Concepts Expressible in Infinitary Logic
In the language of set theory the following statement expresses foundation:
Unlike the axiom of foundation, this statement admits no non-standard interpretations. The concept of well foundedness can only be expressed in a logic which allows infinitely many quantifiers in an individual statement. As a consequence many theories, including Peano arithmetic, which cannot be properly axiomatised in finitary logic, can be in a suitable infinitary logic. Other examples include the theories of non-archimedean fields and torsion-free groups. These three theories can be defined without the use of infinite quantification; only infinite junctions are needed.
Read more about this topic: Infinitary Logic
Famous quotes containing the words concepts and/or logic:
“Science is a dynamic undertaking directed to lowering the degree of the empiricism involved in solving problems; or, if you prefer, science is a process of fabricating a web of interconnected concepts and conceptual schemes arising from experiments and observations and fruitful of further experiments and observations.”
—James Conant (1893–1978)
“The usefulness of madmen is famous: they demonstrate society’s logic flagrantly carried out down to its last scrimshaw scrap.”
—Cynthia Ozick (b. 1928)