Date: Nov 22, 2012 3:54 PM
Subject: Re: Matheology § 154: Consistency Proof!
On 22 Nov., 20:22, William Hughes <wpihug...@gmail.com> wrote:
> Note that I was able to handle your
> "simple" case using induction.
> I consider the following case easy.
> If you disagree maybe you can say
> Consider the sequence of real numbers
> The limit is oo (unbounded)
> According to set theory, the number of 1's in the limit
> is 0. (The limit of the set of positions at which we
> have a 1 is the empty set).
Why should the 1 disappear completely? But let's assume it.
According to analysis the number of 1's in the limit is 1 and the
number of zeros left to the point is infinite. Proof by failure: Try
to establish in analysis the limit oo without 1 by the sequence
> Your contention: This is a contradiction. You cannot get oo
> with only 0's
My contention is same as before: Obviously set theory cannot reproduce
the results of analysis.
> My contention: The two limits are different and there is
> no contradiction.
You try to justify one error by another one. Set theory cannot
reproduce mathematics. That fact is not mended by constructing another
failure. But you see it better here
> > 01.
> > 0.1
> > 010.1
> > 01.01
> > 0101.01
> > 010.101
> > 01010.101
> > 0101.0101
> > ...
where set theory yields no digit and analysis yields inifinitely many
digits left to the point.
Taking the result of set theory seriously, we get a limit less than 1.
And in your example we can get rid of a 1 and have a limit oo with
only zeros. Do you think to defend set theory by that arguing? Should
really somebody take it seriously? And apply it anywhere? I don't