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What Gödel Discovered

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Re: What Gödel Discovered

#221

Post by kfarr »

cryptica wrote: post #173
kfarr wrote: post #149

Earlier quoted context omitted.

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 weigh…

And like any fine metal you can polish it! https://www.youtube.com/watch?v=yiJ9fy1qSFI

Re: What Gödel Discovered

#222

Post by justincredible »

irrational wrote: post #8

> This proof showed that “1 + 1”, does indeed equal “2”. It took 2 volumes to get here. I know this seems logical to mathematicians, but it feels to me like having to take 2 volumes to prove something than any child knows intuitively is... I don't know what word I am looking for... obsessive?

How do we know that 1+1=2 without a proof or circular reasoning? If you think intuition is a good metric of truth, you should look into quantum mechanics.

Re: What Gödel Discovered

#223

Post by stopachka »

vcdimension wrote: post #179

TYPO!!: 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.

Great catch! Updated : ) -- should take a few mins to show up

Re: What Gödel Discovered

#224

Post by drran »

arc0 wrote: post #187
drran wrote: post #178

Earlier quoted context omitted.

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?

The conversion of formulae to Gödel numbers is basically an implementation detail, the fact it can be done allows us to define theorems on the natural numbers that describe properties of the logical system. Encoding an invalid formula is of course possible, but I don't see why that would be a problem, it doesn't inherently 'prove' anything.

For me, it looks like use a program in C to generate a C code, and then prove that a C program can generate an invalid C code, so the C compiler must be invalid. Compilation step is missed. Compiler will refuse to compile the invalid code.

IMHO, the same situation is for Godel numbers. We can use math to generate numbers, which are equivalent to formulas, BUT why these formulas must be accepted? Initial set of axioms was carefully chosen by human, so it not an open set, where anybody can add anything.

Re: What Gödel Discovered

#227

Post by musicale »

Kranar wrote: post #122
cmonnow wrote: post #102

I never understood the fascination people have with self-referential 'paradoxes' like : "This statement is False". Pronouns do not have an independent existence. Until the pronoun 'This' resolves to an actual statement which has a valid true/false property, it is a recurrence without termination i.e. a infinite loop, that has no meaning. Suppose I say: 1. Sky is blue. 2. Previous statement is True. 3. Previous statem…

I think you're too quick to dismiss what is widely considered to be one of the most important and profound results in logic and computability without taking the time to fully appreciate it. Your procedure of repeated substitution in order to resolve a paradox does little to shed light on the situation since at the heart of the halting problem is that there's no formal system that can determine whether this repeated s…

PP seems to be asking for an example that is not obviously self-contradictory yet demonstrates a serious, unsurmountable problem, thereby illustrating the incompleteness theorem.

Is the theorem you describe such an example? If so, you have refuted PP with a non-handwaving result.

If not, then you've provided more evidence supporting PP's complaint.

Re: What Gödel Discovered

#228

Post by redvenom »

legolas2412 wrote: post #220
redvenom wrote: post #118

Earlier quoted context omitted.

This response is not quite right. "It's only completeness [...]" sentence does not make sense. CH is independent of ZFC, period, as proved by Cohen. Talking about 'semantic level' does not make sense. CH is an example of the incompleteness of ZFC. There are models of ZFC in which CH is true and models in which CH is false.

Trying to understand logic. What is a model, and how does it differ from theory? For specific example, what is a model of ZFC? Is it just another theory, one which includes ZFC and few more axioms? Why not call it a derived theory or a subset theory?

This explanation is somewhat informal, but I gets the point across: a theory is a set function symbols, relation symbols, and axioms governing them. A model is a specific set with an interpretation of those function and relation symbols that satisfies every axiom of the theory.

It's much easier to understand if we take an example. An example of a theory is the single sentence:

"There exists an X and there exists a Y such that X is not equal to Y."

(Of course typically in logic you would use logic symbols, but here I am writing out in an English sentence.)

Now, a model of this theory is the set {1,2}. Another model is the set {1,2,3}. More generally: any set with at least two elements is a model of that theory. The "function symbols" and "relation symbols" can be introduced in the language to talk about operations like addition and multiplication.

For example, the theory of groups uses the language of groups with a binary function symbol representing group multiplication. Any group (such as the integers with addition or invertible matrices with matrix multiplication) is a model of that theory.

So: theories are sets of axioms in some language, and models are sets together with actual functions/relations that satisfy those axioms.

Models of ZFC are a little bit counterintuitive. But they are single sets that interpret all the axioms of ZFC, rather than actual sets that we use in informal mathematics. Models of ZFC can be quite unusual because of the incompleteness theorem, and there are infinitely many models because of this (such as some in which CH is true, etc.).

Re: What Gödel Discovered

#229

Post by mike00632 »

ProfHewitt wrote: post #211
greatquux wrote: post #196

Earlier quoted context omitted.

I'm going to need someone to come along and write up a layman's summary of this article. :) Does it change the end result of Gödel? Or is it basically the equivalent of saying what some restricted programming languages are doing, that we can get "good enough" results and not have these problems by restricting what we can do in the language?

Hi GreatQuux! The article linked below explains why [Gödel 1931] did not prove inferential undecidability of Russell's Principia Mathematica and likewise why the formalization of [Gödel 1931] in Lisp proof being discussed is also invalid: https://papers.ssrn.com/abstract=3603021 However, the article linked above does have a correct proof of inferential undecidability (also known as "inferential incompleteness"). Woul…

What do you mean by "inferential" and why are there restrictions on what statements are valid?

Re: What Gödel Discovered

#230

Post by mike00632 »

ProfHewitt wrote: post #212
ethn wrote: post #207

Earlier quoted context omitted.

Secondly, in the PM-Lisp it doesn't necessarily prove that theorem a proves b, it just shows that b can be a successor of the formulas in a

Hi Ethn! Existence of the [Gödel 1931] proposition I'mUnprovable is inconsistent with the following theorem to the effect that theorems can be used in proofs: ⊢∀[Proposition Ψ] (⊢Ψ)⇒Ψ

For a statement so short it would be easy to avoid using jargon...
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