FOL doesn't care whether objects are defined or not. It
assume a universe of discourse U. What you do with FOL
you then define prediates. In set theory the only
predicate, if you don't add further predicates,
is the set membership, can be denoted by the "element
of" sign ∈ . So ZFC is not a theory about objects really,
its a theory about the membership predicate ∈ . Nevertheless
ZFC postulates with the help of the membership predicate
∈ here and then the existence of objects. But this is not
a postulation on the basis of definedness. There is no such
notion in FOL. The postulation is solely done by an existential
sentence and by postulating that there is some object with
a certain property. I have observed in JG and now in WM very
often this confusion that logic works with definedness, or
definability directly sticking to some object. But the ingredient
to postulating existence is not the object but the relationships
or functions that are used to postulate the existence. The object
must then appear in the domain or range of these functions,
or in the extend of the relationship, or in the complement of the
extend of the relationship. The relative complement to the universe
of discourse. Take for example the empty set, and the very simple
postulation of its existence:
exists x forall y (~ y ∈ x)
We see to make this axiom true, models have to be select ednot with
respect that something is contained in the relationship ∈ . Since
the empty set will at least not appear on the right hand side of the
relationship ∈ . It rather forces models of the theory to contain
the empty set in the universe of discourse. Thats all that happens.
The empty set itself doesn't receive some particular definedness
status. Well we can prove from the axiom directly its existence. And
using other axioms, like the extensionality axiom, we can show that
it is unique. But in the language of ZFC there is no "defined".
So the empty set has no special mark beyond how it acts inside
the membership ∈ . Which isn't even a property of the empty set itself
but of the extend of the membership ∈ , how this predicate forms
itself. So definedness is something that belongs to the fairy land,
Where the princess sits before the mirror and asks "Spieglein,
Spieglein an der Wand, Wer ist die Schönste im ganzen Land?« (*). But
in FOL no single object asks for definedness.
Post by WM
Post by email@example.com Post by WM Post by firstname.lastname@example.org
Recall, any *subset* of words must be countable. However, it may not be true
that a subcollection of words must be a subset (E.g. If every subset of integers, equivalently every potentially infinite 0/1 sequence, must be computable then the indexes of the halting Turning machines form a subcollection but not a subset). The f's are s subcollection but not a subset.
If you have a countable set S, then every subcollection T is countable and is a set. This is provable by the Axiom of separation. The required predicate is definable
If we set "definable" = computable (e.g. constrain 0/1 sequences to the computable 0/1 sequences) then the required predicate must be computable.
Usually we do not put definable = computable, at least I never do so. For instance I can define the largest prime number or the smallest positive real number but cannot compute it.
When we talk about such objects, then we know what is meant. Therefore they are defined objects although not existing in mathematics.