Re: an important set theory post

On Sat, 26 Jul 2008 08:08:18 -0700 (PDT), calvin wrote:
On Jul 26, 10:37 am, Dave Seaman <dsea...@xxxxxxxxxxxx> wrote:
On Sat, 26 Jul 2008 07:11:40 -0700 (PDT), calvin wrote:
On Jul 26, 8:43 am, David C. Ullrich <dullr...@xxxxxxxxxxx> wrote:
On Fri, 25 Jul 2008 06:33:52 -0700 (PDT), calvin
So, omega+1 is merely notation for something that
has a clear meaning?

Uh, yes. The clear meaning has been clearly explained
to you here - omega + 1 is the union of omega and
{omega}.
The 'union of omega and {omega}' means something, but
'omega + 1' doesn't (except as notation), because adding
1 to any infinity yields the same infinity.

By the same token, '2 + 2' doesn't mean anything (except as notation).

If so, then it would seem to me
to be in the same category as the notation, 2^aleph_0,
which stands for 'the cardinality of the set of all
subsets of a set that has cardinality aleph_0.'
Uh, yes, it's in the same category in that both
are _definitions_. What's your point?
They're not definitions, it seems to me, but only
notations.

So '@!$*+(#' is "in the same category" as 'omega+1' and '2^aleph_0'?

Infinite powers aren't defined, so 2^aleph_0 means
nothing (except as notation).  

Wrong.  If A and B are cardinals, A^B means (the cardinality of) the set
of all functions mapping B into A.  The definition doesn't say anything
about whether A and B are finite or infinite, because it doesn't need to.

If one didn't already
know that the power set of a particular set is
the set of all subsets of that set, this notation
would be useless.  
Similarly, 'omega + 1' would be
useless if one didn't already know that what was
meant was the result of the successor function
applied to omega.

Can you give an example of a notation that is useful in mathematics, even
though it has no definition?  (I can think of exactly one such notation.)

Any notation that is not misleading is useful, it
seems to me, though I don't know what you mean by
"even though it has no definition".

I was responding to your statement that 'omega + 1' would be useless if
one didn't already know the definition. Of *course* you need to know the
definition; that's the whole point. If things didn't have definitions,
nothing would make any sense. There is one tiny exception, because you
can't quite define *everything* without being circular.

Look, "omega + 1" can't harm me anymore, and "2^aleph_0" never
did, but someone encountering these notations for the first
time, in a usenet thread which may or may not be dominated
by cranks, wouldn't know whether they were gibberish or not.
Your "2 + 2" analogy is not an analogy at all. Those are
simply well-defined symbols that everyone is expected to
understand, before whatever discussion is underway. Nor is
"@!$*+(#" an apt analogy, because that string of symbols does
not even hint at having a meaning (other than a traditional
substitution for profanity).

Every term in mathematics (except one) has a definition. The one
exception is the symbol for "is an element of" from set theory. Any time
you look at a mathematical statement or expression, the first thing you
need to ask yourself is whether you know the definitions of the terms
used. An expression such as 'omega + 1' should make you ask "what does
that mean?" instead of simply claiming that it must be nonsense.

You may have noticed that in the crank threads, the cranks are constantly
being challenged by everyone else to define their terms, which they
seldom do. That's one of the ways to identify a crank. But even in
situations where a known crank says something that looks strange, people
often respond by asking for a clarification rather than just dismissing
the statement outright. The outright dismissals usually arise in
situations where the crank resists requests for clarification and simply
repeats nonsense.

If you look back over the history of this thread, you will find that your
approach has been just the opposite; you make announcements in ignorance
of the definitions. Why didn't you think of asking first?


--
Dave Seaman
Third Circuit ignores precedent in Mumia Abu-Jamal ruling.
<http://www.indybay.org/newsitems/2008/03/29/18489281.php>
.

Relevant Pages

  • Re: an important set theory post
    ... The clear meaning has been clearly explained ... to you here - omega + 1 is the union of omega and ... The 'union of omega and ' means something, ... The notation omega + 1 _does_ mean something. ...
    (sci.math)
  • Re: Cantor Confusion
    ... That definition nowhere implies that "omega" is reached, ... Not more legitimate than asking for fbecause at noon also ... So the transformation does not simplify notation, ... What alleged error it that? ...
    (sci.math)
  • Re: an important set theory post
    ... to you here - omega + 1 is the union of omega and ... The 'union of omega and ' means something, ... which stands for 'the cardinality of the set of all ... nothing (except as notation). ...
    (sci.math)
  • Re: an important set theory post
    ... to you here - omega + 1 is the union of omega and ... which stands for 'the cardinality of the set of all ... nothing (except as notation). ... with the meaning of the notation "2 + 2". ...
    (sci.math)
  • Re: an important set theory post
    ... The clear meaning has been clearly explained ... to you here - omega + 1 is the union of omega and ... which stands for 'the cardinality of the set of all ... nothing (except as notation). ...
    (sci.math)
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