|Set Theory: set theory is the system of rules and axioms, which regulates the formation of sets. The elements are exclusively numbers. Sets contain individual objects, that is, numbers as elements. Furthermore, sets contain sub-sets, that is, again sets of elements. The set of all sub-sets of a set is called the power set. Each set contains the empty set as a subset, but not as an element. The size of sets is called the cardinality. Sets containing the same elements are identical. See also comprehension, comprehension axiom, selection axiom, infinity axiom, couple set axiom, extensionality principle._____________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. |
Set theory/Bigelow/Pargetter: is a child of the union of arithmetics and geometry. Descartes has done some preliminary work, the meta language was invented in the coordinate system.
It allows us to expand the correlation between points in the coordinate system, a line corresponds to a set of number pairs, etc.
Equations: many such quantities can be adequately described by equations.
Example: set of points on a circle line
(x - a)2 + (y - b)² = c².
for fixed numbers a, b and c. This corresponds to a unique set and this is unambiguously equivalent to an equation.
Set theory/Bigelow/Pargetter: reduces not only geometry to numbers and sets, but also numbers to sets. This cleared pure mathematics from empirical concerns.
Modal Realism/Bigelow/Pargetter: pro: for each logically consistent universal, there will be possibilia that instantiate it.
Instantiation/Bigelow/Pargetter: guaranteed by logical consistency.
Platonism/modal realism/Bigelow/Pargetter: our platonism is determined by the fact that we allow actualized uninstantiated universals. ((s) Not instantiated in the actual world).
N.B.: then we do not need set theory to guarantee instantiations of geometric proportions a priori. They can be studied whether or not they are instantiated in the real world.
Set theory/Bigelow/Pargetter: nevertheless, we say that there are sets of numbers that correspond to possible objects. One and the same geometric figure corresponds to an infinite number of different sets of pairs of numbers. ((s) The figure can be moved in the coordinate system).
These different sets of number pairs have something in common, even if they do not have two pairs of numbers in common: a universal.
Sets/Bigelow/Pargetter: they exist whether or not one detects them.
Universals/Bigelow/Pargetter: also exist, e.g. if you discover that two equations are in the same relation to pairs of numbers: they have the same extension._____________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.
J. Bigelow, R. Pargetter
Science and Necessity Cambridge 1990