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What Gödel Discovered
171–180 of 271 posts
Re: What Gödel Discovered
#172Earlier quoted context omitted.
The constitution is BS anyway. For example, Article I, section 10 says "No state shall...coin money, emit bills of credit, make any thing but gold and silver a tender in payment of debts..." And look what the US has today; fiat money. No only is it not gold or silver, but it's not event backed by it. Based on this definition, nobody in the US earns any money so nobody should have to pay any income tax since basically…
Key word is “state”. The point is that the states don’t coin money - the federal government does.
This is yet one more attempt to distort the original meaning of the law by drawing attention to specific words that are immaterial to the original intent of the law.
The states are in fact printing money which is not gold or silver no matter how you look at it... It doesn't matter that they are all printing the same currency. This activity is happening within the physical confines of individual states and they are using that money to pay their debts.
The simplest interpretation is the correct one. Especially when you consider the history of the US.
If the law was merely about not allowing states have sovereignty over their own fiat currencies then they would not have mentioned the words 'gold' or 'silver'.
Re: What Gödel Discovered
#173Earlier quoted context omitted.
The constitution is BS anyway. For example, Article I, section 10 says "No state shall...coin money, emit bills of credit, make any thing but gold and silver a tender in payment of debts..." And look what the US has today; fiat money. No only is it not gold or silver, but it's not event backed by it. Based on this definition, nobody in the US earns any money so nobody should have to pay any income tax since basically…
Aside from your gold bug commentary, I do find your poop example intriguing as it's probably one of the worst examples possible for currency: - literally anyone can make it - it is unsanitary and can spread disease - it is difficult to measure On the plus side: - it does represent (some amount of) labor
I think it would be an amazing currency.
- Good alternative to Universal Basic Income since people can produce it themselves directly.
- The unsanitary aspect of it will help prevent too large accumulation of it. People will have a stronger incentive to put their money to work instead of hoarding it.
- Not difficult to measure since it can be weighed and banks could print paper bills that are backed by it using weight as the units of account.
The main downside I can see is that it will encourage people to eat more and demand for laxatives would grow significantly.
Re: What Gödel Discovered
#174This is really excellent. Amazing how much clearer the author’s Lisp code makes things (even to a non mathematician and non lisper). It’s rare to find explanations of difficult topics that are this well communicated, but I hope it becomes a trend.
Thank you for the kind words! :)
I agree and I want to add that I particularly appreciated how you consistently, in programming terms, communicated the type of things. Eg:
(proves a b)
In my experience, most mathematicians would have instead written "So far saying 'a proves b'". Repeating the type of the parameters consistently made it significantly easier for me to follow along. I hope this becomes more common.
Re: What Gödel Discovered
#175Very interesting. I'm wondering if this has implications in physics ? Since "the book of nature is written in mathematical language", does this mean that attempts to find a unified theory of everything are doomed ? I'm thinking not necessarily, since we wouldn't need to prove all possible physical phenomena, only those that actually happen, but how do we know if they are among the one that can be proven ?
The implications are really on the concept of language . It proves that there is no language that can be complete; as such, we cannot express a complete system, even if it existed. So a unified theory of everything might well exist, but you wouldn’t be able to describe it. Which is a nice philosophical riddle.
I think that if you allow some axioms, you can describe it.
But I'm not sure if it will be called complete.
I think because "(next axiom)" is not defined, it will make your language not mappable to integer numbers and you avoid Godel proof.
Re: What Gödel Discovered
#176There are two things I will argue with on this otherwise great explanation. First, I've said this before and I'll bang this drum across every Godel post I see. Please don't introduce the notion of truth into an introductory post on Godel's Incompleteness Theorems (the Power of Numbers section). Introductory posts on Godel's Incompleteness Theorems like to say things like "there are true things you cannot prove" or in…
Great points
Re: What Gödel Discovered
#177This is amazing.
Two thoughts:
1) It seems that the black/white formalism of ‘ a statement can be only true or false ‘ causes issues. Have people tried introducing additional options for verifiability, e.g. ‘true / false / maybe?’ or something that points to uncertainty à la Heisenberg?
2) It would be really interesting for the author to continue the line of reasoning to help add insight to Gödel’s ontological proof. ( https://en.m.wikipedia.org/wiki/Gödel%27s_ontological_proof )
Re: What Gödel Discovered
#178This is an exceptionally good explanation of Gödel's incompleteness theorem for the general reader.
Thank you, glad you liked it :)
Well, I disliked it. I see proof that every valid formula can be converted into a Godel number. However, every invalid formula can also be converted into a Godel number. So, if we are able to construct a number, then it proves what?
Re: What Gödel Discovered
#179TYPO!!: In the section "Suspicious use of subst" it says that b is being replaced, but in the second formula box a has been replaced, not b.
Furthermore the number that replaced a is not the same as the number used in the previous subst statement.
Re: What Gödel Discovered
#180People act like Godel’s Incompleteness Theorem is all about how formal systems are limited in their level of “insight about truth”. But actually it’s just the same observation as the Halting Problem: you have a system that tries to reason about the behavior of other systems (like Turing Machines or axiomatic set theory), but it can’t possibly always introspect about its own behavior because you can also configure it to behave differently based on its own conclusion about its behavior.
What if I ask you to predict whether you’re going to go left or right, but I also tell you that I’m going to force you to go the opposite direction as what you predict? Then the set of things you can accurately predict is Incomplete. But that’s not a profound limitation of human consciousness, it’s just the familiar paradox of any fortune teller or time travel scenarios.