Re: Well Ordering the Reals

albstorz@xxxxxx writes:

> David Kastrup wrote:
>> albstorz@xxxxxx writes:
>>
>> > William Hughes wrote:
>> >> albstorz@xxxxxx wrote:
>> >> > David Kastrup wrote:
>> >> > > albstorz@xxxxxx writes:
>> >> > >
>> >> > > > David Kastrup wrote:
>> >> > > >> boink <boink@xxxxxxxxxx> writes:
>> >> > > >>
>> >> > > >> > that's funny, because in some sense that's exactly what
>> >> > > >> > happens, but you don't get it. after omega many steps,
>> >> > > >> > you get the infinite ordinal omega which is the set of all
>> >> > > >> > finite ordinals. and omega is infinite and contains no
>> >> > > >> > infinite ordinal.
>> >> > > >>
>> >> > > >> Nonsense. With that kind of logic, with aleph_1 many steps,
>> >> > > >> you get the infinite ordinal omega_1, but there is no such
>> >> > > >> thing. Steps don't get you omega. Omega is an
>> >> > > >> _inexhaustible_ supply of sequential steps. Either you have
>> >> > > >> it, or you don't. If you don't, you can't put it together
>> >> > > >> using finite steps. A sentence like "you get omega, if you
>> >> > > >> just exhaust an inexhaustible supply of finite steps..."
>> >> > > >> does not make sense.
>> >> > > >
>> >> > > >
>> >> > > > It's really funny. You have infinitely many natural numbers
>> >> > > > but you have not infinitely many steps (or you have it but
>> >> > > > you can't supply the natural numbers with it).
>> >> > >
>> >> > > You have an inexhaustible supply of steps. Since it is inexhaustible,
>> >> > > you can't perform "all of those".
>> >> >
>> >> > This must hold in the exact same manner for the inexhaustible
>> >> > supply of natural numbers. Since it's inexhaustible you can't
>> >> > have all of those. The set N doesn't exist. That's what I'm
>> >> > talking about the whole time.
>> >>
>> >> No, you cannot get all the numbers from the inexaustible supply.
>> >> But what about the entity "the inexaustible supply". Does this
>> >> entity not exist?
>> >
>> > It's no entity in any useful sense of the word. So it exists, but it
>> > is no entity and it is no set.
>>
>> A set is something that either contains any given prospective
>> element or not. As long as any given entity is either a natural
>> number or not, the set is existent.
>
> That's inacceptable.

Feel free to whine and stomp your feet and cry. Which is pretty much
what you are doing.

> What's about the "Aussonderungsaxiom"?

What about it?

> There is no universe A u notA in ZF!

Even if this sentence made remotely sense, its relevance would appear
to be your secret.

>> Not by cobbling all elements together, but
>> a set is not defined as the result of cobbling something together.
>>
>> > Maybe it's a potential.
>> >
>> > "Infintum actu non datur."
>> > Infinity is just a "facon de parler".
>>
>> Mathematics is just a "facon de parler". Infinities are no
>> difference.
>
> You misinterpret the term "facon de parler". If the whole math would
> be just a "facon de parler", it would terminate to be a science.

Mathematics has never been a science. It has traditionally been one
of the artes liberales (well, two actually).

It forms the most essential base and structuring agent for pretty much
every empiric science, but is not in itself science since it makes no
observations about the real world.

--
David Kastrup, Kriemhildstr. 15, 44793 Bochum
.

Relevant Pages

  • Re: Well Ordering the Reals
    ... >> David Kastrup wrote: ... of the set of those elements which hold the propertie notA (which is ... > Mathematics has never been a science. ... (modern) math is a kind of art. ...
    (sci.math)
  • Re: Well Ordering the Reals
    ... >> David Kastrup wrote: ... > Mathematics is not, and never has been, a "science" in the modern ... this sense, but to develop math, that's pure science in this deep ...
    (sci.math)