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 hoʻokomo ʻana i nā helu Gödel (ka palapala unambiguous o nā formula i nā helu kūlohelohe) a me ka diagonalization (ka hoʻololi o ka manuahi manuahi i nā hana me kā lākou helu Gödel) ʻelua mau manaʻo nui a Gödel i hōʻike ai i kāna hōʻoiaʻiʻo. ʻO ka manaʻo hōʻoia hoʻoholo e hoʻohui ai ʻo Gödel i kēia mau manaʻo hiki ke kākau ʻia i lalo:
$$P(p) \, \text{wahr} \Leftrightarrow p \in \, \overline{B}^* \Leftrightarrow d(p) \in \overline{B} \Leftrightarrow d(p) \notin B \Leftrightarrow g(P(p)) \notin B \Leftrightarrow P(p) \, \text{unbeweisbar}$$
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 ʻana, paipai mākou iā Gödel's 1931 paʻi a me ka ʻatikala heluhelu maikaʻi na Stepan Parunashvili . Ma waho aʻe o nā kumumanaʻo o ka piha ʻole, ua hoʻokō ʻo Gödel i nā mea i loaʻa hou i ka honua, e like me ka irrefutability o ke kuhiakau mau o Cantor a me ka hōʻoia ontological o ke Akua ma ka ʻōlelo o ka loiloi modal.