Re: Cantor's definition of set
- From: Jan Burse <janburse@xxxxxxxxxxx>
- Date: Fri, 26 Oct 2007 14:07:56 +0200
G. Frege schrieb:
That's right. BUT we might claim that all elements in the set
{123, apple, fritz the cat}
have a certain "property" in common, namely to belong to the set
{123, apple, fritz the cat}.
I guess that's what Halmos meant with his statement:
"It is ubiquitous mathematical practice to identify a property with a
set, namely with set of all objects that possess the property [...]."
(Paul R. Halmos, Naive Set Theory)
Halmos starts with property, and not with set.
What you stipulated is that a set implies a
property, i.e. the converse.
Which is not necessarely true. Where do you draw the
properties from? Ok, if you allow a set to be a property,
this will work. So for each Set S, we have the property
P_S, such that:
P_S(x) <-> x in S.
Actually I don't see that a lot is gained there.
Can somebody give examples where this leads to
something of interest?
But halmos quote is doubtful in the object level.
It would necessiate naive comprehension:
For a property P, we have { x | P(x) }.
This ain't working. So a property gives probably rise
to "class" but not to a set.
On the set theoretic meta level on the other hand
extensionally we have the domain of discourse U, and
P subset U is the interpretation of a property.
Bye
.
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- Cantor's definition of set
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