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.
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.
