Re: Is continuum completely filled up?

toshiaki wrote:
"Virgil" <virgil@xxxxxxxxxxx> wrote in message
news:virgil-F50F52.20450007012007@xxxxxxxxxxxxxxxxxxxxxxxxxxx
In article <45a18eed@xxxxxxxxxxxxxxxxxxx>,
Tony Orlow <tony@xxxxxxxxxxxxx> wrote:


Von Neumann's folly!
Those who are incapable of understanding their betters, try to malign
them.

I don't malign von Neumann, just his limit ordinals as a model of the
naturals.


I actually think the H-riffics may be a well ordering of the reals after
all...
What TO thinks and what mathematics shows rarely make contact.

You would argue if I said 2+2=4.


First of all, you tell me whether there is at all an infinite set. And
if such one exists, are we elligible to talk about its properties?
No "pure" infinite set. There is always order where infinitude's
implied.
False! One does not need an order relation to imply infiniteness of a
set.

I think an order relation is equevelent to AC .
Without this, uncountable sets cannot be dealt with ,and with AC , I think ,
We can give them order .

Yes, I think AC is used as an alternative, nonspecific means to impose
order on a set. It seems to me to describe a dimensional situation,
where we have a set of n sets of values, dimensions, and sets
constructed using n-tuples of values, each taken from one of those sets.
Those n-tuples represent points in that set of dimensions. The actual
application of "choice" seems to be somewhat different, and vaguer, as
far as I can tell.

The successor operation, which is enough to imply an infinite set, is
not actually an order relation, though it it commonly used to generate
one.

So, this comes "after" that is not a statement of the order of this and
that? Hmm...sounds like one to me. Is succ(x) not equivalent to y such
that x<y and not exist z such that x<z and z<y? "Element of" is
sufficient for finite sets. Infinite sets require "less than".


Cannot we reinterplet of the meaning of the set in the case related to
infinity ?
What cardinality implies is not just the same as what the number does .
In the case of infinite objects , We may think of the set not as a container
,but as a rule like a statement of Axiom of Infinity ?

Regards OT


An infinite set is really a structure like a sequence or tree, or a
process represented as such.
.

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