Kurt Gödel's Philosophical Remarks

[Humanista]

1. The world is rational.
2. Human reason can, in principle, be developed more highly (through certain techniques).
3. There are systematic methods for the solution of all problems (also art, etc).
4. There are other worlds and rational beings, who are of the other and higher kind.
5. The world in which we now live is not the only one in which we live or have lived.
6. Incomparably more is knowable a priori than is currently known.
7. The development of human thought since the Renaissance is thoroughly one-sided.
8. Reason in mankind will be developed on every side.
9. The formally correct is a science of reality.
10. Materialism is false.
11. The higher beings are connected to the other beings by analogy, not by composition.
12. Concepts have an objective existence (likewise mathematical theorems).
13. There is a scientific (exact) philosophy {and theology} (this is also most fruitful for science), which deals with the concepts of the highest abstractness.
14. Religions are, for the most part, bad, but Religion is not.

[eremita]

Já que o OP só tá copiando-e-colando a tradução da Eva-Maria Engelen, direta ou indiretamente, sem sequer dar-se ao trabalho de compartilhar o troço na metalíngua do lugar (português), então vou usar inglês mesmo. Assim não preciso re-traduzir nada; e dá menos margem para "ackshyually" e outras falácias de olha-o-aviãozinho. Caso outros foristas queiram, traduzo excertos do que escrevi, ou do que o Gödel escreveu no alemão.

Also, OP is "conveniently" omitting that those are axioms. Axioms are shit you're supposed to take as true without questioning, because they're either obviously true or already shown to be true. Neither is the case here so I'm not playing along — instead I'm showing the axioms themselves are a slopfest (easy to dispute, ungrounded) on the same level as the plot of Skibidi Toilet.

1. The world is rational.

The original (Die Welt ist vernünftig) is already bad, and the translation follows fashion.
Does the world follow consistent rules? Most likely, yes.
Do those rules have "existence"? Hard to say, probably not. It's more like we interpret certain patterns as if they were rules.
Can humans comprehend those rules partially? It's more like we're forging the rules, but if you pretend they're inherent to reality, likely yes.
Can humans comprehend those rules completely? Most likely, no.

2. Human reason can, in principle, be developed more highly (through certain techniques).

"Human reason" is just an abstraction dammit. A convenient one, like algorithm, but it shouldn't be confused with something real.
That said you can, at least in principle, develop human reason "more highly", if "more highly" means at least one of those two things:

• less likely to follow incorrect or poorly formed premises
• less likely to output an fallacious conclusion out of true premises

Note one thing does not automatically lead to the other.

3. There are systematic methods for the solution of all problems (also art, etc).

Emphasis mine. I'm calling bullshit on that "all".

4. There are other worlds and rational beings, who are of the other and higher kind.
5. The world in which we now live is not the only one in which we live or have lived.

Those two axioms are on the same tier as "let's assume the tooth fairy exists, and she lives in her castle".

6. Incomparably more is knowable a priori than is currently known.

Emphasis mine, again. That's qualia-like babble.

7. The development of human thought since the Renaissance is thoroughly one-sided.

I'm not digging through the original to know WTF this muppet is referring to, when he says the development was one-sided (einseitig). But it's likely something like "waaah, they developed that thing that really matters, instead of that thing that ⟨blink⟩I⟨/blink⟩ want it to!"

8. Reason in mankind will be developed on every side.

Please tell me this is a translation error, and Gödel isn't vomiting certainty about the future. Pleeeeease. /me checks original Nah, he is. *rolls eyes*
This axiom is exactly on the same level as "trust me" = "you're gullible rubbish, aren't you? If I smear anything on your snout you'll swallow it. Open your mouth, aaaah~". Yes, it's that bad; not only he's vomiting certainty on something he cannot show to the reader, but that he himself cannot know for sure.

9. The formally correct is a science of reality.

Okay, this one is on Engelen. The original is a wee bit clearer: "Das formal Richtige ist eine Wirklichkeitwissenschaft." He's basically saying formal thought is a science dealing with real things.
Still heavily contestable, though. Refer to what I said under #1 and #2.

10. Materialism is false.

You know what's false? This whole list of axioms.

11. The higher beings are connected to the other beings by analogy, not by composition.

You can argue cosmic coxinha is connected to mundane coxinha by analogy (as in, both are structured the same way) instead of composition (as in, both contain shredded chicken), but this still begs the question:
Does cosmic coxinha exist? (Probably not.)
Same deal with "higher beings" (höheren Wesen).

12. Concepts have an objective existence (likewise mathematical theorems).

You know what has objective existence? Dirty toilet paper. Like the one Gödel wrote this shite on. But unlike the shite he wrote on it.

13. There is a scientific (exact) philosophy {and theology} (this is also most fruitful for science), which deals with the concepts of the highest abstractness.

{and nihilogy, and centaurology, and santaklauslogy} — srry, I couldn't resist.
This one was already shown to be heavily disputable, given previous ones.

14. Religions are, for the most part, bad, but Religion is not.

First off, note how every single other axiom in this list is epistemic (claims something is true or false), but this one is deontic (claims something is good or bad). You could translate this crap as "I dun liek relijunz lol but I luuuuv teh religion lmao " and it would be still semantically equivalent to the original (die Religionen sind zum größten Teil schlecht, aber nicht die Religion).
Probably an artefact of how this list of axioms is compiled, but this creates an implicit fallacy called "wishful thinking" — "it's good so it's true lol lmao haha". (It goes without saying "X is not bad" +> "X is good" is not the same as "X is not false" +> "X is true".)
But even if we disregard this crap (we shouldn't), the axiom is still used toilet paper. It's creating one of those meaningless distinctions between "religions" and "religion".

*yawn*

[Fernando Silva]

4. There are other worlds and rational beings, who are of the other and higher kind.
5. The world in which we now live is not the only one in which we live or have lived.

E nós sabemos disto porque ...?

[Gorducho]

E nós sabemos disto porque ...?

E.g.:
4.

Dans les mondes arrivés à un degré supérieur, les conditions de la vie morale et matérielle sont tout autres, même que sur la terre. La forme du corps est toujours, comme partout, la forme humaine, mais embellie, perfectionnée, et surtout purifiée. Le corps n'a rien de la matérialité terrestre, et n'est, par conséquent, sujet ni aux besoins, ni aux maladies, ni aux détériorations qu'engendre la prédominance de la matière ; les sens, plus exquis, ont des perceptions qu'étouffe ici-bas la grossièreté des organes ; la légèreté spécifique des corps rend la locomotion rapide et facile ; au lieu de se traîner péniblement sur le sol, il glisse, pour ainsi dire, à la surface, ou plane dans l'atmosphère sans antre effort que celui de la volonté, à la manière dont on représente les anges, ou dont les Anciens se figuraient les mânes dans les Champs Elysées. 

5.

172. Nos différentes existences corporelles s'accomplissent-elles toutes sur la terre ?
« Non, pas toutes, mais dans les différents mondes : celle d'ici-bas n'est ni la première ni la dernière, et c'est une des plus matérielles et des plus éloignées de la perfection. »

[eremita]

[citações do Livro dos Espíritos; vide post acima para contexto.]

i.e. savons pas tabarnaque; les gens prétendent savoir car ils ont trop foi (et peu de neurones, pêh).
Falando em português mesmo. Eis uma tradução porquinha destes dois excertos… quero demonstrar algo.
NB: caso alguém fale francês aqui, dê uma olhada para ver se não traduzi algo errado.

Nos mundos que chegam a um grau superior, as condições da vida moral e material são completamente outras, mesmo em comparação com a Terra. A forma dos corpos é a mesma, como em todo lugar, [é] a forma humana; mas melhorada, aperfeiçoada, e acima de tudo purificada. Os corpos não têm a materialidade terrestre e, por consequência, não estão sujeitos às necessidades, doenças, nem à degeneração causada pela predominância da matéria; os sentidos, mais refinados [exquis], têm percepções que são estranguladas pela grosseria dos órgãos; a leveza específica [?] dos corpos faz com que a locomoção seja rápida e fácil; em vez de arrastar-se penosamente pelo solo, deslizam, por assim dizer, sobre a superfície, ou planam na atmosfera sem outro esforço além da vontade, do mesmo jeito que representam os angos, ou como os Antigos [sic] imaginavam seus espíritos nos Campos Elísios.

P 172: Todas nossas diferenças existenciais corporais são realizadas na Terra? [NT: tipo, "será que na Terra todas nossas diferenças corporais aparecem? Ou será que tem umas que não aparecem?"]
R: Nem todas, mas nos mundos diferentes; esta daqui de baixo não é nem a primeira nem a última, é uma das mais materiais e mais longe da perfeição.

Repararam que ele não prova merdíssima nenhuma? O Rivail / "Kardec" só alega? (...devo fingir que o cara incorporou um espírito ou asneira do tipo?)
A coisa inteira é motivada por fé, ou alguma falha de carácter similar.

[Fernando Silva]

E nós sabemos disto porque ...?

E.g.:
4.

Dans les mondes arrivés à un degré supérieur, les conditions de la vie morale et matérielle sont tout autres, même que sur la terre. La forme du corps est toujours, comme partout, la forme humaine, mais embellie, perfectionnée, et surtout purifiée. Le corps n'a rien de la matérialité terrestre, et n'est, par conséquent, sujet ni aux besoins, ni aux maladies, ni aux détériorations qu'engendre la prédominance de la matière ; les sens, plus exquis, ont des perceptions qu'étouffe ici-bas la grossièreté des organes ; la légèreté spécifique des corps rend la locomotion rapide et facile ; au lieu de se traîner péniblement sur le sol, il glisse, pour ainsi dire, à la surface, ou plane dans l'atmosphère sans antre effort que celui de la volonté, à la manière dont on représente les anges, ou dont les Anciens se figuraient les mânes dans les Champs Elysées. 

5.

172. Nos différentes existences corporelles s'accomplissent-elles toutes sur la terre ?
« Non, pas toutes, mais dans les différents mondes : celle d'ici-bas n'est ni la première ni la dernière, et c'est une des plus matérielles et des plus éloignées de la perfection. »

Então não sabemos.
É apenas parte da doutrina espírita.

[Gorducho]

Sr. eremita: eu apenas respondi ao questionamento do Sr. Fernando Silva.
Sabemos porque, entre outras possíveis fontes, é Informado na Codificação Kardequiana.
Minha área de Estudos predileta. Então respondo (evidentemente) dentro do m/âmbito de conhecimentos.

Do those rules have "existence"? Hard to say, probably not. It's more like we interpret certain patterns as if they were rules.

[NT: tipo, "será que na Terra todas nossas diferenças corporais aparecem? Ou será que tem umas que não aparecem?"]

se refere a saber se vivemos (pretérito) e viveremos sempre cá na Terra.
Resposta Doutrinária = Não.

[eremita]

Sr. eremita: eu apenas respondi ao questionamento do Sr. Fernando Silva.

Relaxa — percebi, por isso respondi às "citações" e não a você diretamente.

[NT: tipo, "será que na Terra todas nossas diferenças corporais aparecem? Ou será que tem umas que não aparecem?"]

se refere a saber se vivemos (pretérito) e viveremos sempre cá na Terra.
Resposta Doutrinária = Não.

Ah, tá. Pra te falar a verdade fiquei bem perdido ao traduzir aquilo, meu francês já foi melhor…