05
Dec
2018
BMS Ep. 7: Godel Made Easy
My latest episode is the podcast equivalent of the narrator/Tyler Durden telling the guys to get off the porch because they’ll never be in Fight Club. This is surely way too hard for you. You can’t understand mind-blowing math at this level of amazingness. Don’t even try.
This was a great episode! Far different than my expectations of a libertarian communist podcast. I have a degree is mathematics and had never heard of this theorem.
Steve Patterson would be sad to see you implicitly assume infinite natural numbers and therefore state that prime numbers are infinite. But I’ll stick with you and the correctness of calculus…
Communist?
Bob, the Christian and communist.
Ah, I was thinking about that line.
wasn’t*
Thanks comrade Ethan! BTW that amazes me that you never heard of Godel’s theorem, and you majored in math? They cite it all over the place in philosophy of mind / computer science discussions. I even used it in my doctoral dissertation; it comes up with the knowledge problem etc.
I did well in my first semester of Honors Analysis, but I absolutely hated it. I didn’t do well in the second semester of the course.
I enjoyed abstract algebra and number theory more, but those courses didn’t attempt to build out the natural numbers in the way analysis did.
I ended up taking a more applied math route and now work as an actuary.
I may have heard it, but at the time in college my interest shifted more to politics and economics due to Ron Paul 2012.
I was somewhat unimpressed by Steve Patterson arguing against the Liar’s Paradox by claiming that the language is insufficient to describe this paradox.
It’s kind of implausible that Patterson seems to understand the paradox, and the listener also understands the paradox, but at the same time Patterson attempts to convince the listener that there’s no way of describing it. I mean if it cannot be described, then how come people can be told about it and understand?
If you accept that named statements can exist then just write down a pair:
Statement A says statement B is false.
Statement B says statement A is true.
Now you cannot apply a simple dumb evaluator, because it goes round and round between the two statements. You need an evaluator that can attempt to detect the repetition and then apply some alternate logic (like NULL values or “unknowable” values if you accept those).
It’s easy to demonstrate practical examples of Godel’s Theorem but very difficult to comprehensively prove it, because a proof comes up against effectively unlimited cognitive power, including the ability to self-extend your worldview as necessary. That’s the whole barrier of self awareness that separates dumb handle cranking machinery from real intelligence.
I agree, there does need to be a third or NULL statement. Which is essentially how NULLs function in programing. At least in SQL data structures all three of the following statements are false:
– True = NULL
– False = NULL
– NULL = NULL
Null means neither true or false, or undefined. Which is a slight answer to the liars paradox. I think Patterson can’t accept that an answer isn’t possible.
I have greatly enjoyed some of Patterson’s output. His article Faith and Reason is great, and even some of his things on infinity are interesting. He loses me on the base unit math stuff.
I will fully grant that he may know more about the fundamentals and metaphysical structures of math, but calculus is not that difficult to understand. I don’t even think “space” needs to be infinitely divisible to mathematically model calculus as if the universe was infinitely divisible.
And his opinions on Bitcoin…
Those comparisons to NULL evaluate to NULL themselves (not false). That’s important because there’s no way to recover back out of it again.
False = NULL => NULL
NOT( False = NULL ) => NULL
NOT( NOT( False = NULL )) => NULL
Once the information is destroyed, the destruction continues to spread into anything that touches your original NULL. But you want that, because it’s modeling the propagation of ignorance.
There’s two special operations that mask the spread of ignorance:
True OR NULL => True
False AND NULL => False
The same thing happens in logical simulations with the “X” value, but if you go buy digital logic circuits and really wire up a Liar’s Paradox in gates, then it either oscillates or goes into genuinely unknown logic values which can burn out the circuits in some cases.
Hey Bob,
Thanks. I’ve walked through multiple overviews of Godel and this was simply the best one. Better than the guys at Numberphile or The Mathologer on YouTube.
It would have been great if you could have explained Godel’s actual system that allows him to assign numbers to axioms in such a way that arithmetically operating on those numbers produces derived propositions. This is the piece I still don’t understand.
As a side, I take it you know that Godel produced his own version of Anselm’s Ontological argument.
Jim
Hi Jim,
Thanks so much, that means a lot. That’s why I thought I could contribute, because yeah I literally studied this topic for years before finally getting it.
I would have to go find a good book on it (the one that initially gave me the aha moment when I was in grad school) to remember how his encoding system worked. It had to do with prime numbers that you multiplied together, so you were sure each proposition would point to a unique integer.
The other thing that has always bugged me about the “Godel Sentence” is the presumption that it’s true.
If you apply [1] the statement cannot be false because we don’t allow inconsistencies, and [2] all statements that are not false must be true… then you have indeed derived the sentence from the axioms. Those are axioms after all, the decision that true and false are the only options, and the belief in global consistency.
So it’s a bit of a dodge I think around the Liar’s Paradox by taking things 3/4 to their logical conclusion and then saying “Stop here!”
The point is that the Liar’s Paradox (and other related systems of statements, of which you can easily find more) cannot resolve to EITHER true or false. That’s the core of the incompleteness, that true and false do not adequately describe the world (not even an abstract mathematical world). There’s other examples with sets and stuff like suppose you define an operation that says “I take a set of numbers and pull out the maximum value” so that’s fair enough but what happens when you feed an empty set to that operation? What is the maximum value of an empty set? Well there isn’t one, so you need a placeholder to represent the lack of a thing (not an unknown value, not zero either, but a value that cannot exist, the absence of a value).
Tel wrote:
The other thing that has always bugged me about the “Godel Sentence” is the presumption that it’s true.
If you apply [1] the statement cannot be false because we don’t allow inconsistencies, and [2] all statements that are not false must be true… then you have indeed derived the sentence from the axioms.
I understand where you’re coming from, but I don’t think that’s right, Tel. Strictly speaking Godel’s theorem says “In any system free from contradiction…” He doesn’t actually assume the Godel sentence is true.
Oh! Tel, I missed the easiest way to respond to you: You’re totally right, if we could actually prove *from the axioms* that we didn’t want there to be a contradiction in the system, then we could also prove that the Godel sentence had to be true. But that’s exactly what Godel’s 2nd theorem does. So it’s not as if everybody has missed your insight; it’s codified in his 2nd theorem.
The way we convince ourselves that the Godel sentence can’t be false in the first theorem is coming from outside the system; we assume that there should be a consistent axiomatic framework for arithmetic. But we can’t actually prove such consistency from within the system itself. It’s not coming from the axioms, it’s coming from our hope in a rational universe.
Perhaps I’m just a little suspicious about the distinction of what’s “in the system” and what’s “outside the system”.
If a belief in self-consistency is your starting point, and you cannot intrinsically prove self-consistency based on anything else… then this is sounding a lot like an axiom of the system. Maybe that depends on how you want to look at these things, but those are the type of characteristics that define an axiom.
There’s also some hidden assumptions that you believe every answer must be either TRUE or FALSE thus you have presumed every statement has an answer, or in other words everything that is syntactically correct must also be semantically meaningful. This hidden assumption is a little bit questionable, but maybe that’s the whole point of Godel’s proof.
Not sure if you guys listened to this one. Goes through the common paradoxical statements.
https://www.localmaxradio.com/episode/39
Bob,
Is this just a broader version of Russell’s set theory paradox?
Wow Murphy, this is the first time I actually understand Godel!
Murphy, perhaps you can explain the answer to a question that has been bugging me ever since I first read the incomplete theorems.
What happens when we apply Godel’s conclusions about axiomatic/mathematical systems, TO GODEL’S CONCLUSIONS?
Self-referential logic.
Godel paraphrased:
All complete axiomatic/mathematical systems are inconsistent, and all consistent axiomatic/mathematical systems are incomplete.
OK, what then of Godel’s own conclusions?
Ostensibly they too are “proven” using only the finite number of axioms that Godel utilized. OK, then by Godel’s own conclusions while indeed his conclusions are “provable” within the axioms, we can’t make the jump to conclude the conclusions are true with a capital T.
Major_Freedom statement of Godel statement: In a sufficiently complex axiomatic system, this statement “Godel’s conclusions cannot be proven using all of the axioms he utilized” is provable.
If that statement is false, then Godel’s conclusions are wrong. If true, then Godel’s conclusions are wrong.
I always thought Godel’s theorems when subjected to themselves results in a contradiction.
What I think about when thinking this:
Godel always relied on his own understanding of his own theorems for them to be shown as proven, but he never (note I am not saying he should have, I actually think this is impossible) formalized his own understanding of the statements he made that we read. The praxeological “Verstehen” component.
His understanding of how the axioms lead to those conclusions he made about integers is not present in whatever set of axioms he used to construct his proofs.
Sure, we can always IMAGINE counting from 1 to 2 quadrillion to 50 septillion to an arbitrarily gigantic number of statements and if we’re all geniuses we too can prove that there are true statements not provable by these axioms.
But wait a minute, we never formalized THAT understanding into the axioms.
In other words, we made a conclusion purporting to be provable by such and such axioms written down, but we smuggled in unstated statements to arrive at such a conclusion.
What if Godel never proved what he thought he proved, but what he really did was show that knowledge is truth not only requires, but presupposes, an active…I hesitate to use this word, but “force” or “energy” from our own minds that is never completely formalized on paper but must remain and could only ever be, an understanding, a verstehen?
Not an arbitrary subjective intuition or belief that is advanced ex cathedra, but a necessary action-logical gathering and organizing component that must be identical for all other minds in order for the on paper conclusions to be in principle universally understandable by all?
In other words, plop Godel’s theorems on a table and tell me “All of us genius mathematicians understand that these theorems are true”, and I’ll say “where in these documents does it show that?” If the answer is they don’t, then by what other means did the genius mathematicians arrive at their understanding, and what makes them think they can claim these theorems are true when you used unstated statements that they couldn’t formalize even if you wanted to?
It seems Godel’s theorems refute Godel’s theorems.
Or at least it seems that way to me.
“Godel’s conclusions cannot be proven using all of the axioms he utilized” is provable.”
There was a fairly recent post at Landsburg’s. The conclusion was
“More generally, the moral is this: As a matter of pure logic, If there is anything that you can prove you can’t prove, then you can prove anything.”
In your statement was “In a sufficiently complex axiomatic system, this statement “Godel’s conclusions cannot be proven using all of the axioms he utilized” is provable”
If true, then you have proved your system is inconsistent and you can prove anything. Black is white, God exists, God doesn’t exist. Godel’s conclusions cannot be proven and Godel’s conclusions can be proven.
Part of LS’s reply to Bob:
“Now let’s make things more interesting by changing your statement to “I can prove that I can’t prove that 0=1″. If this statement is true then you can prove that there’s something you can’t prove, which means that you can prove your own consistency, which means you are inconsistent, which means you can prove that 0=1. So in this case the statement “I can prove that I can’t prove that 0=1″ is true, the statement “I can’t prove that 0=1″ is false, and the statement “0=1″ is of course false. [This analysis continues to assume that your axioms include enough about arithmetic to make Godel’s theorem apply.]”
In that example, the statement 0=1 was false, even though you could prove it.
Harold who is “LS”? Am I missing something?
MF I will come back to this (and remind me if it seems I forgot!) but I’m bogged down with day job stuff right now.
Sorry, SL, not LS – Steve Landsburg
Post here
http://www.thebigquestions.com/2018/07/26/thursday-solution-4/
There was discussion of the difference between “it can be proven” and “I (as a logic machine) can prove” which may be significant.
My comment above was supposed to be a reply to MF long comment above so the quotes are from him/her.
ookee