?url_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adc&rft.title=A+hierarchy+of+languages%2C+logics%2C+and+mathematical+theories&rft.creator=Kastner%2C+Charles+W.&rft.subject=Historical+Linguistics&rft.subject=Philosophy+of+Language&rft.subject=Logic&rft.description=We+present+mathematics+from+a+foundational+perspective+as+a+hierarchy+in+which+each+tier+consists+of+a+language%2C+a+logic%2C+and+a+mathematical+theory.++Each+tier+in+the+hierarchy+subsumes+all+preceding+tiers+in+the+sense+that+its+language%2C+logic%2C+and+mathematical+theory+generalize+all+preceding+languages%2C+logics%2C+and+mathematical+theories.++Starting+from+the+root+tier%2C+the+mathematical+theories+in+this+hierarchy+are%3A+combinatory+logic+restricted+to+the+identity+I%2C+combinatory+logic%2C+ZFC+set+theory%2C+constructive+type+theory%2C+and+category+theory.++The+languages+of+the+first+four+tiers+correspond+to+the+languages+of+the+Chomsky+hierarchy%3A+in+combinatory+logic+Ix+%3D+x+gives+rise+to+a+regular+language%3B+the+language+generated+by+S%2C+K+in+combinatory+logic+is+context-free%3B+first-order+logic+is+context-sensitive%3B+and+the+typed+lambda+calculus+of+type+theory+is+recursively+enumerable.++The+logic+of+each+tier+can+be+characterized+in+terms+of+the+cardinality+of+the+set+of+its+truth+values%3A+combinatory+logic+restricted+to+I+has+0+truth+values%2C+while+combinatory+logic+has+1%2C+first-order+logic+2%2C+constructive+type+theory+3%2C+and+categeory+theory+omega_0.+++We+conjecture+that+the+cardinality+of+objects+whose+existence+can+be+established+in+each+tier+is+bounded%3B+for+example%2C+combinatory+logic+is+bounded+in+this+sense+by+omega_0+and+ZFC+set+theory+by+the+least+inaccessible+cardinal.+%0A%0AWe+also+show+that+classical+recursion+theory+presents+a+framework+for+generating+the+above+hierarchy+in+terms+of+the+initial+functions+zero%2C+projection%2C+and+successor+followed+by+composition+and+m-recursion%2C+starting+with+the+zero+function+I+in+combinatory+logic+%0A%0AThis+paper+begins+with+a+theory+of+glossogenesis%2C+i.e.+a+theory+of+the+origin+of+language%2C+since+this+theory+shows+that+natural+language+has+deep+connections+to+category+theory+and+since+it+was+through+these+connections+that+the+last+tier+and+ultimately+the+whole+hierarchy+were+discovered.++The+discussion+covers+implications+of+the+hierarchy+for+mathematics%2C+physics%2C+cosmology%2C+theology%2C+linguistics%2C+extraterrestrial+communication%2C+and+artificial+intelligence.%0A&rft.date=2003-03&rft.type=Preprint&rft.type=NonPeerReviewed&rft.format=application%2Fpdf&rft.identifier=http%3A%2F%2Fcogprints.org%2F2875%2F1%2FClassRecTheory0.pdf&rft.identifier=++Kastner%2C+Charles+W.++(2003)+A+hierarchy+of+languages%2C+logics%2C+and+mathematical+theories.++%5BPreprint%5D+++++&rft.relation=http%3A%2F%2Fcogprints.org%2F2875%2F