# Thread: Godel and Boolean logic

1. Originally Posted by Vkothii
Therefore we cannot use arithmetic to prove the universe is'nt lying, it might be.
Math doesn't prove anything directly about the universe. The universe proves or disproves the mathematical models we use to discuss its nature.

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.

2. Originally Posted by sherlok
...
anyway, if you understand what a proposition is, that's a good start to understanding propositional calculus.

For instance, removing the pejorative from your quoted statement, it says: "feel free to explain your statements": so then IF it is the case that I "feel free... " etc THEN "explain your statements" implies "I feel free to explain.. " etc.
...
You seem to have little training in logic.
What you're describing here is a conditional material implication.
Which is not the same thing as a proposition.

Hopefully, whilst you're banned, you'll take some time to study.

3. Originally Posted by swarm
Math doesn't prove anything directly about the universe. The universe proves or disproves the mathematical models we use to discuss its nature.

Well said swarm.
This is exactly what I was saying in Posts #4 and #20.
I think that Vkothii/sherlok is confused about the purpose of mathematics/logic (or any closed system for that matter..).

Originally Posted by swarm
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.
This may be an interesting idea to investigate.

4. 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.

5. Originally Posted by noo b
5) Can you describe, perhaps, a "closed system"
(are you aware there is no such thing?)
Closed System

Definition

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).

http://www.imprint.co.uk/thesaurus/closed%20system.htm

6. Originally Posted by noo b
1) Why do you think the purpose of mathematics, or "any closed system" is confusing

I don't; it's you who are confused Vkothii

Originally Posted by noo b
6) can you describe the difference between a "proposition" and "a conditional material statement which is an implication"
I can only take from this that you've never encountered a dictionary of philosophy.

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.

7. Originally Posted by Vkothii
But if "the universe is lying", and we have good reason to suspect that either we have theories that "lie" to us, or a universe that "lies" when we interrogate it with the theories, then it might be useful to investigate this apparent lie, or falsity of evidence.
Hence the scientific method or... sanity. It's easiest to presume the universe can't lie and if we encounter an anomaly it's the theory that's short of explanation power rather than the universe 'lying'.

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.
True dat, but alas, you have what you have to work with. Really I think godel touched directly upon the circumstance of perspective, and in a weird way, contradicts his own argument by having established a universal truth! just my opinion of course.

8. Originally Posted by swarm
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.
Translation: There's no way to know if you missed something.

9. Originally Posted by wesmorris
Translation: There's no way to know if you missed something.
Indeed.
What's more, Godel demonstrated that you can't use your methodology for determining whether or not you've missed something, to determine whether or not that methodology has missed something.

10. Originally Posted by glaucon
Indeed.
What's more, Godel demonstrated that you can't use your methodology for determining whether or not you've missed something, to determine whether or not that methodology has missed something.
Actually its more subtle than that.

If you have a formal system for knowing then either you have missed something and/or you are crazy.

11. Lol, awesome.

12. endless loopy stuff

13. Originally Posted by swarm
Actually its more subtle than that.

If you have a formal system for knowing then either you have missed something and/or you are crazy.
lol

True enough.
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.

Cool.

14. Originally Posted by Search & Destroy
endless loopy stuff
Just make sure you don't mention TRI****ES, they are forbidden.

15. Originally Posted by Vkothii
Therefore the statement: "I am a liar" is a proposition which cannot be proven in any propositional formal logic, it can only be negated (inverted).
No, the statement "I am a liar" is not problematic. What you want is "this statement is false." That one is where the issue lies.

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.

Originally Posted by Vkothii
Therefore it is a tautology that no formal logic system is complete.
It's not a tautology, because it only applies to systems of formal logic sufficiently powerful to encode the statement "this statement is false." I.e., you need to be able to make statements about the truth value of other statements in the system. Systems that are too weak to do that don't suffer from this flaw - they can be shown to be consistent and complete.

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.

16. Originally Posted by Captain Kremmen
Just make sure you don't mention TRI****ES, they are forbidden.
LOL ...it is good to know.

17. Originally Posted by Captain Kremmen
Just make sure you don't mention TRI****ES, they are forbidden.

18. Did I do good, or bad?
What's bumping?

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.

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