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:
“Once one is caught up into the material world not one person in ten thousand finds the time to form literary taste, to examine the validity of philosophic concepts for himself, or to form what, for lack of a better phrase, I might call the wise and tragic sense of life.”
—F. Scott Fitzgerald (18961940)
“... We need the interruption of the night
To ease attention off when overtight,
To break our logic in too long a flight,
And ask us if our premises are right.”
—Robert Frost (18741963)