Re: Godel proved maths inconsistent not incompleteness theorem

On Mar 26, 4:55 pm, Chris Menzel <cmen...@xxxxxxxxxxxxxxxxxxxx> wrote:
On Mon, 24 Mar 2008 23:26:43 -0700 (PDT), Charlie-Boo
<shymath...@xxxxxxxxx> said:





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.

Hm, so you think leaving certain details to the reader implies that the
proof doesn't use ZF?

Yes, the entire proof is supposed to be in ZF. Any part of it would
be part of a formal proof in ZF.

They have not substantiated your claim that the proof can be formally
carried out using only the ZF axioms.

 What an odd thing to think.  But fine.  Have a
look at any of the hundreds of theorems that are proved in detail in
that book.  Try, e.g., the definition by recursion on well-ordered
classes on page 40.  Or the Baire Category Theorem on paper 212.  Or the
Stone Rep Theorem as indicated:

Your other references were bogus. This is a shell game - you keep
giving bogus references. At what point do I stop and simply call you
a liar who wastes people's time?

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

You mean the proof is left to the reader.  So you were not able to fill
in the details of the proofs of even these elementary lemmas?  Small
wonder CBL is such a mess.

and the third is proven informally without any reference to ZF.

Because it is a simple proof by induction -- whose validity in ZF is
proven earlier in the text in the section on recursion.  You didn't
realize this?  

It does not show this in ZF. The claim is false.

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.

Sorry, what I claimed is easily verified through the examples given.

Where are the lines of the formal ZF proof? My ARXIV paper contains
several examples of formal proofs - axioms, rules, theorems - every
line is a premise, or is justified as being one of the axioms or is
produced by a rule from earlier lines.

You don't show this for ZF. You have not substantiated your claim.

C-B

Sad you seem unable to do so, even at such an elementary level.- Hide quoted text -

- Show quoted text -

.

Relevant Pages

  • Re: primitive recursive: obsolete?
    ... we have to add more axioms. ... all true sentences of arithmetic in the language of PA. ... logic we define the set of sentences true in the standard model of PA ... theorems" or not has no bearing on the fact that what I said is ...
    (sci.logic)
  • Re: Godel proved maths inconsistent not incompleteness theorem
    ... from ZF + other axioms. ... are all theorems of pure ZF or ZFC. ... language, in this case, of course, the language of set theory. ...
    (sci.logic)
  • Re: Godel proved maths inconsistent not incompleteness theorem
    ... from ZF + other axioms. ... are all theorems of pure ZF or ZFC. ... language, in this case, of course, the language of set theory. ...
    (sci.logic)
  • Re: proof of undecidability of halting problem
    ... I said theorems, not theories. ... If you have "fewer axioms" and e.g. PA with no redundant (otherwise ... We're making use of all the axioms of Z set theory. ... Turing Machines) and the single set of the recursive functions or r.e. ...
    (sci.logic)
  • Re: proof of undecidability of halting problem
    ... I said theorems, not theories. ... If you have "fewer axioms" and e.g. PA with no redundant (otherwise ... We're making use of all the axioms of Z set theory. ... Turing Machines) and the single set of the recursive functions or r.e. ...
    (sci.logic)