Math Forum / Mathematics / Mathematical Logic / August 2005

.999... = 1. As famous as this theorem is, I haven't been able to find
a rigorous proof (one that does not use infinite strings in additive
columns as if the strings were finite) and I haven't been able to prove
it for myself. I'd like to have a proof that:

Daryl wrote:

HAH!   Very cute.  (TF is clearly no constructivist in this sense!)

In the web-based Stanford Encyclopedia of Philosophy there is a section
  second-order and higher-order (Herbert B. Enderton)
but no clickable link!  Will there be a link to a web-based article
soon?

I'm starting a new thread because the other one got
out of control.
I want to thank those of you who gave me some useful
feedback on my article (you know who you are :-).

In chapter 2 of Shoenfield's "Mathematical Logic", there is the following
exercise:
5.  Let T be the theory with no nonlogical symbols and no nonlogical axioms.
a)  Show that ~~(x = x) OR ~(x = x) is a theorem of T not provable without

It's often said that:
(1.a) If GC [the arithmetic Goldbach Conjecture] is true, we probably
      would not be able to prove it in [1st order] PA.
(1.b) If GC is false, then for sure it's provable in PA.

I have several questions concerning how logicians feel about "friendly
premises" -- who discusses their use, what is the correct terminology,
etc.  I've read a great deal in books on logic for the layman but
haven't come across anything that corresponds to this issue.

Can someone explain to me what completeness and soundness are ?
Thank you billion times.

Let S = R x [0,1] where R is the real numbers.
Define +,*: S x S -> S
as
(x,p) + (y,q) = (x+y,(p+q)/2)

This must surely be a really simple question; but I don't yet "click"
well enough with constructive math to know the answer, so I ask.
It is a trivial theorem of orthodox math that
  "any sequence which is a permutation of the harmonic sequence,

My oh my, I feel in a creative mood this morning, having invented
p-adic midpoints I want to create something of a game to play. I was
thinking recently how airheadedness are crossword puzzles and that we
need a game that is educational and not so vapid and empty as a

In the "nonfirstorderizability" thread, the following comments
were made about "branching quantifiers", a topic that resurrects
every so often here.        [apologies for losing the attributions]
______________________

Many thanks to all those who have contributed to this thread,
especially Keith Ramsay, for his lengthy and helpful essays.
Keith inquires:

alt: Life's simple, why make it complicated?
alt: Life's simple, don't make it complicated

Much is made of educational failings in the United States, and efforts made
and self- congratulations passed all around about those efforts to improve
American education.
But how many States require, and how many "educational reform" efforts