$\forall Axiomatic_System \\=> (!Completeness) OR (!Consistency)$

There are inherent limitations of every formal axiomatic system containing basic arithmetic.

Completeness:

for any statement in the axioms’ language, that statement or its negation is provable from the axioms

Consistency:

there is no statement such that both the statement and its negation are provable from the axioms

First incompleteness theorem:

The conclusion is: there will always be elements whether

• both belong and don’t belong to a set => Breaking consistancy.
• can not be classified => Breaking the completeness.

Any consistent formal system F within which a certain amount of elementary arithmetic can be carried out is incomplete; i.e., there are statements of the language of F which can neither be proved nor disproved in F.

Gödel sentence => The above mentioned statements that either unprovable or could be proved to be both true/false.

• The Gödel sentence is designed to refer, indirectly, to itself.
• Define itself as something that’s not itself in itself.

• Construction:

  • $T \leftrightarrow \lnot True(T)$

• With inconsistency $B\lor\lnot B$, we can prove anything:

  • $B\lor\lnot B$
  • $def: \lnot A \rightarrow (B \lor \lnot B)$
  • $\lnot(B\lor \neg B)\rightarrow \neg \neg A$
  • $B \land \neg B \rightarrow A$
  • $B \land \neg B$

Second incompleteness theorem:

Solution:

The Type Theory, Where operators are put into types and they can not be used to describle those that are in the same type, including themselves. This way, the Russell Paradox, using an operation in a statement pointing to the statment itself as a Godel sentence, is simply invalid.

Reference: Wikipedia, Zhihu1, Zhihu2

2017/11/21