|Sets: a set is a summary of objects relating to a property. In the set theory, conditions are established for the formation of sets. In general, sets of numbers are considered. Everyday objects as elements of sets are special cases and are called primordial elements. Sets are, in contrast to e.g. sequences not ordered, i.e. no order is specified for the consideration of the elements. See also element relation, sub-sets, set theory, axioms._____________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. |
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Sets/Quine/Goodman/Bigelow/Pargetter: we may no longer need any other universals if we allow sets. Because you can do almost anything with sets that mathematics needs.
Armstrong: he believes in universals, but not in sets!
BigelowVsQuine/BigelowVsGoodman: for science we need more universals than sets, for example probability and necessity.
Universals/Sets/Predicates/Bigelow/Pargetter: if a predicate does not correspond to a universal, e.g. dogs, we assume that they correspond to at least one set.
Predicate/Bigelow/Pargetter: but even then we cannot assume that each predicate corresponds to a set!
Set/Bigelow/Pargetter: For example, there is no set X containing all and only the pairs
Universal Set/Universal Class/Bigelow/Pargetter: can also not exist.
Predicate: "is a set" does not correspond to a set that contains all and only the things it applies to! (Paradox, because of the impossible amount of all sets).
Set theory/Bigelow/Pargetter: we are still glad if we can assign something to most predicates, and therefore set theory (which originates from mathematics and not from semantics) is a stroke of luck for semantics.
Reference/Semantics/Bigelow/Pargetter: set theory helps to impose more explanatory force on the reference in order to formulate a truth theory (WT). It remains open which role reference should play.
Existence/sets/set theory/axiom/Bigelow/Pargetter: none of the following axioms secures the existence of sets: pair set axiom, extensionality axiom, union set axiom, power set axiom, separation axiom: they all only tell us what happens if there are already sets.
Axioms/Zermelo-Fraenkel/Bigelow/Pargetter: their axioms are recursive: i.e. they create new things from old things.
Based on two axioms:
Infinity axiom/Zermelo-Fraenkel/Bigelow/Pargetter: (normally formalized to contain the empty set axiom). Stands for the existence of a set containing all natural numbers according to von Neumann.
Omega/Bigelow/Pargetter: according to our mathematical realism, the sets in the sequence ω are not identical to natural numbers. They instantiate them. That is why the infinity axiom is so important.
Infinity axiom/Ontology/Bigelow/Pargetter: the infinity axiom has real ontological significance. It ensures the existence of sufficient sets to instantiate the rich structures of mathematics. And physics.
Question: is the axiom true? For example, suppose a quality of "being these things". And suppose there is an extra thing that is not included. Then it is very plausible that there will be the qualities of being "those things" that apply to all previous things plus extra things. To do this, these properties must first be available. Moreover, if we are realists about such properties, such a property can count as an "extra thing"!
This ensures that if there is an initial segment of, the next element of the sequence also exists.
Infinity: but requires more than that. We still have to make sure that the whole of ω exists! I.e. there must be the property "to be one of these things", whereby this is a property instantiated by all and only by Neumann numbers. That is plausible in our construction, because we use sets as plural essences (see above) to understand.
Problem: we only have to guarantee a starting segment for the Neumann figures. That should be the empty set.
Empty set/Bigelow/Pargetter: how plausible is their existence in our metaphysics?
J. Bigelow, R. Pargetter
Science and Necessity Cambridge 1990