---------- Forwarded message ----------
From: Andrii (lOkadin) Zvorygin
<andrii.z@gmail.com>
Date: Mar 19, 2007 2:16 AM
Subject: Russell's Anti-Paradox -- for translating to Lojban
To: lojban@lojban.org
The
Russell's paradox has been around a long time. I really wonder why
no one has noticed that "barber shop" example is a false analogy to
the Universal Set. I won't get into the barbershop example as it would
only distract from the logic but I do have a full thread
da can not be an element of da. If da contained da it would be da.
It is actually inconcievable that da is an Element of da. It is concievable that da is da.
where ny = all natural numbers.
- say we have a set da du(=) { ny }
- now we have a set de du(=) { ny , { ny } }
- da
is an element of de. Note da is not equal to de here. We can tell as
they have a different number of elements as well as transfinite cardinality. Note the { } are necessary to
add the piece of information distinguishing this as a set rather than
an arbitrary list of unrelated numbers.
- da is da, but da is not an Element of da.
I
just can't concieve of da being an element of da. Maybe there is a flaw
to my reasoning. Please reveal to me my flaws so I can become
greater..ui.ai.a'u(happy intent interest)
note .o = IFF = <-->
Let Φ(x) be any formula of
first order logic
in which x is a free variable.
Definition. The collection a, denoted {x : x∈a .o Φ(x)}, is the individual a satisfying ∀x [x∈a .o Φ(x)].
to which
Russel says:
∀x [x∈da .o x∉x]
da∈da .o da∉da
So instead of having da∈da -- as it is inconcievable. We would have just da = da.
da = da .o da∉da
da .o da∉da
da∉da is actually a fundamental truth.
As we have just shown da can not concievable be an element of da.
So da∉da can be subsititutied with JETnu which means True.
da .o JETnu