I ka wā i paʻi ai ʻo Kurt Gödel i kāna puke The Incompleteness Theorems kaulana i 1931, ua hoʻoluliluli ia i nā kumu o ka ʻike makemakika: Ua hōʻole ʻo ia ʻaʻole hiki i nā axioms āpau ke hoʻonohonoho ʻia ma ke ʻano he kumu kūpono no ka hōʻoia ʻana i nā ʻōlelo āpau e pili ana i nā helu - a luku ʻia kēlā. ʻO kā Hilbert moeʻuhane e hōʻoia ai i ke kūlike o ka makemakika makemakika.
ʻO ka hoʻolauna ʻana o nā helu Gödel (ka hoʻohālikelike kūikawā o nā haʻilula i nā helu kūlohelohe) a me ka diagonalization (ka hoʻololi ʻana o ka loli manuahi i nā hana me kāna helu Gödel) ʻelua mau manaʻo nui a Gödel i hoʻolauna ai i kāna hōʻoia. ʻO ka manaʻo hōʻoia koʻikoʻi kahi e hoʻohui ai ʻo Gödel i kēia mau manaʻo hiki ke kākau ʻia penei.:
$$P(p) \, \text{wahr} \Leftrightarrow p \in \, \overline{B}^* \Leftrightarrow d(p) \in \overline{B} \Leftrightarrow d(p) \notin B \Leftrightarrow g(P(p)) \notin B \Leftrightarrow P(p) \, \text{unbeweisbar}$$
ʻOiai ʻo \(P(p)\) ʻaʻole hiki ke wahaheʻe (ʻoiai he mea ʻē aʻe ia e hōʻoia ai a no laila he ʻoiaʻiʻo), \(P(p)\) a no laila ʻaʻole pono. No laila aia mau kahi huaʻōlelo maoli i kahi ʻōlelo (me nā koho axioms) ʻaʻole hiki ke hōʻoia ʻia. Eia \(g\) ka Gödelization, \(p\) ka helu Gödel o ka predicate \(P\) , ʻo ia ka archetype komplemente \(\overline{B}^*\) o \(B\) (ka pūʻulu o nā mea āpau Nā helu Godel o nā manaʻo kuhi hewa āpau) ma lalo o ka diagonal function \(d\) .
No ka heluhelu hou aku , ua paipai ʻia ka puke pai a Gödel i ka makahiki 1931 a me ka ʻatikala naʻauao a Stepan Parunashvili . Ma waho aʻe o nā kumumanaʻo incompleteness, ua hāʻawi ʻo Gödel i nā haʻawina hou ʻē aʻe, me ka hiki ʻole ke hōʻole ʻia o ka kuhiakau hoʻomau a Cantor a me ka hoʻopaʻapaʻa ontological no ke ola ʻana o ke Akua ma ka ʻōlelo o ka logic modal.