Re: Godel proved maths inconsistent not incompleteness theorem

On Thu, 13 Mar 2008 10:44:50 -0700 (PDT), Charlie-Boo
<shymathguy@xxxxxxxxx> said:
On Mar 12, 11:36 pm, Chris Menzel <cmen...@xxxxxxxxxxxxxxxxxxxx>
wrote:
On Wed, 12 Mar 2008 19:55:01 -0700 (PDT), Charlie-Boo
<shymath...@xxxxxxxxx> said:

...
Wffs refer to relations (sets.)  Can they refer to non-sets there?
Wffs contain operations on relations.  Can those operations be done on
nonrelations?

The problem was they figured every wff defined a set based on the wff
being true when a value is substituted for a free variable.  Then they
thought of {x|~(x e x)} and there was a wff that was not a set.  So
they said, ok, a wff is not always a set.

My god.  What must it be like to be so profoundly muddle-headed?  I
really do feel badly for you.

But there are only aleph-1 wffs

Goodness me.  (aleph_1 is uncountable.  You probably mean aleph_0.)

and aleph-2 sets of numbers alone,

I give a cardinality of aleph-0 to finite sets,

So, in your mouth, "aleph-0" means "finite". Bizarre. It would make
just about as much sense to use, say, "seventeen" for that purpose.
"aleph-0" has a fixed and longstanding meaning in mathematics and it is
perverse, or clueless, in the extreme to decide to use it to mean
something entirely different.

but whatever numbers you want is fine with me -

What numbers I want has nothing to do with it. The issue is your use of
"aleph-0" to mean something other than its universally accepted meaning
in modern mathematics.

so there are plent more sets than wffs and we can say that there is in
fact a set for every wff.  It is counter-intuitive to say sets do not
include what all wffs define.

In any case, they set out to define what any set can define, when the
real problem was they needed to be exact about terms such as wff.

If you first decide if a wff can make a reference to something that
is not a set, then if yes, then naturally these wffs are not sets,
and if no, then the {x|~(x e x)} example is NOT a set and we still
have each wff defining a set. So we don't really need to worry
about what's in a set as we do in ZF. We just need to be careful
about the formal system - what is the real syntax of a wff? Can it
contain references to nonsets?

The depth of your confusion is truly great. In this light your
profound ignorance of the history and content of set theory is more
understandable. I will read your posts with more compassion in the
future.

You haven't answered the question either (the source of the problem
with the Russell Paradox.)

The generally acknowledged source is implicit in ZF -- though you would
have to understand elementary ZF to appreciate the point. The source --
in the sense of the principle most central to its derivation -- is the
unrestricted principle of comprehension: That, for any formula "F(x)" in
the language of set theory with "x" free there is a set y such that, for
all x, x in y iff F(x). This principle was replaced by the schema of
separation, which does not permit one to prove the existence of sets ex
nihilo.

A wff can reference relations (sets) e.g. (all x)P(x) refers to
relation P.

What are you talking about? Let, e.g., P be "x=0", i.e., "x is the
empty set". So you are saying that "(all x)x=0" refers to the relation
"x is the empty set", i.e., the set {0}? It doesn't *refer* to anything
at all; it is simply the false statement that everything is identical to
the empty set.

May a wff contain a reference to something instead of the relation and
use a non-relation there? If f(some relation) is a wff then is
f(something not a relation) necessarily a wff as well?

Your question is ill-formed as stated. It is allegedly a question about
set theory. So formulate your question in the language of set theory.
Can you even do that?

(FWIW, on the simplest way of cashing out your question, the answer is
trivially "yes" in ZF. Let s1 be a relation (i.e., a set of n-tuples,
for some n>0). Let s2 be a set that is not a relation. Then, if f is a
wff with a free variable x, then "f(s1)" and "f(s2)" are obviously both
wffs, where they are the result of substituting "s1" and "s2" for "x" in
f, respectively. Somehow I don't think this is what you have in mind.)

When we see in a wff (all x) ... must the ... be a reference to a
relation e.g. P(x) above or can wffs contain a non-relation for the
... ?

More incoherent gibberish. When we see a wff "(all x)...", the "..." is
obviously itself a wff -- this is just a trivial fact about formal
languages. Wffs don't ever "contain" relations or non-relations, they
are syntactic entities that contain other pieces of syntax. Some of
those pieces of syntax might themselves indicate relations in the
intended semantics of the language. You appear to be badly confusing
syntax generally with semantics; or something.

The fact of the matter is, naive set theory is right,

If by naive set theory you mean the theory consisting of the axiom of
extensionality and every instance of the principle of of comprehension,
that theory is provably inconsistent. Though it does appear that you
have trouble distinguishing "inconsistent" from "right", so your claim
is understandable.

they were just confused as to how to define wffs after it was
discovered that there were things that are not sets. You are not
showing any better ability to grapple with the question, even with a
guide.

Mmm hmm. Zermelo, von Neumann, Gödel, et al were confused. And you're
not. You *might* want to rethink that.

If (all x)_(x) is a wff for something in place of the _ could that
something be other than a relation?

Really, for your own good, go educate yourself.

.

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