Re: Godel proved maths inconsistent not incompleteness theorem

On Apr 16, 6:03 pm, Charlie-Boo <shymath...@xxxxxxxxx> wrote:
On Apr 16, 7:24 pm, MoeBlee <jazzm...@xxxxxxxxxxx> wrote:



On Apr 16, 3:07 pm, Charlie-Boo <shymath...@xxxxxxxxx> wrote:

On Apr 16, 3:42 pm, MoeBlee <jazzm...@xxxxxxxxxxx> wrote:
On Apr 16, 12:19 pm, Charlie-Boo <shymath...@xxxxxxxxx> wrote:

On Apr 16, 1:55 pm, MoeBlee <jazzm...@xxxxxxxxxxx> wrote:
On Apr 15, 6:28 pm, Charlie-Boo <shymath...@xxxxxxxxx> wrote:

On Apr 15, 6:07 pm, MoeBlee <jazzm...@xxxxxxxxxxx> wrote:
The claim that virtually all of ordinary mathematics can be formulated
in ZFC is not itself a formal mathematical claim. I've read a modest
amount of set theory, and then introductory abstract algebra,
topology, analysis, graph theory, and computability theory to see how
it can be formulated in ZFC.

Oh BS. Nobody has published a formalization of computability theory
proofs (though trivial in CBL.)

AGAIN you change the terms of the discussion. I said that the
mathematics can be formulated; I didn't say that someone belabored the
matter by printing out every single formula in the pure language of Z
set theory.

If you can formally prove prove any computability theorem using ZFC,
then no matter how big or complex it is, you could give a simple
(short) proof as an example, and give a detailed explanation of how
the big ones work. Saying it's too big to give doesn't make it.

At the level of computability theory, there probably ARE NO simple
examples, since by the time we reach that level we will have defined
dozens and dozens of terms to get to the defined terms 'recursive'

You say that you know it can be done because you know how to do it.
Then explain how you would define a set being recursive and a set
being r.e.

What do I get for regurgitating for you ordinary mathematical
definitions? And such definitions are in your Boolos book. Oh well,
since I just love typing as a command performance for Prince Charlie-
Boo Himself: Ordinary definition of a set S being recursive is that
that the characteristic function of S is recursive (and you know what
a recursive function is, and what a characteristic function is,
right?). Ordinary definition of a set S being recursively enumerable
is that S is the range of a recursive function (and you know what a
range is, right?)

Then by what logic (rules) would you prove the theorem?

First order predicate logic with identity. How many times do you need
to be told that?!

This entire post is talking about definitions
(including a circular
definition of recursive)
and how proofs work in general.

You ASKED for something about those definitions! Sheesh!

And no, not circular. First we define 'is a recursive function' then
we define simply 'is recursive'. The definition of 'is a recursive
function' is by the ordinary inductive definition. And previously we
will have defined 'characteristic function'. THEN we define simply 'S
is recursive' by 'S is recursive iff the characteristic function of S
is a recursive function'. This is first semester computability. Gawd,
what an ignoramus you are.

And you mentioned before that all you were asking about is the general
structure (or whatever word you used), so I gave you that. Sheesh!

We all know
that. You are not saying a word about how you would accomplish this
proof in particular.

You're a liar. "Not a word". You even quoted what I said about this
proof in particular.

What is the proof that you would formalize? (By this I mean proof in
the ordinary sense of that found in a Mathematics book.)

I told you! The one in the Boolos book! Are you really so lacking in
basic reading skills?

What rule
allows you to conclude that some set is recursive - how would that
work?

Ultimately, the rules would be universal instantiation then modus
ponens, if we wish to be so pedantic. From a "for all" definition, by
universal instantiation we get "S is [fill in definiens of 'recursive'
here] -> S is recursive. And we prove "S is [fill in definiens of
recursive here]". And we conclude, by modus ponens, "S is recursive".
(Then we use universal generalization to conclude for all S.)

What is the formal representation of the theorem?

Depending on the specifics of our formal language and of the
formalization, it could look something like this,

Ax((Mx & MCx) <-> Rx)

Where we previously had defined the 1-place predicate 'M' (for
recursively enumerable) and the 1-function symbol 'C' (for relative
complement among n-tuples). But that's a bit of a gloss since there'd
also be clauses mentioning that we're talking about natural numbers or
n-tuples of them or finite sequences of them, blah blah, technical
details boring to type out in a post to you but easily accomplished.

These are questions concerning the particulars of the proof. Can you
say anything about that - how this proof would actually work?

Your turn. Just take the definitions and the outline I gave you below.
And see how to do the Boolos proof in such a context. It's really
mundane. Please take my unwillingness to do it for you not as an
indication that it can't be done but rather as a reflection of the
fact that I've already jumped through enough of these silly hoops for
you while YOU are the one, not me, who needs to learn this basic
stuff.

You have to have a real proof that you are formalizing. What is the
real proof?

Depending on the specific formulation it's either a Hilbert style
sequence of formulas, or an ND tree, or an ND sequence of pairs of
formulas and sets of "charged formulas", or sequents, etc.

I am not talking about tedious complex details. I am asking what the
principles are by which you can define these concepts, what the
intuitive (real) proof being formalized is, and what rules are used to
derive that theorem.

I've already told you a hundred times already! The terms are defined
in the language of set theory, such as we move through a sequence of
definitions, from 'pair' to 'ordered pair' to 'relation' to 'domain'
and 'range' to 'function' to 'natural number' to 'primitive recursive
function' to 'recursive function' and 'characteristic function'. And
the rules are as little as just modus ponens (in a suitable Hilbert
style) or as many as a bunch of natural deduction rules (plus derived
rules too, if we like). And the intuitive argument that we're
formalizing is just as in the Boolos book or some other ordinary
variation of it.

And in a suitable natural deduction system, the structure would be
something like this:

1. S is recursive
...

n. S is recursively enumerable & the complement of S is recursively
enumerable

n+1. S is recursive -> (S is recursively enumerable & the complement
of S is recursively enumerable)

n+2. S is recursively enumerable & the complement of S is recursively
enumerable.

...

n+m+2. S is recursive

n+m+3. (S is recursively enumerable & the complement of S is
recursively enumerable) -> S is recursive.

n+m+4. S is recursive <-> (S is recursively enumerable & the
complement of S is recursively enumerable)

n+m+5. AS(S is recursive <-> (S is recursively enumerable & the
complement of S is recursively enumerable))

Now PLEASE just learn this basic stuff YOURSELF!

If you need help along the way, then there are lots of people who'll
help you out; but you have to at least read the books and pay enough
attention to let SOMETHING into your brain.

MoeBlee
.

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