The Liar's Paradox

[Silent Roar]

I'm sure that many of you have heard of the Liar's Paradox; a phrase or paragraph that denotes itself. The simplest version is as follows:

This sentence is a lie.

You see, if the sentence is in fact a lie, it would prove to be true, which would make it a lie, so on and so forth.
What the point of this thread is, is the debate of the existence of a Liar's Paradox in mathematics. If you have ever heard of Kurt Gödel, you know what I am talking about. If not, I suggest you look him up.
Anyways, Gödel was famous for his invention of the two Incompleteness Theorems.

Incompleteness Theorem Number One:
For any consistent formal, computably enumerable theory that proves basic arithmetical truths, an arithmetical statement that is true but not provable in the theory can be constructed. That is, any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete.

Incompleteness Theorem Number Two:
For any formal recursively enumerable (i.e. effectively generated) theory T including basic arithmetical truths and also certain truths about formal provability, T includes a statement of its own consistency if and only if T is inconsistent.

(For an in-depth analysis of the two theorems, click here.)

So, in conclusion, I would like to know some opinions on the Incompleteness Theorems and how they relate to not only mathematics and philosophy, but to the world.

 

[bluecherry]

*tilts head* Interesting. I have not yeard of this before. I'd like to give thoughts and/or observations on this, but I think I'd need to see some examples to get a better idea of it first perhaps and also I clicked the link you provided and it told me that such an article did not exist. sweatdrop I don't know if maybe something happened to the page or if maybe the link's address had a typo or something, but it's not there. A shame, I actually had plenty of time right now to read it even if it was a really long article.

 

[Kecitich]

hi, bluecherry and everyone.

I think he meant this link? http://en.wikipedia.org/wiki/Gödel's_incompleteness_theorem  

[engineer-of-doom]

I clicked the link in the first post and it seems to be working properly now.

3nodding  

[bluecherry]

Oh good! It is working now. 4laugh Wikipedia was telling me it didn't exist or something the other day. emo I'll read that when I get around to it, maybe sometime today or tomorrow, and then edit this post to put my thoughts on it. 3nodding

 

[engineer-of-doom]

Okay, I'll take a crack at this one.

If one accepts that the universe itself is included in these theorems, then the first one implies that there are unprovable truths. There is no indication of what type of truths these might be. The second theorem, regarded in a similar fashion, pretty much kills any religion, philosophy, or science which claims to explain everything. So, is the universe knowable? These theorems would say No.

As I prefaced this, I made the statement that one had to accept that the universe was included to make these claims... as I think about it... if mathematics is part of the universe, and mathematics itself cannot be both consistent and complete, then how could the universe of which it is a part be both consistent and complete?

More discussion please.  

[bluecherry]

I was going to reply to this topic loooong ago. eek Sorry I kind of neglected it after saying I would reply. I still am going to, but right now I'm a little busy. Reading the above post especially makes me curious about this subject.