Re: Godel proved maths inconsistent not incompleteness theorem

On Mar 24, 2:14 pm, Chris Menzel <cmen...@xxxxxxxxxxxxxxxxxxxx> wrote:
On Mon, 24 Mar 2008 10:10:55 -0700 (PDT), Charlie-Boo
<shymath...@xxxxxxxxx> said:





On Mar 17, 11:23 am, Chris Menzel <cmen...@xxxxxxxxxxxxxxxxxxxx>
wrote:
On Mon, 17 Mar 2008 06:18:34 -0700 (PDT), Charlie-Boo
<shymath...@xxxxxxxxx> said:

On Mar 16, 11:01?pm, Chris Menzel <cmen...@xxxxxxxxxxxxxxxxxxxx>
wrote:
On Sat, 15 Mar 2008 01:25:35 -0700 (PDT), Charlie-Boo
<shymath...@xxxxxxxxx> said:

...
...the ZF axioms aren't used to prove anything outside of simple,
fairly obvious, statements about sets.

Ignorant codswallop. I gave you three examples (of thousands):

1. Every singular limit ordinal k with cofinality < card(k) lacks the
Souslin property.

2. The Stone Representation theorem (every Boolean Algebra is isomorphic
to a field of sets)

3. Every normal function on the ordinals has arbitrarily large fixed
points.

(Refutations welcome.)

Where "welcome" = "ignored".

You also ignored my request for a proof of any of these

You want me to *copy* the proofs from well known texts because you are
too indolent to go look them up for yourself?  I'll give you titles and
even page numbers, but you'll have to do your own homework.

using only the ZF axioms.

Oh, right, you've got this nutty idea that all significant proofs are
from ZF + other axioms. The three theorems above, and thousands of
others, are all theorems of pure ZF or ZFC. Go learn something.

Ok, help me learn.  What are the formal representations of the
theorems

As with most all mathematics, their formal representations are their
translations from mathematical English into an appropriate first-order
language, in this case, of course, the language of set theory.  Do it
yourself if for some reason you think it important to do so; sorry, but
your spoon-feeding will only go so far.  However, IIRC, there is one set
theory text whose proofs are all expressed entirely in the language of
first-order set theory, Takeuti and Zaring's Axiomatic Set Theory, where
you will find many theorems worked out in excruciatingly formal detail.

and where (page of last line of proof) are they proven?

The first two are Levy's Basic Set Theory.  There's a nice cheap Dover
edition so you've got no excuses (http://tinyurl.com/254fz3).  I'll give
you the pages the proofs *start* on, which seems to make more sense
assuming you actually want to study them:

1. p. 307ff (proof sketch -- you get to work the details yourself)

I work out the details? I thought you said that they provide the
proof using only ZF.

2. p. 257ff

For the third, I recall learning the proof from a precursor to Devlin's
The Joy of Sets.  Looking at its index on Amazon, "normal function" is
defined on page 72, so I'm sure the theorem in question is within a page
or two of that.

Same here as above: The proof uses 3 lemmas, 2 of which are not proven
and the third is proven informally without any reference to ZF.

Congratulations. None of the proofs refers to ZF and most aren't even
given. You have just proven the exact opposite of what you claimed.

C-B

Have fun.  Glad to see you're actually planning to learn some set theory
instead of pontificating about it in abject ignorance.- Hide quoted text -

- Show quoted text -

.

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