Type theory

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'''Type theory''' (presumably, as opposed to type practice) is a part of metamathematics that distinguishes an object and the set to which it belongs, in a formal way. Type theory was first developed by [[Bertrand Russell]] as his solution to a foundational crisis in [[mathematics]] that he started.

==Motivation in detail==
In 1874, {{wpl|Georg Cantor}} proved that the cardinality of a power set is always greater than that of the original set. In other words, you can always make a set bigger than any set you care to name even if that set is infinite. This led to the creation of modern set theory, and many mathematicians were worried that the freedom created by this new theory would lead to contradictions. In order to lay this worry to rest (and in the service of a pedantic [[Immanuel Kant|Kantian]]-[[Gottfried Leibniz|Leibnizian]] idea of a universal language){{efn|The idea of a universal language is one conception of language and logic: as a Universal Medium vs the conception of language and logic: as Calculi}}<ref>[https://www.jstor.org/stable/20114564?seq=1 Logic as Calculus and Logic as Language originally published in Synthese (1967): p.324-330]. Van Heijenoort, Jean</ref><ref>[https://b-ok.cc/book/2244023/cd1674 Calculus Ratiocinator vs Characteristica Universalis? The two traditions in Logic, Revisited, (1997)]. Hintikka, Jaacko</ref>, {{wpl|Gottlob Frege}} formalized the theory in 1879 and more extensively in 1884. In spite of Frege's accomplishments, a devastating blow was dealt to the formalization by Bertrand Russell's self-reference [[paradox]]es {{efn|Technically Russell's Paradox is an antinomy: the paradox is neither ''veridical'': a paradox whose conclusion is true despite its air of absurdity, nor is it ''falsidical'': a paradox whose conclusion not only appears absurd but is also false, there being a fallacy in the proof. There is no way, presently, to becloud antinomies either by convincing us that there conclusion is true or by finding the fallacious step in the argument.}}<ref>(Secondary Source([https://philpapers.org/rec/WAGSMQ Review of: Quine, New Foundations, and the Philosophy of Set Theory by Sean Morris)) published in the Journal for the History of Analytical Philosophy, Vol 8, No.9, page 4. Wagner, Henri (2020)]</ref>
Is the man who loves all men who do not love themselves self-hating? Self-loving?<ref>This particular variation is taken from Jaakko Hintikka's "On Gödel."</ref> Though the great [[skeptic]] Russell never brought it up, the old [[Omnipotence paradox|theological question]] of whether [[God]] can lift a stone so heavy he can't lift it is another variation on [[Russell's paradox]].

To put it into a form more directly useful for skepticism, consider if God (or any [[omnipotent]] being) can create something so wonderful that He cannot create it.{{efn|Creation is not special and virtually anything can be substituted for it.}} If he cannot, he is not omnipotent (by definition), and if he can, then there is something he cannot create and He is not omnipotent. Unlike omnipotence, mathematics can be rescued from paradox by many tactics.

Already in Frege's work there is a distinction between objects, predicates and predicates of predicates, and so on; a mixing of which is essential to Russell's paradox in its abstract form.{{efn|Or at least the abstract form in Frege's formalization.}} If Frege had been more pedantic in this distinction, as he was in the distinction between sense and reference, then perhaps we never would have found these paradoxes. As it stands, Russell had to create type theory in order to separate the wheat from the chaff and strengthen the foundation of mathematics.

==How does type theory deal with the paradox?==
By definition, the primitive objects have type 0.{{efn|It is not necessary that there are any such objects for there to be a type for them.}} Sets that contain these objects have type 1.{{efn|There is here, at least, the empty set.}} Sets that contain sets of type 1 have type 2. Ordered pairs, for instance, are of this type. Sets that contain objects of type 2 have type 3. And so on and so on, as far beyond the small infinities as we can take. Thus one cannot ask about a set that contains itself, for this set would have to be higher than its own type.{{efn|More indirect forms of self-reference are still possible and still have deep effects on mathematics.}} There is no person who loves everyone who doesn't love themselves and [[Omnipotence paradox|no being that can do everything that it cannot do]].

==Other Uses==
Like all real [[science]], type theory did not burst from the head of one man fully grown, but instead evolved with time. The above explanation is adapted from a simplification used by {{wpl|Kurt Gödel}} during the proof of the {{efn|Gödel's incompleteness theorems|incompleteness theorems|,}} and was improved by {{wpl|Alonzo Church}} while working on the [[Alan Turing|Decision Problem]]. {{wpl|Per Martin-Löf}} began the exploration of type theories in non-classical logics, and both his and Church's work are important in the theory of functional programming.

==See also==
* [[Logic]]
* [[Computer]]
* [[Typology]] - an alternative rabbit-hole for the theologically inclined

==Notes==
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[[Category:Logic]]

Categories: Logic