Re: Is continuum completely filled up?


"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 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.

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 .
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.

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

.

Relevant Pages

  • Re: An uncountable countable set
    ... formal proof of the equivalence between the H-riffics and the reals. ... is a node in the tree along a finite-length path. ... I presented the H-riffics in "Well Ordering the Reals" as a proposed well ordering. ... Every H-riffic corresponds to a node in an infinite, but countable, ...
    (sci.math)
  • Re: An uncountable countable set
    ... Every H-riffic corresponds to a node in an infinite, but countable, ... along a path in a binary tree. ... In order to prove that the H-riffics really cover the reals I have to use a Cauchy- or Dedekind-like method to prove that any element in the continuum can be specified, even if it requires an infinite specification. ... You are carrying that over from the standard notion of sequences as always countable. ...
    (sci.math)
  • Re: An uncountable countable set
    ... Tony Orlow wrote: ... Every H-riffic corresponds to a node in an infinite, but countable, ... along a path in a binary tree. ... produce infinitely recursively-defined H-riffics from your existing ...
    (sci.math)
  • Re: An uncountable countable set
    ... Every H-riffic corresponds to a node in an infinite, but countable, ... Tony Orlow wrote: ... along a path in a binary tree. ... produce infinitely recursively-defined H-riffics from your existing ...
    (sci.math)
  • Re: An uncountable countable set
    ... then we'll talk about "covering the reals". ... formal proof of the equivalence between the H-riffics and the reals. ... That's something I have to get back to, I suppose, but Dave Rusin had confirmed that a base-2 H-riffic representation of 3 was a repeating string, much like 1/3 in decimal. ... That's a countably infinite set of strings, each being finite in length but representing infinite values. ...
    (sci.math)
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