Re: Set existence
- From: tchow@xxxxxxxxxxxxx
- Date: Thu, 14 Feb 2008 18:30:04 +0000 (UTC)
In article <fovgfv$jni$1@xxxxxxxxxxxxxxxx>, <malcobe@xxxxxxxxx> wrote:
What does it mean that ZFC has an uncountable model? Doesn't it mean
that from ZFC axioms we can formally prove the existence of a set (or
class) M and a relation (or set-theoretic relation) R such that every
axiom of ZFC is true under the interpretation?
It means that there is an uncountable set (not a class) and a relation
such that every axiom of ZFC is true under the interpretation.
It does *not* mean that *from the ZFC axioms we can formally prove [blah]*.
That assertion would be "ZFC can prove that ZFC has an uncountable model,"
not "ZFC has an uncountable model." "ZFC proves x" is a distinct assertion
from x itself.
--
Tim Chow tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences
.
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