3
votes
2answers
91 views

If $T$ proves any incorrect $\forall$-rudimentary sentence, then $T$ is inconsistent

A theory $T$ in the language of arithmetic is called $\omega$-inconsistent if for some formula $F(x)$, $\exists x F(x)$ is a theorem of $T$, but so is $\neg F(n)$ for each natural number $n$. ...
4
votes
1answer
199 views

Understanding the syntactical completeness

A formal system is syntactically complete if for each sentence (closed formula) $\varphi$ either $\varphi$ or $\lnot \varphi$ is provable. A formal system is semantically complete if every ...
4
votes
3answers
344 views

Aftermath of the incompletness theorem proof

This is somewhat of a minor point about the incompletness theorem, but I'm always a little unsure: So one proves that there is a formula which is unprovable in the theory of consideration. Okay, at ...
1
vote
1answer
150 views

Definition and meaning of “Proof Schema”, “Class Sign”

I'm a newbie in advanced mathematics, and I'm trying to understand Godel's theorem. I came across these two words which I couldn't understand clearly. "Proof Schema" and "Class-Sign" Can anybody ...
2
votes
2answers
364 views

How it is posible that $\omega$-inconsistency does not lead to inconsistency

After wikipedia: Theory is $\omega$-inconsistent if, for some property P of natural numbers, T proves P(0), P(1), P(2), and so on (that is, for every standard natural number n, T proves that P(n) ...