
Re: ZFC and God
Posted:
Jan 23, 2013 11:39 AM


WM <mueckenh@rz.fhaugsburg.de> writes:
> On 23 Jan., 14:36, "Jesse F. Hughes" <je...@phiwumbda.org> wrote: >> WM <mueck...@rz.fhaugsburg.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 webpublished 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"

