On 1/24/2015 1:23 AM, Jason Resch wrote:
On Fri, Jan 23, 2015 at 10:15 PM, meekerdb <[email protected]
<mailto:[email protected]>> wrote:
On 1/23/2015 8:16 AM, Jason Resch wrote:
On Friday, January 23, 2015, John Clark <[email protected]
<mailto:[email protected]>> wrote:
>
> On Thu, Jan 22, 2015 at 4:46 PM, Jason Resch <[email protected]
<mailto:[email protected]>> wrote
>
>>> >> Do we know that? Do we know that such a digit exists?
>>
>> > It follows from the axioms that there is a certain definite digit.
>
> They show you how to generate terms in a sequence and if you add up
enough of
them you'd get the the 10^(10^(10^100))th decimal digit of pi; but it
assumes that
there is no barrier that makes doing that impossible and states that
assumption
with 3 little dots (...). I don't know for certain but those 3 little dots
*might*
be saying something that is logical nonsense, I do know for certain that
the first
mathematicians who used those 3 little dots knew nothing about quantum
mechanics or
the computational limit of the universe, and that gives me pause.
>
What I explained is that if you think there is a largest number *that* is
what
definitely leads to logical nonsense.
Say there is a largest number N, such that N+1 is not a bigger number, but
is still
N. That means N+0 = N+1. Now subtract N from both sides.
Why should subtraction of the biggest number not obey special rules, e.g.
Subtracting N from any normal number yields -N. Subtracting N from any Big
number
yields zero.
I still don't know if that would escape the problems. Let's say M is the number right
before the biggest number (or any "Big numbers"). Then using your rules you find that:
M - (M+1) = 0 --- or is it -N?
but
(M - M) + 1 = 1
It might be possible to come up with axioms that allow you to have a biggest number that
operates in a consistent way, but I think it would be very difficult, and probably not
very useful.
It may be useful in computer science, c.f.
http://www.impan.pl/~kz/files/MKKZ_ArFM.pdf
Nor do I see the point of hobbling a theory (supposedly about the infinite natural
numbers) by declaring some aspects of that theory to be strictly off limits and beyond
the possibility of discussion.
I'm not proposing that anything is off limits. I'm proposing that there are different
axiomatic systems which may equally correspond with our finite observations, therefore it
is not justified to pick one of them and claim that it is empirically proven. Axiomatic
systems may contradict one another and that is why "X provable in A1" only implies "X true
relative to A1" and not "exists X". Plato was led to suppose that perfect forms existed
because in his day it seemed there was only one possible mathematics which was something
like arithmetic plus Euclid.
Brent
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