Re: Godel proved maths inconsistent not incompleteness theorem

On Fri, 14 Mar 2008 02:16:50 -0700 (PDT), Charlie-Boo
<shymathguy@xxxxxxxxx> said:
...
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".

This is all stupid ***.

I confess I don't see why. I simply pointed out that you are using the
names of the infinite cardinals completely incorrectly. How do you
expect to communicate your ideas to others if you are using words that
have universally understood meanings to mean something else entirely?

I just said that when you are examining issues regarding infinite
sets, there are situations where all finite sets are treated the same.

A perfectly reasonable point. My point was simply that the word you
were using to make your point was the wrong one. You used "aleph-0" as
a synonym for "finite". Since aleph-0 is the first infinite cardinal,
it will surely hinder your ability to make points like the one above if
you continue to use it incorrectly. You're welcome.

We never went into any detail about what I was talking about

I went into great detail in the remainder of the post. You even
included it in this reply, but you didn't respond to it. I give you
another chance below.

and you waste time trying to compare it to the cardinality of finite
sets,

That's certainly an odd take on what I was doing. (I'm not even sure
what that take is, frankly.)

which has no place in what I was talking about.

I myself find it important that my interlocutors and I agree on the
meanings of the words we use. Don't you?

Work on showing specific results rather than a feeble attempt at
character assassination.

Ah, so in addition to "aleph-0" = "finite", we also have "pointing out
mistakes" = "character assassination". I'll try to bear that in mind.
Meanwhile, the remainder of the post addressed several specific issues
you'd raised in connection with Russell's paradox. You are hostage to a
number of serious confusions that I have tried to sort out for you. You
might want to give it a read. I've edited so as to leave only
substantive discussion.

-cm

*****

....
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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