Infinite Cardinals and Pseudo-Mathematics

[Moshe Zadka]

[A Parody of TomC's Post about Quoting Strategies]

The following message will be posted periodically until observed
clue-levels in these parts improve, or until the heat death of the
Universe arrives.

EXECUTIVE SUMMARY: 
    To send more accurate messages about infinities, please try to 
    make sure you do not make a mess out of the fairly simple concepts
    of mathematical infinities. This isn't History. People expect accurate
    statements, backed by proofs not wild assumptions.

LONG STORY:

Wouldn't you like to make your messages corrrect? If so, I have some
posting tips for you. If not, just ignore this. (Of course, if you don't
want your messages to be accurate, it's not clear why to bother to send
them in the first place. :-) I'm going to take a bit of time to explain
this, because newcomers to comp.lang.python often lack the necessary 
mathematical skills were I to send a superbrief message.

Here's the issue: you appear to have sent a mish-mash of mathematical
facts. Worst of all, you have done so by not having a firm grasp of
the definitions of the terms you used. That is what mathematics is
about, you know. :-)

If all you want to do is appear mathemtically knowledgable, that's one thing,
but here you seem to be spouting nonsense just to fill up a message. This
is neither an on-topic post, nor one with any useful information. Here's
why.

First of all, this spreads mathematical disinformation about infinities --
you're supposed to know the basic definitions of which you're talking.
Otherwise you'll probably seem like a clueless newbie to the rest of us.
This isn't really an issue of exclusivity (I know that more then a handful
of people know this stuff nowadays), so much as it is an issue of keeping
things accurate.

Second, putting forth statements like "there might be only two infinities"
does little good. It misleads others to think that this was what Goedel
or Cohen proved, when what they proved that there might be no infinities
between aleph null and the continuum.

For example, here's how an explanation about infinities should look if
you'd like them to be more effective.

    A cardinality of a set is usually taken to be the minimal ordinal
    which is equi-potent with the set. This, of course, implies the Axiom
    of choice. In particular, we have that any set is equi-potent with its
    cardinality, and thus when a set is infinite, the cardinality is
    infinite.
    
    Of course, with no axiom of choice the mechanisms to prove the axiom
    of cardinality are so ugly, we wouldn't want to think about it.
    However, a short look at the relevant texts will show that at least
    the last statement of the previous paragraph is still true.

    On the other hand, this has nothing to do with whether a Turing
    Machine needs infinite memory: it does not. After $n$ steps, a Turing
    Machine (say, for simplicity, with one ended strip), will only need
    the first $n$ places of the strip. This implies that every Turing
    Machine that stops needs only a finite amount of memory. Of course, it
    will need a different amount of memory for every input, and in
    general, that amount is infinite.

Notice how in the text above, sound mathematical arguments are presented,
with a reference to relevant texts where it would be to encumbered to 
present the argument. This is the way it should be.

If you are receiving this message in response to a news posting, please
understand that all universities provide courses about this subjects, 
so it is seldom necessary to make wild guesses. Some high schools also
offer classes on these subjects.

Here's a section about the basic mathematical infinities. Perhaps you were
unaware of these facts before you got swallowed by an arguement. It's not
only a good read; it's critical to understanding the terms you are talking
about.

  Two mathematical sets, A and B, are said to be equi-potent if there is a
  one-to-one and onto function from A to B. Equi-potency is easily seen to
  be reflexive (A is equi-potent to A), symmetric (if A is equi-potent to
  B, B is equi-potent to A) and transitive (if A is equi-potent to B, and
  B is equi-potent to C, then A is equi-potent to C). A is said to be no
  more potent then B is there is a one-to-one function from A to B. A
  clever argument, which we will not repeat here, shows that is A is no
  more potent then B and B is no more potent then A, then A and B are
  equi-potent. B is said to be more potent then A if A is no more potent
  then B, but they are not equi-potent.

  Cantor's diagonal argument proves that A is less potent then the set of
  all functions from A to the set {0, 1}. Aleph null is the set of natural
  numbers. A set is said to be of cardinality aleph null is it is
  equi-potent with the natural numbers. The continuum is the set of
  functions from the natural numbers to {0, 1}. A set is said to be of the
  power of the continuum if it is equi-potent to that set. The continuum
  hypothesis is that there are no sets which are less potent the the
  continuum and more potent then aleph null. This has been proved to be
  independant of the other axioms of set theory.

It's even more annoying when people say obviously incorrect things, or
agree with previous such things. Please don't do that.

I'm honestly not trying to annoy you!  I'm just trying to give tips
about what works well in mathematics, and what doesn't. This
used to be standard fare before one said anything said anything about
infinities, but now something seems to be lost.  
-- 
This is a parody, and should not be taken seriously. However, the gist of
it is true: please try to have a mathematical clue before saying anything
about infinities. There are traps for the uninitiated!