ZFC is a liar.
ZFC is a liar.
Source: http://sci.tech−archive.net/Archive/sci.math/2006−11/msg06487.html
• From: "zuhair"
• Date: 25 Nov 2006 12:19:29 −0800
Hi All,
ZFC is a liar, and all sets in ZFC are lies.
I said this in a prior post to this forum, yet I didn't explain it
fully.
The idea came into my mind, when I recognized the similarity between
Russell's paradox and the liar paradox.
"the set of all sets that are not in themselfs" and" all of my
statements are lies", make's one think that "a set that is not in
itself" is equal to saying " a statement that is a lie"
IN nutshell, a set that is not in itself IS a lie.
To clarify all of this, rather unclear informal speach. let us define
again the two words "liar" and"truth teller".
M is a liar M:As by M −> s is false.
M is a truth teller M:As by M −> s is true.
Now it is clear that a liar cannot make a true self reference of his
statements.
a liar always lies, so he should lie about the authenticity of his
statements.
Imagine yourself in a world were all people are either liars or truth
tellers, and you are walking with a person that you do not know weather
he is a liar or a truth teller.
suppose that person made a statement _S.
to know weather he is a liar or a truth teller, you can simply ask him:
did you made the statement
S. if he say yes then he is a truth teller, if he say NO, then he is a
liar.
ZFC is a liar. 1
ZFC is a liar.
The point is that we KNOW that statements s,k,f,g,... are produced by
the person M. and when we tell him are these your statements, then if
he say no then he is a liar.
That's why I say a liar as defined above will always deny the
authenticity of his statement. i.e deny himself being the stater of his
statement.
While a truth teller always confirm the authenticity of his statements,
i.e. always confirm himself being the stater of his statements.
Now the question is simply DO ZFC deny or confirm its sets.
you see ZFC is the producer of models called sets. this is like a
person who produce statements.
we can say that sets consistent with ZFC are statements produced by
ZFC.
But how do we check self reference of ZFC. i.e. ZFC is not a man so
that we can ask him if he produced his statements or not.
In fact, the test which reveal ZFC's self reference about its sets, is
weather ZFC has a universe or not.
first we know sets produced by ZFC.
Now the set of all sets produced by ZFC is the set who is defined like
this:
U={x|x is produced by ZFC}
so every x if it is "produced by ZFC" will be in U. this is the best
test of self reference.
IF U exists in ZFC then this is equivalent to saying that ZFC is
confirming the authenticity of its set being produced by it, i.e by
ZFC.
IF U do not exist in ZFC then this is equivalent to saying that ZFC is
denying the authenticity of its sets being produced by it.
It is clear that there is no universe in ZFC. because this rises
Russell's paradox.
therefore ZFC never states ( have as a set ) every set in it being
produced by it.
Accordingly ZFC is a liar.
Accordingly every set in ZFC is a lie.
ZFC is a liar. 2
ZFC is a liar.
What is the importance of all of this, I said above that ZFC is a liar,
but I didn't say that ZFC is inconsistent. In fact a liar as defined
above ( what continouselly lie) is a consistent system of statements.
i.e such a liar is consistent with itself. That's why ZFC is
consistent.BUT NOT COMPLETE.
what I mean by not complete is that it cannot deal with sets that has
universal nature. for example, the set of all sets in ZFC, the set of
all well founded sets, the set of all ordinals, the set of all sets not
bijectable to 3. etc.....
why because ZFC is a liar and a liar cannot deal with these statements,
since they imply a true self reference that liar cannot performe, any
attempt to make such statements (sets) would be inconsistent with the
very basic nature that defines a liar i.e he continouselly lies.
Therefore we need a set theory that is comparable to the truth teller.
This set theory would be both consistent and complete.it can quantify
over universal sets.
That was the objective behind formulation of "Z.irregular set theory"
that I have presented in a topic by this name. and also under another
topic"why regularity"
If this theory or any theory that has all its sets in themselfs, proves
to be complete, and has more applications than ZFC, i.e. every thing in
ZFC has a comparable concept in this new set theory, but not the
converse. then this theory would make a better theory than ZFC.
Zuhair
.
ZFC is a liar. 3