Re: Cantor's definition of set

G. Frege schrieb:
@JJ: In ZFC (our "standard" set theory) we actually have that everything
(in the domain of discourse) *is* a set. (No urelements.)


F.

Yes, but you can easily read the axioms in a many sorted
first order language, and then when it says:

Two sets are the same if they have the same elements

forall x forall y forall z ((z in x <-> z in y) -> x=y)

The corresponding axiom in the many sorted first order
language is then:

forall x (
Set(x) -> forall y(
Set(y) -> forall z((z in x <-> z in y) -> x=y)))

Or assume that your urelements will end up in sets. Like
integeres will be special sets, reals will be special sets,
fritz the cat will be a special set etc..

For integer numbers and reals this goes quite well, and
properties of sets automatically transfer to the properties
of the integeres and reals (dedekind cut etc..).

Also note that urelements can be structured. Nothing
prevents you from having sequences or hierarchies
as urelements, and these urelements being further
decomposable.

So maybe the better name is non-sets than urelements.

http://en.wikipedia.org/wiki/Urelement
http://mathworld.wolfram.com/Urelement.html

Bye

.

Relevant Pages

  • Re: Cantors definition of set
    ... but you can easily read the axioms in a many sorted ... The corresponding axiom in the many sorted first order language is ... and U (urelements). ...
    (sci.logic)
  • Re: tommy1729 set axioms update
    ... Suppose he didn't have the added axioms that make his ... He still has extensionality (not ... so there are no urelements (at ... to read and study math, 'cause he can make up his own ...
    (sci.math)
  • Re: ZFC means?
    ... >> And so we've shown that a set of ZFC's non-logical axioms (including ... > Axiom of Extensionality _in that theory_. ... I don't have an immediate point in mind, but just for perspective, as I ... way to distinguish among the urelements? ...
    (sci.logic)
  • Re: Skolems Paradox and why is math the way it is?
    ... >> through recursive calls to finite strings and the ZF axioms), ... >> is clearly a subclass of any other class of any model of the reals, ... I thought the word "first order formula" ... ONE has proven theorems about them (since I haven't seen new axioms ...
    (sci.math)
  • Re: Skolems Paradox and why is math the way it is?
    ... order proof counting the universe. ... reals, but that does NOT tell us which specific reals are in that set. ... > is that it is a set according to the axioms of ZFC. ... used IF-logic instead of first order logic or secdon order logic. ...
    (sci.math)
Quantcast