There are other kinds of systems which fall out of the purview of Godel's theories. For instance, if you could somehow make a system which had a non-recursively enumerable set of axioms, it could potentially be complete and consistent. However, such a system is not realistic for humans, since it would contain an infinite number of axioms.
The assertions themselves like 1+1=2 are provable within the context of your theory. It's the theory as a whole which cannot be proven to be consistent.
You already couldn't speak with certainty. Just because your axioms are consistent doesn't mean that they're correct.
Wittgenstein was a faggot for saying that. Notice Godel didn't tell mathematicians to stop doing math, cause that would be stupid.
Godel's theorems don't actually provide "new" information. They mainly just btfo formalists like Bertrand Russell.
Fucking embarrassing when some autodidact tries to think.
1+1=2 is equally as valid as 2-1=1
This guy’s a retard
ohhh that sounds hurt oh no you wouldnt want to hurt me would you daddy?
there isn't anything wrong with being unfamiliar with a subject and asking a question. But there is something wrong when you spout shit trying to get answers because you are lazy instead of doing research yourself.
reported for >lewd
Wasn't Russell the first one to put forward mathematical paradoxes (does the set of all sets contain itself or whatever)?
How is that a paradox
Read Godel's Proof. Don't be a pseud. It's a good book and is accessible to nearly anyone.
Sorry I don't understand how that is a paradox I'm low IQ
no there were paradoxes before that. He pretty much ended Frege's logicist project though with his Russell paradox which is the question of whether or not the set of all sets that are not members of themselves is a member of itself. If it is a member of itself than it isnt the set of all sets that arent members of itself and if it isnt a member of itself then then the set of all sets that arent members of themselves is missing an element.
Does the set of all sets that are not members of themselves contain itself?
That's Russell's paradox. It doesn't actually matter what the paradox is, but I'm interested in why you think Godel in any way attacked Russell with his work.
what does it mean for a set to be a member of itself?
Can't really explain it better than that math.stackexchange.com
>[if] I use the laws of that system to create other assertions (like 2 - 1 = 1), those assertions could be unprovable or contradictory
No this is the exact opposite of what the 1st and 2nd incompleteness theorem say. Any assertions that you derive from the system will be true (soundness) and provable by definition. The problem is that there exist statements which must be true on the model (i.e. on the semantic interpretation of the theory defined on the particular "L-structure", or language, that you're working with).
Additionally, Godel's theorems don't in any way imply that arithmetic is inconsistent. What they do imply is that (1) In any system strong enough to capture Peano arithmetic, there will either exist undecidable sentences, i.e. sentence which can neither be proven true nor false on the basis of the axioms of the system (e.g. the so-called Godel sentences that he constructs in his famous theorems), OR the system will be inconsistent (since an inconsistent system can trivially prove every sentence, and (2) no consistent system can prove it is consistent (because if it could, then you could construct a decision procedure that would be capable of deciding any and all statements in the L-structure, including the statements that are undecidable).
Finally, undecideability is connected with incompleteness because (1) you have sentence which the theory can neither prove true nor false, while (2) at the same time any interpretation or model has to assign a truth value of either "true" or "false" to every sentence of the l-structure. Thus, since every sentence is either true or false on the model, but certain sentences can't be proved (nor can their negation), thus their existence sentences which are true but unprovable, i.e. the system is incomplete.
The fact that Veeky Forums doesn't even know enough logic and proof theory + model theory to explain the basics of incompleteness is really unsettling (but not really surprising).
Yeah but Russell just thought that it demonstrated the inadequacy of our current method of formalization. That's why he developed type theory.
Also Russell was a logicist, not a formalist - but Godel btfo out of both logicists (first incompleteness theorem) and formalists (seecond incompleteness theorem), so I guess it doesn't really matter.
1+1=2 is not an axiom
Why do logicians bother talking about the semantics of sentences and words, as if they are hypostatized entities somehow existing outside or independent of human interpretation and deployment?
The whole logical enterprise makes no fucking sense to begin with. Neither words nor sentences are stable in meaning. No meaning is ever stable, because meaning is always immanent in use, and in the context of every application.
How the logical positivist project ever got off the ground is just mind-boggling. It's like they went out of their way to ignore a hundred years of philosophy, anthropology, linguistics, everything.
Well said
>It's like they went out of their way to ignore a hundred years of philosophy
Classic Anglos
>reddit spacing
>autodidact tier 'cant kno nuffin muffin!' intellectualism as a response to a well rounded post
you need to go back
>I dont know anything and I dont know how to think or know
>therefore you cant possibly either
here's a perfectly sound logical inference:
if the blue flares of midnight light upon the dusky mote in gods eye then bix nood fucked ur bitch ass mom lol
the blue flares of midnight lit upon the dusky mote in god's eye
therefore bix nood fucked ur bitch ass mom lol by modus ponens
what has been 'said' here? is this true?
'valid'**
>the blue flares of midnight lit upon the dusky mote in god's eye
sauce
my own pure purple brain friend
there is true as in tautological language meaning, and math, self defined consistencies... or is that sound...
and there is true, as in, a gradiation of preceise accuracy in relation to the comprehensive agreement on the exactness of relations to mind/paper/word/total map of interrelating definition-strings-pictures.... to the world as it exists in and of itself.
A blue horse with 3 legs just trotted by my home with a blade of green grass in its mouth wearing a 3 inch tall fedora.
That statment may not be 100% true, or 100% false: but something of the sort could have actually occurred: and my description, recollection, accuracy of spotting and judgement of details: could be more and less 'off' the mark, of perfect truthful accuracy.
the issue you bring up may be the issue of knowing: it is possible "the blue flares of midnight lit upon the dusky mote in god's eye " occurred, and you can believe it is true: but you have no proof or evidence: and it may actually be true: but can you 'actually know it is true', even if you say "I know its true", even if you yourself have never seen or experienced evidence..
for example: I know that another intelligent life form has lived and died on a planet very far away from ours. ...I cant possibly know this right... even if it is true...because I have never experienced the truth of that, that can allow me to say 100% it is true..
thats the fancy dancy issue
Except the tradition was centered in Vienna.
It leaves us with a conscious realisation that we have to cement our own social structures and philosophical systems to lord over us. Just because there is no platonic metaphysical certainty, doesn't change the fact that those structures and beliefs function in a society and can last for centuries. This is the real lesson of "postmodernity".
>doesn't know what Gödel numbering is
ugh geez
Bertrand Russell was a platonist
Well, logical empiricism was mostly inspired by a massive post-Hegel butthurt and inability to cope with continental philosophy.
Note that analytic-oriented universities, and even textbooks, still characterise the whole 20th century continental tradition as incoherent and not worth reading. I had several professors that were just like this, completely dismissing authors they wouldn't even read, and it was incredible to me that people like this call themselves philosophers. The anglo conception of philosophy is intellectually barren and dishonest as a whole.
I would kiss a girl
Yeah fucking Aristotle amirite
hey me to!
wrong
>Why do logicians bother talking about the semantics of sentences and words, as if they are hypostatized entities somehow existing outside or independent of human interpretation and deployment?
>The whole logical enterprise makes no fucking sense to begin with. Neither words nor sentences are stable in meaning. No meaning is ever stable, because meaning is always immanent in use, and in the context of every application.
None of this negates logic. Logic can account for it.
>logic is a transcendental signified
what is this, 1890?
do you not understand what indeterminacy of meaning and translation means for a predicate calculus? it relocates "validity" entirely to the realm of intersubjectivity
points, lines, shapes, substance...what else could there be?
>to
user you don't want to be the cute one, the girl is supposed to be the cute one.
That's a play on words. Not a paradox.
Like:
"This sentence is a lie."
Is not a paradox.
The sentence is true, otherwise it can't be a lie.
If it's a lie, it stops being true, and so can't be a lie.
It's always true.
Truth can be known but not incapsulated in a system. Truth is known through synthesis not analysis. Read some Guenon. Leibniz also said it well (paraphrasing from memory): "A system is true in what it affirms and false in what it denies".
Godels incompleteness theorem isnt that big of a deal. Its like a lot of the stuff in quantum physics for example. Pseudo intellectuals who don't understand it just like to play word games
love
but if it's true then it is a lie
There is. Even Eubulides figured this shit out:
A conclusion can still be patently incorrect even if watertight inferences were taken to arrive at it.
is a chemical reaction in your brain.
You are not there.
wow man like woah
if I'm not there then why is it my brain?