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Mathematics self-proves its own Consistency (contra Gödel et. al.)

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Re: Mathematics self-proves its own Consistency (contra Gödel et. al.)

#31

Post by EGreg »

jfarmer wrote: post #21
EGreg wrote: post #12

"(There is a very weak theory called Provability Logic that has been used for self-referential propositions coded as integers, but it is not strong enough for the purposes of computer science.) " weak theory? not strong enough for computer science, eh? strange ways to talk about mathematical theorems that are either true or not. let's see him define what he means by any of these terms lol

"Strong" and "weak" have well-understood meanings in mathematical logic and model theory. For example, imagine there's a true and "important" statement S about Turing machines for which we could prove "it is impossible to prove S using provability logic." In that sense PL would be "too weak" to do computer science.

Okay, but what is "very weak"?

Also, I remember you - you did adonomics back in the day for facebook apps :)

Re: Mathematics self-proves its own Consistency (contra Gödel et. al.)

#32

Post by EGreg »

This is all kind of meta, but I think that, to prove anything, he would need to fix a system in which he could operate, and prove it in that.

Kind of reminds me of the following proof I have that something in this world is indeed absolute.

    1) "Nothing is absolute" is self contradictory.

    2) Therefore, something is absolute.
Neat, huh?

By the way, one needs to elaborate a little bit about the above proof. The reason (2) follows from (1) is because the negation of 1, "something is absolute", is not self-contradictory. However, if both (1) and (2) were shown to be self-contradictory, then you wouldn't have the excluded middle.

Re: Mathematics self-proves its own Consistency (contra Gödel et. al.)

#33

Post by jfarmer »

EGreg wrote: post #31
jfarmer wrote: post #21

Earlier quoted context omitted.

"Strong" and "weak" have well-understood meanings in mathematical logic and model theory. For example, imagine there's a true and "important" statement S about Turing machines for which we could prove "it is impossible to prove S using provability logic." In that sense PL would be "too weak" to do computer science.

Okay, but what is "very weak"? Also, I remember you - you did adonomics back in the day for facebook apps :)

He's just speaking colloquially and I don't know anything about the system of logic he's describing. He means something like "weak relative to all the other ones we'd consider on a day-to-day basis" or "so weak we'd never seriously consider it except as a specialized tool."

That's not really the point, though. The point is that "strong" and "weak" are well-defined terms in mathematical logic and he was almost certainly using them in that sense. :)

Re: Mathematics self-proves its own Consistency (contra Gödel et. al.)

#34

Post by EGreg »

jfarmer wrote: post #33
EGreg wrote: post #31

Earlier quoted context omitted.

Okay, but what is "very weak"? Also, I remember you - you did adonomics back in the day for facebook apps :)

He's just speaking colloquially and I don't know anything about the system of logic he's describing. He means something like "weak relative to all the other ones we'd consider on a day-to-day basis" or "so weak we'd never seriously consider it except as a specialized tool." That's not really the point, though. The point is that "strong" and "weak" are well-defined terms in mathematical logic and he was almost certain…

I think he was saying that their math-fu was weak, too weak for comp sci :)

Re: Mathematics self-proves its own Consistency (contra Gödel et. al.)

#36

Post by snowwrestler »

I don't think that Godel's theorem says that math is necessarily inconsistent.

My understanding is that it says math is either inconsistent or incomplete. So if you construct a mathematical system that is consistent (allowing proofs by contradiction), there will be some set of statements that are true, but cannot be proved true by that mathematics.

Re: Mathematics self-proves its own Consistency (contra Gödel et. al.)

#37

Post by gizmo686 »

runarberg wrote: post #26

I've always been fascinated with naïve set-theory, and how it was refuted after it's apparent inconsistency, the famous Russell's paradox of the set of all sets that don't contain it self, and only contains it self if it doesn't etc. So supposing it's False, yields True, and supposing it's True yields False. But what fascinated me was not the Russell's set, but it's inverse, the set of all sets that contain it self,…

S doesn't seem that interesting. Assume that S does not contain itself. Consider the set P, which can be constructed by adding P as an element to S. Note that P contains itself. Therefore, P is in S. Also note that every other element in P is also in S, and that every element in S is in P. Therefore P and S are the same set. Therefore S contains itself, contradicting our assumption that S does not contain itself.

If we assume that S does contain itself, then I do not see a way at arriving at a contradiction.

Re: Mathematics self-proves its own Consistency (contra Gödel et. al.)

#39

Post by becauseICan »

Gödel walks into the Church and proclaims, "You cannot prove that you are good and therefore you cannot prove that there is a good God but you need not fear for I have developed a system whereby you can still be ethical despite knowing not of your goodness."

Whether or not Gödel is correct soon becomes irrelevant as he is promptly ejected from the Church. As he walks away he hears the sound of the hymn "I Cannot Be Sung in the Church" being pelted out at at full volume by the congregation.

Re: Mathematics self-proves its own Consistency (contra Gödel et. al.)

#40

Post by [deleted] »

EGreg wrote: post #34
jfarmer wrote: post #33

Earlier quoted context omitted.

He's just speaking colloquially and I don't know anything about the system of logic he's describing. He means something like "weak relative to all the other ones we'd consider on a day-to-day basis" or "so weak we'd never seriously consider it except as a specialized tool." That's not really the point, though. The point is that "strong" and "weak" are well-defined terms in mathematical logic and he was almost certain…

I think he was saying that their math-fu was weak, too weak for comp sci :)

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