🚀 go-pugleaf

On 3/13/2011 5:03 PM, Transfer Principle wrote:
> On Mar 13, 12:07 pm, Jim Burns
> <burns...@osu.edu> wrote:
>> On 3/12/2011 10:31 AM, Graham Cooper wrote:

>>> SHOW ME A DIGIT ON THE DIAGONAL THAT
>>> EQUALS D AND EQUALS D+1
>>> ---------------------------------
>>> THAT is your entire argument!
>>> THAT is your proof of BIGGER THAN INFINITY!
>> That obviously cannot be done. All of us
>> "standard theoreticians" agree with you,
>> although Transfer Principle refuses to jump
>> one way or the other, as near as I can tell.
>
> If _both_sides_ actually agree on something,
> then I _don't_ disagree with the _unanimous_
> consensus. I only "refuse to jump one way or
> the other" when there's at least one _actual_
> poster on each side of the debate. I don't
> disagree with a unanimous consensus because
> I'd have _no_one_ to defend.

I am unexpectedly saddened by your admission.

Where do YOUR beliefs go in all of this?

>> The rest of the argument shows why you have to
>> be able to do this impossible thing in order to
>> list all of the real numbers -- which is why
>> it is "obvious" (after some thought) that
>> THE REAL NUMBERS CANNOT BE LISTED.
>>
>> You apparently understand all this.
>>
>> Why then are you trying to disprove a proven
>> thing?
>
> ... proven in ZFC, that is.

If you mean ONLY proven in ZFC, you are very wrong.

Perhaps you mean ONLY in standard set theories.
That is a bit harder to speak to, especially as
you may be defining "standard set theories" to
exclude those which can prove that the real
numbers cannot be listed. That wouldn't be very
sporting of you, if that were so.

The structure needed to prove that the real numbers
cannot be listed (given that they exist) is much less
than what is needed to prove that a complete, ordered
field, the real numbers, exists. Or so it seems to
me. I admit that I have not looked into the question.
If that is so, then I conjecture that any system
in which we can prove there is a list of real numbers,
we can prove that it is incomplete.

All of this talk of ZFC or other set theories is
beside the point, though, if what we are interested
in is what Cooper means. He has not formally sworn
allegiance to ZFC, but what he has used in his
attempts to DISprove Cantor's theorem (infinite
decimal representations of real numbers, and so on)
is sufficient to PROVE Cantor's theorem.

>> You might as well add to 2 to 2 over and
>> over again until you get something other than 4.
>
> Every time someone mentions "2+2=4," someone else
> usually mentions piano interval arithmetic, in
> which a major 2nd plus a major 2nd is a major 3rd.
> For some reason, those on the standard side are
> _more_ willing to accept a theory in which 2+2 is
> something other than 4 (such as 3, for pianos)
> than they would accept a theory where there are
> as many reals as naturals.

I'm sure you understand that, in piano interval
arithmetic, "2 + 2 = 4" means something else
other than what it means in "standard" arithmetic.
Why do you pretend to be ignorantly treating them
as though they were the same?

It is not very difficult to come up with a theory
where "there are as many reals as naturals", if
you're willing to misrepresent what people mean
to that extent. Just call the naturals "the reals",
and "there will be as many reals as there are
naturals". Happy now?

>> The ironic thing about all this is that
>> EVEN IF YOU DID disprove Cantor (Let's say
>> you do this in some unimaginably bizarre
>> alternate universe)
>
> ... or an unimaginably "bizarre" _theory_,
> like NFU, or zuhair's ND, and so on ...

Are you saying that Cantor's diagonal theorem
can be disproven in NFU or zuhair's ND? I
would like to see that. Obviously, given what
I have written above, I would not expect that.

Or, are you just giving examples of "bizarre"
theories? Then... <YAWN/>

>> Why not give it a rest, and go do something
>> where there is at least a chance of success,
>> like teaching pigs to fly?
>
> The existence of NFU and zuhair's ND show that
> disproving Cantor _does_ have more a chance of
> success than making pigs fly or spotting UFO's.

Well, yes, you SAY that, but SHOWING me some
mathematics (a disproof of Cantor's diagonal
theorem, for example) would so much more interesting
than more CLAIMS.