Re: TM Tape is Always Finite

From: Kent Paul Dolan (xanthian_at_well.com)
Date: 01/07/04 

Date: Wed, 7 Jan 2004 02:27:42 +0000 (UTC)

"Russell Easterly" <logiclab@comcast.net> wrote:

> I will the the first to admit that the output of TM2 is
> paradoxical.

No, it isn't the least bit paradoxical. The paradox is only
in your faulty mental model of what it means for some
process to be repeated an infinite number of times; once you
get that fixed, everything else becomes simple. The problem
is that so far you have refused to face that your mental
model is plain _wrong_, trying to work it back into the
argument in a number of different disguises. It is easy to
tell all by yourself, without me ranting at you, when you
have done this: you again get to the _wrong answer_. Since
your answer is well known to be wrong, when you arrive at it
via another route, it is _your_ responsibility to confront
that you have once again done something wrong, and go find
the place where you have let your erroneous model intrude
once more to lead you astray. It is not appropriate for you
to come back here to insist that because of your newly
incorrect formulation, what was false is somehow suddenly
magically made true. Math doesn't work that way.

> There can't be a 0 at the end of the tape,

A right-infinite tape doesn't _have_ an "other end", so your
statement is _meaningless_. This is part of your faulty
mental model of infinity, and has tripped you up at least
half a dozen times so far in this thread. Making arguments
using meaningless concepts won't steer you away from your
wrong answer.

Logicians tell us "from false premises may any answer
whatsoever be deduced", so larding your arguments with false
premises like "infinite lists have last members" isn't
helping you learn to think.

> yet there must be at least one 0 somewhere on the tape.

That is not true because it is not a complete description.
You are trying to describe a dynamic process as if it were a
terminated process.

There need only be a 0 after the current 0 for the current 0
to be overwritten, but that is a _transient_ condition, not
one that holds for that zero for all time. At some later
time it is some later 0, and still later, still a later
zero, but never is it a _last_ zero; with an infinite number
to be overwritten, one is never overwriting either the last
zero, or the next to the last zero. Each and every zero
gets overwritten, thus it is a contradiction in terms to say
there is somehow "one left" (nonsense) when the process is
"finished" (meaningless). _Any_ specific zero (trivially
obviously) gets overwritten with a 1, therefore by
induction, all of them do.

> Other posters have suggested looking at the "limit" of the
> output tapes.

I haven't. I've suggested you realize that process is all
you have available for consideration, not some final
product. You consistently confuse "in the limit" with
"after the process is done"; they aren't related, and with
that confusion, you should avoid any path that goes past an
"end point" of the calculation. You don't need it to solve
the problem.

> If we give TM1 a finite string of 0's it will output a
> finite string of 1's. TM2 will output a finite string of
> 1's followed by exactly one 0.

That is not the problem you posed, and is unworthy of
consideration. It feeds your false mental model, but
provides nothing of value to make up for misleading you.
Put it down, don't pick it up again, even if you dress it
in a new suit of clothes.

> We can say that TM1 will produce an infinite string of 1's
> "in the limit". But, the 0 at the "end"

There is no "end"; by your own statement of the problem,
TM2's tape contains an infinite number of zeros. An
infinite list doesn't _have_ an end. That's what it _means_
to be infinite, after all: "without end". Any chain of
thinking you do that implicitly or explicitly contains that
"end" is pure nonsense.

> of TM2's tape doesn't go away "in the limit".

There is no "limit", an infinite process (at least at a
steady pace) goes on "forever", and you cannot step _past_
"forever" to look at what comes after.

> The limit of TM2's tape is still finite.

It didn't even start out finite, much less does it end that
way. You said:

: Assume we give TM2 a tape that contains
: an infinite string of 0's.

That isn't some triviality, that is _the definition of the
problem_, and considering any other situation, such as
starting with a finite string of zeros, is _meaningless_
in finding the answer to _this_ problem. Stop doing that,
it is annoying and wastes time and patience.

> If TM1 can write an "infinite" number of 1's then we have
> to assume that TM2 can as well.

Unamazingly, that is the conclusion any competent
mathematician _does_ reach: TM2 will write an infinite
stream of ones; at the point where it writes any specific 1,
it falls farther and farther behind TM1 writing a 1 at the
corresponding spot on the other tape, but that doesn't
matter, because TM2 _also_ has an infinite number of steps
available, and will sooner or later write a 1 in each spot
corresponding to a spot TM1 wrote a 1. In fact, you should
be able with a fairly simple calculation, given the step N
at which TM1 wrote that 1, to determine exactly the step M
at which TM2 writes the corresponding 1; since you can do
that for any N, TM2 must write a 1 everywhere that TM1
writes a corresponding 1. Do that exercise, perhaps you can
use it to talk yourself out of your delusions.

Please! I've already asked this at least twice:

1) Do not answer _at all_ until you understand how this
works, you have been told many times by many correspondents
exactly where your errors occur, and you have ignored each
and every such input, simple repeating your stale arguments
in new words, never confronting that if the previous
argument was demolished by an argument that _proves_ the
exact opposite of what you maintain, that eliminates all
chances that you can somehow derive that false conclusion
correctly with a newly worded argument.

That way lies insanity.

2) Do answer (only) when you at long last catch a clue.

As you correctly stated, you aren't very good at this math
stuff. Eventually you will learn, as James Harris probably
never will, that a math argument isn't won with debating
skills, but only with math skills, and that merely talking
your opponents into a bored stupor doesn't give you a win,
it just leaves you in firm possession of an erroneous
opinion, to the benefit of no one, and the amusement of
everyone but you.

xanthian.

You might want to consider a hobby where your skills are a
match for what you want to do. Just a thought. I'm not any
good at math (any more, I was once quite competent), so I
try to stick to arguing people out of errors in the limited
parts of math I understand, as a hobby, and don't waste my
time trying to do innovative research, completely beyond my
skills. 

-- 
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