continuum hypothesis

From: Alex Mizrahi (udodenko_at_hotmail.com)
Date: 09/20/04 

Date: Tue, 21 Sep 2004 02:46:03 +0300

Hello, All!

i'd like to clarify some stuff about continuum hypothesis for me.. i'm not
an expert in set theory, but CH makes me sick - i cannot imagine that
there's unimaginable stuff lile this 8-]

so, as i understand on form of expressing CH is aleph1 = c. aleph1 is
cardinality of the set of all countable ordinals. countable ordinals are
order types for countable sets. that ordinals define classes (sets) of
methods of well-ordering natural numbers where each two methods in that set
are isomorphic.
we can define method of well-ordering as a relation on NxN. this relation
can be represented as infinite square table. each such table corresponds to
one real number from [0,1] in binary form. so cardinality of set of all
countable ordinals is not more than continuum. however, not every table (or
number) defines well-ordering (some define some bull***). and some
well-orderings are isomorphic, so there's 'less' ordinals than such
well-orderings.
however, stuff looks quite well-defined here - we can make predicate that
tells us if table (or number) defines well-ordering(1). we can define
equivalence relation (2) that tells us if well-orderings are isomorphic -
correspond to one ordinal. and then just count (3) equvalence classes in
such relation..

but CH can be axiom - it can be true or false. so, question is - which
part(s) will be different if we take CH true or false - (1), (2), and/or
(3)? in other words, if CH is false, we cannot build a method to see if
relation is well-ordering, compare if orderings are isomorphic, or count
classes of isomorphic orderings? or am i missing something?

With best regards, Alex 'killer_storm' Mizrahi.

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