|Type theory: The type theory is a restriction of formal systems to a kind of reference which prevents symbols of a level (of a type) from referring to symbols of the same level (the same type). This is intended to avoid paradoxes arising from a self-reference of the signs or expressions used. Original proposals for type theories are given by B. Russell (B. Russell, “Mathematical logic as based on the theory of types”, in American Journal of Mathematics, 30, 1908, pp. 222-262). See also self-reference, circularity, paradoxes, Russell's Paradox, branched type theory._____________Annotation: The above characterizations of concepts are neither definitions nor exhausting presentations of problems related to them. Instead, they are intended to give a short introduction to the contributions below. – Lexicon of Arguments. |
Christian Thiel on Type Theory - Dictionary of Arguments
Thiel I 324
VsType Theory: among its complications was not only the fact that such a theory has to consider not only types but also orders, and also the more than annoying fact that now, for example, the upper limit of a non-empty set of real numbers (whose existence is assumed in all continuity considerations in classical analysis) is of a higher order than the real numbers whose upper limit it is.
The consequence of this is that one can no longer simply quantify using "all real numbers", but only using all real numbers of a certain order. Unacceptable for field mathematics, and a huge obstacle for the "arithmetic program" of classical basic research.
All the more so for the logicism that follows._____________Explanation of symbols: Roman numerals indicate the source, arabic numerals indicate the page number. The corresponding books are indicated on the right hand side. ((s)…): Comment by the sender of the contribution. Translations: Dictionary of Arguments The note [Concept/Author], [Author1]Vs[Author2] or [Author]Vs[term] resp. "problem:"/"solution:", "old:"/"new:" and "thesis:" is an addition from the Dictionary of Arguments. If a German edition is specified, the page numbers refer to this edition.
Philosophie und Mathematik Darmstadt 1995