Post by Ross FinlaysonPost by FromTheRaftersPost by WMPost by FromTheRaftersPost by joesPost by joesWhat does „complete” mean?
It means that no natural number can be added to {0, 1, 2, 3, ..., ω}
Duh, the set of all natural numbers N contains all of them.
Post by joesWith which numbers do you describe the sizes of N and N_0?
Most of them are dark and cannot be used as individuals.
Not their elements. I was asking for their number, how many
of them there are.
There's a number of them, however, how many there are is not a number.
Either there is a complete and fixed set ℕ with a fixed number |ℕ| of
elements.
If so, then 0.999...*10 = 9,999... where the number of nines has not
changed by more than 0.
Or this is not the case.
But assuming completeness and simultaneously claiming different
numbers of nines is stupid.
The number of nines is not a number.
I thought .999... = 1, ....
You know, or it "goes" to.
When I saw this thread subject the first thing that came to
mind was "casting out 9's", which is according to this fact
that for integers in base b, that b-1 or roots of b-1 can
be provided a divisibility test by recursively summing the
digits and testing when convenient the recursion for divisibility.
Here though it's about the radix, often a name for the base,
eg. "the hexadecimal radix", yet here mostly about the
modular unit, it's the decimal point, the positional
placeholder between > 1 and < 1, the radix point.
The idea of, "10.1", then, vis-a-vis, 1, gets into,
for example, "100.01", and so on, or for writing
"1|1" instead of "." (the radix, the modular radix).
or, "... times 1 plus 1 divided by ...".
Now, this would be inconvenient in a usual sense,
to have sort of an integer part and non-integer part,
connected with addition, instead of the most usual
sort of field, where the placeholder is what moves,
when multiplying or dividing by the radix.
https://en.wikipedia.org/wiki/Radix
The Wiki page doesn't really address this other quite
usual notion of the definition of "radix".
It links to "non-standard positional number system"
which is usually just about calendar and clock
arithmetic and still quite all in the modular,
yet the definition of "decimal point" makes
for that the placeholder at the end of the integer
part and beginning of the non-integer part,
is also called the "radix".
On the Wiki, the entry for "decimal point" points
to "decimal separator".
https://en.wikipedia.org/wiki/Decimal_separator
It has a mention of "radix point".
https://en.wikipedia.org/wiki/Decimal_separator#Radix_point
The radix _point_ is part of the character of the radix,
it sort of is the "unit radix".
So, imagine a fixed-width machine fixed-point number,
where the radix point is in the middle. Then the least
(non-zero) value, for example, looks like
0000.0001
https://en.wikipedia.org/wiki/Fixed-point_arithmetic
In fixed point arithmetic, the _increment_ of the
underlying machine number, usually is considered
to represent this "dialing the knob", if you recall
the discussion this year here about models of arithmetic
that fulfill having these finite means, then, in
the unbounded of those.
So anyways, having equal densities of 0's and 1's
can be said to be so for infinite sequences of 0's
and 1's, _among all of them_, and folding them
in half about the radix point _keeps this also so_,
and running out in the unbounded that "restricted
sequence element interchange" runs out, _does run out_.
Thusly, in a greater setting, such natural statements
like "half the integers are even" are quite so.
You're welcome to say "they have the same cardinality
the integers and even integers", and not welcome to
say "the density of the even integers being 1/2 is not so".
Then, as a course-of-passage _goes_ from 0 to 1,
through each point between and in order, like
time-ordering what results modular clock arithmetic,
it's most totally usual in all systems parameterized by "t".