Re: An uncountable countable set

Virgil wrote:
In article <45205fa9@xxxxxxxxxxxxxxxxxxx>,
Tony Orlow <tony@xxxxxxxxxxxxx> wrote:

Virgil wrote:
In article <45203919@xxxxxxxxxxxxxxxxxxx>,
Tony Orlow <tony@xxxxxxxxxxxxx> wrote:



Since ordinals are, by definition, well ordered, they cannot contain any
endlessly decreasing sequences, which TO's models require.
Neither can the reals.
How about the set of negative integers? How is that not an endlessly decreasing sequence of reals?
The origin is at a finite location. Order starts from the bottom, if "decreasing" has any meaning.

The set of negative integers has no "bottom".

TO seems to be changing his tune when it is used against him.

According to TO every set of numbers has a natural order, and it is within that natural order that we must view it, but now he wants to reject the natural order because it runs counter to another of his claims.

TO blows hot and cold with the same reath.

I'm saying the if you iterate the negative integers starting at 0, in that order, there is no infinite descending sequence. On the other hand I don't know why I said "neither can the reals". In any case, the only way the ordinals manage to be "well ordered" is because they're defined with predecessor discontinuities at the limit ordinals, including 0. That doesn't seem "real", and the axiom of choice aside, I don't see there being any well ordering of the reals. The closest one can come is the H-riffic numbers. :)
.

Relevant Pages

  • Re: An uncountable countable set
    ... Tony Orlow wrote: ... endlessly decreasing sequences, which TO's models require. ... Neither can the reals. ... According to TO every set of numbers has a natural order, and it is within that natural order that we must view it, but now he wants to reject the natural order because it runs counter to another of his claims. ...
    (sci.math)
  • Re: An uncountable countable set
    ... Tony Orlow wrote: ... endlessly decreasing sequences, which TO's models require. ... Neither can the reals. ... According to TO every set of numbers has a natural order, ...
    (sci.math)
  • Re: An uncountable countable set
    ... endlessly decreasing sequences, which TO's models require. ... Neither can the reals. ... According to TO every set of numbers has a natural order, and it is within that natural order that we must view it, but now he wants to reject the natural order because it runs counter to another of his claims. ... In any case, the only way the ordinals manage to be "well ordered" is because they're defined with predecessor discontinuities at the limit ordinals, including 0. ...
    (sci.math)
  • Re: An uncountable countable set
    ... endlessly decreasing sequences, which TO's models require. ... Neither can the reals. ... According to TO every set of numbers has a natural order, ... with predecessor discontinuities at the limit ordinals, ...
    (sci.math)
  • Re: An uncountable countable set
    ... Tony Orlow wrote: ... endlessly decreasing sequences, which TO's models require. ... Neither can the reals. ... How about the set of negative integers? ...
    (sci.math)
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