Date: Jan 23, 2013 11:39 AM Author: Jesse F. Hughes Subject: Re: ZFC and God WM <mueckenh@rz.fh-augsburg.de> writes:

> On 23 Jan., 14:36, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:

>> WM <mueck...@rz.fh-augsburg.de> writes:

>

>> > I know. But if you have read the discussion, you have seen that two

>> > matheologians claim just this. Why do they? Because they cannot answer

>> > the question: What paths are (as subsets of the set of nodes) in a

>> > Binary Tree that is the union of all its levels? Are there only the

>> > finite paths? Or are there also the infinite paths?

>> > Try to answer it, and you will see that you need the omegath level or

>> > must confess that it is impossible to distinguish both cases. Hence,

>> > Cantor's argument applies simultaneously to both or to none.

>>

>> I'm not interested in the web-published claims of two individuals on a

>> different topic than we're discussing.

>

> You are in error. Pause for a while and think it over.

I will not be distracted from the topic at hand until we've completed

the discussion.

>>

>> Once again, let me remind you what you claimed. You claimed ZF was

>> inconsistent, and in particular that ZF proves that the union

>>

>> U_n {1,...,n}

>>

>> is both finite and infinite.

>>

>> Now, we've had two competing definitions of infinite in this

>> particular discussion.

>>

>> (1) A set S is infinite if there is no natural n such that |S| = n.

>>

>> (2) A set S is infinite if it contains a number greater than every

>> natural n.

>>

>> The first definition is what mathematicians almost always mean, and

>> they *never* mean the second, but this is mere semantics. Let's talk

>> results.

>

> You are right, mathematicians prefer (1). But matheologians use (2).

> An infinite set contains a number of elements, at least aleph_0, which

> is greater than every finite number.

You seem to have changed claims here.

Which of the following do you claim ZF proves?

(1) The set U_n {1,...,n} contains an element k greater than every

finite number.

(2) The set U_n {1,...,n} has cardinality greater than every finite

number.

The second is not controversial, and the first is unproved.

>>

>> We both agree that, using definition (1), the above union is infinite

>> and (I think) we agree that we cannot show it is finite (=not

>> infinite). If I'm mistaken on this point, then please show me.

>>

>> On the other hand we both agree that, per definition (2), the union is

>> "finite", but I have seen no contradiction result, since you have not

>> shown that the union is "infinite" in this sense. Nor can you find a

>> single publication in which a mathematician has claimed the union

>> above (i.e., the set N of natural numbers) contains an element larger

>> than every natural.

>

> You confuse the things. ZF claimes that the *number of elements* is

> larger than every finite number. Just this causes the contradiction. A

> union of finite initial segments cannot have a number of elements that

> is larger than every finite number.

Perhaps I misunderstood you then, but let's work with this. Here's

the new contradiction you claim ZF proves, correct?

(1) |U_n {1,...,n}| > k for every natural number k.

(2) |U_n {1,...,n}| is not greater than every natural number k.

More precisely, when we say "the number of elements of S is k", we

mean nothing more or less than "|S| = k", right? And thus we are back

to the claim that ZF proves both of the following statements:

(1) (Ak in N)(|U_n {1,...,n}| > k)

(2) NOT (Ak in N)(|U_n {1,...,n}| > k)

We may take the proof of (1) for granted. I will ask once again for a

proof of (2). I don't want to talk about paths. I don't want to

discuss anything aside from a proof of (2). Please show me that

argument or explain to me where I misinterpreted your claim and let's

move on with this discussion.

--

Jesse F. Hughes

Playin' dismal hollers for abysmal dollars,

Those were the days, best I can recall.

-- Austin Lounge Lizards, "Rocky Byways"