Re: Galileo's Paradox

Eckard Blumschein wrote:
On 12/6/2006 5:27 AM, David Marcus wrote:
Eckard Blumschein wrote:
So it would not make any difference when the order aleph_0 and aleph_1
were reversed or more reasonably the correct denotation was preferred:
countable instead of aleph_0 and uncountable instead of aleph_1.

Yes, yes, I know you like to take standard words and give them new
(vague) meanings.

This is a praxis introduced by mathematicians to an extent that even
rich languages get short of terms without distorted meaning.

Nonsense. We aren't about to run out of words any time soon. You've
shown you are quite good at making up new terms. The problem is you use
standard terms with your own personal meaning and you use nonstandard
terms without telling anyone what they mean. So, the problem is all you!

But, I was asking you whether the cardinality of the
reals is greater than the cardinality of the integers if we use the
standard meanings for the words "cardinality", "reals", and "integers".

I consider cardinality nonsense. Just say countably infinite instead of
a_0 and uncountable instead of a_1 and forget the rest.

I wasn't asking you what you "consider". Let's try a simpler question.
Is there an injection from the integers to the reals? Is there an
injection from the reals to the integers?

As for aleph_2 itself, because of the
independence of the continuum hypothesis, it doesn't come up too often
in mathematics.

= never?

Of course, not.

However, there were some interesting articles recently
in the Notices of the AMS that discussed axioms to add to ZFC. There
seemed to be good reasons to add an axiom which would make the
cardinality of the reals equal to aleph_2.

And which role has been envisioned for aleph_1?

Kind of a silly question. aleph_1 is the first cardinal after aleph_0.
That's its "role".

--
David Marcus
.

Relevant Pages

  • Re: Distinct linear orderings on Z
    ... >>And then at some point this led us to deciding that cardinality and ... >>meant for an infinite set to have a certain number of elements. ... always goes hand in hand with subtraction. ... Does the interval on the reals have to include 1? ...
    (sci.math)
  • Re: An uncountable countable set
    ... has cardinality larger than the natural numbers. ... complete ordered field (he uses reals to exemplify). ... I don't disagree that the first is infinite while the second is ... Considering that the set of finite naturals is ultimately finite, ...
    (sci.math)
  • Re: An uncountable countable set
    ... There are sets with a cardinality larger than that of the natural ... The first part of his first proof shows that a complete ordered field ... complete ordered field (he uses reals to exemplify). ... sich in jedem vorgegeben Intervalle ...eine Zahl ... ...
    (sci.math)
  • Re: Anti-diagonalist page
    ... >>than the cardinality of the naturals. ... >>On the assumption that "the cardinality of the reals is greater than ... there still is that objection. ...
    (sci.logic)
  • Re: Anti-diagonalist page
    ... >than the cardinality of the naturals. ... >On the assumption that "the cardinality of the reals is greater than ... there still is that objection. ...
    (sci.logic)
Quantcast