Incompleteless in Logic

[Xei]

I recently encountered Godel's incompleteness theorem (there's two technically). What it basically states is that there are some statements in mathematics, many of them very simple, which are impossible to actually prove using mathematics. As physics is based upon mathematics, is the objective world equally flawed in its logic? Do some logic systems become too complex to be predicted logically? Is determinism therefore flawed? Is it impossible to simulate neural activity and hence consciousness? Would this allow for free will? If you are an advanced mathematician then your contributions would be greatly appreciated, of course. I personally know nothing of set theory so I can't say anything at all about the technical details. Discuss!

[Universal Mind]

I think the laws of logic are completely solid, but only the real laws of logic. I am not talking about mere human concpetions of them. I mean the ones that were part of reality before humans ever existed. There is a law of logic, according to humans, called the "law of the excluded middle". It says, "Either p is the case or p is not the case." In other words, every supposition is either true or false. I think they are wrong about that, and my conclusion is based on one paradox. Consider this statement... This statement is false. Now think about the law of the excluded middle. Based on it, either the sentence is true or the sentence is not true. But in reality, it cannot possibly be either.

[Keeper]

sometimes, however, that law does work. Either is is raining or it is not / Either you can count to ten or you can't I disagree about it being a true/false dichotomy. I think of it more as a logical/illogical one. To say that the law of the excluded middle [LEM] is just a case of true or false doesn't seem logical to me. This statement is false This can be regarded as LEM with p = nonsense

[skysaw]

I recently encountered Godel's incompleteness theorem (there's two technically). What it basically states is that there are some statements in mathematics, many of them very simple, which are impossible to actually prove using mathematics. I'm not a mathematician, but I am familiar with Godel's theorems. Your description is not quite accurate in what they imply. The gist of the theorems is that any mathematical or logical system that is complex enough to be complete must contain inconsistencies. In set theory an example is the set of all sets that do not contain themselves. It is impossible to show whether or not this set contains itself. It simultaneously must and must not contain itself.

[Universal Mind]

sometimes, however, that law does work. Either is is raining or it is not / Either you can count to ten or you can't I disagree about it being a true/false dichotomy. I think of it more as a logical/illogical one. To say that the law of the excluded middle [LEM] is just a case of true or false doesn't seem logical to me. This can be regarded as LEM with p = nonsense The law of the excluded middle almost always works, but it is apparently not absolute. There is at least one exception. Would you say the statement I posted (This statement is false.) is true or not true?

[Keeper]

Neither but then again, does it have to be a choice of true and not true, instead of two others?

[Taosaur]

I think the laws of logic are completely solid, but only the real laws of logic. I am not talking about mere human concpetions of them. I mean the ones that were part of reality before humans ever existed. Logic is a human tool for examining phenomena and our own thoughts and activity, and communicating our results. Like our other technologies, we can benefit from bringing it more closely in conformity with nature, but it's a mistake to take logic as "part of reality."