02-28-09, 09:20 AM #21
I think what you are trying to get at is that no formal system is sufficient to completely and coherently discuss everything there is to discuss about the universe.
02-28-09, 07:26 PM #22
02-28-09, 07:32 PM #23
02-28-09, 10:24 PM #24
I think its a foregone conclusion that all formal systems are subsets of the universe and Godel showed any formal system capable of self description is at least either incomplete or incoherent therefor applying a system which can't fully handle the subset is not going to result in success with the superset.
02-28-09, 10:27 PM #25
An isolated system having no interaction with an environment.
A traditional cybernetics term connoting a system which is hermetically isolated with respect to environmental influence (e.g., impervious to external "feedback"), and hence not responsive or adaptive to its medium over time. The opposite of an "open system" (in the same traditional terminology).
02-28-09, 10:36 PM #26
I don't; it's you who are confused Vkothii
A proposition is any implicative statement. So, for example, 'p' can be a proposition.
Unlike a proposition, a conditional material implication is a form of argument, generally of the form: If 'p' then 'q'.
Note how the two are completely dissimilar in form.
Note also how the latter necessarily contains at least 2 (though usually 3) of the former.
Seriously man... take some time off... go study.
03-04-09, 04:00 PM #27
As Godel demonstrated, we can only know the truth of our ability to construct workable solutions, we can't know if these are universal solutions and we can't assume they will be able to answer any question, or that we can ever build such a machine.
03-04-09, 04:07 PM #28
03-04-09, 04:21 PM #29
03-05-09, 05:12 AM #30
03-05-09, 06:04 AM #31
03-05-09, 01:00 PM #32
03-05-09, 03:15 PM #33
That's what I was haphazardly trying to say, but the formulation just wouldn't come out right.
I think Godel's work can only be really understood correctly when taking into account how it was a reaction to the challenge posted by Hilbert (one of them at least..). In particular, Hilbert was seeking a proof that a formal system can be complete and decidable; Godel sought to accomplish that, but succeeded in the opposite.
05-08-12, 06:04 PM #34
05-08-12, 07:03 PM #35
Well, I guess that's the same thing if you define "liar" to mean "someone who always and only makes false statements." But that's not the normal definition of "liar," and the point is more clear if you just refer to the statement directly.
Also, even for strong systems, your statement isn't true as written. I can most certainly construct a formal logic system that is complete by simply adding the required statements ("this statement is false," etc.) to it. The problem I'll encounter if I do that - and this is the meat of Godel's theorem - is that the resulting system will be inconsistent. I'll be able to both prove and disprove any statement I can come up with.
Godel's theorem goes like this: any system of formal logic powerful enough to model the natural numbers cannot be both consistent and complete.
Note that there exist interesting systems below the power threshold in question. Euclidean geometry is known to be both consistent and complete. To the thread title: Boolean algebra is not subject to Godel, and has been proven to be complete.
And you can always construct a system of formal logic that escapes Godel and still proves any first-order statement you want it to, simply by making the axioms an exhaustive list of all first-order properties you want to be true. Granted, that's kind of trivial (and bloated), but there it is.
Last edited by quadraphonics; 05-08-12 at 07:27 PM.
05-08-12, 07:30 PM #36
05-09-12, 12:54 AM #37
05-09-12, 06:59 AM #38
Did I do good, or bad?
As far as I can see, the ban on discussion of TRI*****S has not been rescinded.
It is written.
What's 3 years in the history of the universe, after all?
If bumping means going off topic, then I apologise.
In future, please no-one even mention not being allowed to mention ***ANGLES.
It is clearly off limits.
Last edited by Captain Kremmen; 05-09-12 at 09:00 AM.