Re: Help. What is a model?

On Sun, 26 Jul 2009 06:32:51 -0700 (PDT), T H Ray <thray123@xxxxxxx>
wrote:

On Jul 26, 8:14 am, David C. Ullrich <dullr...@xxxxxxxxxxx> wrote:
On Sat, 25 Jul 2009 12:09:59 -0700 (PDT), Tom <thray...@xxxxxxx>
wrote:



On Jul 25, 6:51 am, David C. Ullrich <dullr...@xxxxxxxxxxx> wrote:
On Fri, 24 Jul 2009 08:34:44 EDT, "T.H. Ray" <thray...@xxxxxxx> wrote:
David Ullrich writes

On Thu, 23 Jul 2009 10:52:59 EDT, "T.H. Ray"
[...]

But we digress. The question is the following -
it's not clear to me whether you're still confused
about this. Consider (i) and (ii):

(i) The definition of "theorem" is "true mathematical statement".

(ii) Something is a theorem if and only if it is a true mathematical
statement.

The question is whether you agree that (ii) follows from (i).

Clearly not!  The condition ii is necessary but not
sufficient for the definition.  I am weary of being
called "confused."

That's too bad. But insisting that (ii) does not follow from
(i) shows that you're very confused about something.
Possibly the definition of "definition", or maybe the
problem is with the meaning of "A follows from B",
hard to say. But it's something very basic, and you
have it simply wrong.

So you say. I have no deficiency in understanding that A
follows from B. It is you who fail to comprehend that your
formulation of A and B is an empty tautology that amounts
to "A theorem is true IFF a theorem." That is not a strong
enough condition to define a theorem. It does not follow from
"A theorem is true IFF proven," that all true mathematical
statements are theorems.

I have not said you are wrong--simply that you fall short of
the definition of theorem. Reread Nathanson's essay on proof
and tell me where you think he is wrong.

Huh? This makes no sense. It seems to be some sort of
comment on my position regarding what a theorem is.
But I've said nothing at all about what a theorem is -
the only thing I've commented on is your bizarre
claim that (i) does not imply (ii).

The possibility that the problem is with the meaning
of "A follows from B" just occurred to me, when you
said that (ii) does not follow from (i) and _also_
said that (ii) is necessary but not sufficient for (i).
If A is necessary for B then A follows from B.

Can you parse "neceessary but not sufficient?"
Statement ii should be "something is a theorem IFF
proven true." As I have said consistently, proof
conditions theorem.

If you want to say something is a theorem IFF proven
true that's fine with me. I'm not saying I agree, but
it's not a simply incoherent position.

But IF you insist that something is a theorem IFF
proven true then you cannot also say that the
definition of "theorem" is "true mathematical
statement".

Tom

David C. Ullrich

"Understanding Godel isn't about following his formal proof.
That would make a mockery of everything Godel was up to."
(John Jones, "My talk about Godel to the post-grads."
in sci.logic.)

David C. Ullrich

"Understanding Godel isn't about following his formal proof.
That would make a mockery of everything Godel was up to."
(John Jones, "My talk about Godel to the post-grads."
in sci.logic.)
.

Relevant Pages

  • Re: Help. What is a model?
    ... Statement ii should be "something is a theorem IFF ... proven true then you cannot also say that the ... "Understanding Godel isn't about following his ... "Understanding Godel isn't about following his formal ...
    (sci.math)
  • Re: not anymore ?
    ... David C. Ullrich ... "Understanding Godel isn't about following his formal ... "Understanding Godel isn't about following his formal proof. ...
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  • Re: Intuitionistic logic
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  • Re: My talk about Godel to the post-grads.
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