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Topic: The uncountability infinite binary tree.
Replies: 118   Last Post: Dec 17, 2012 8:49 PM

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Rupert

Posts: 3,806
Registered: 12/6/04
Re: The uncountability infinite binary tree.
Posted: Dec 16, 2012 4:01 PM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

On Dec 16, 9:49 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
> On 16 Dez., 21:34, Rupert <rupertmccal...@yahoo.com> wrote:
>
>
>

> > It is mathematics. Mathematics includes discourse about uncountable
> > sets.

>
> But it does not include discourse using uncountably many characters,
> because for that sake infinite strings of bits would be required. By
> finite strings of bits only countably many characters could be used.
> And if we restrict our conversation to those usable characters and
> omit the others, then we discuss in a countable language.
>


You're missing the distinction between the metatheory and the object
theory. The metatheory in which mathematical discourse takes place, in
which we discuss various object theories, is in a countable language.
But the object theories could be in an uncountable language.

> > The fact that it would not be possible for a human to use such a
> > language is irrelevant

>
> It would not only be impossible for a human but impossible per se,
> because there might be sentences that do never end.
>


I don't know what the distinction is between "impossible for a human
to use" and "impossible per se".

In metamathematics, we can study languages other than the languages
which are in fact possible for humans to use.

> Mathematics may contain many foolish ideas, They can be discussed. But
> the language applied to discuss them must be free of foolish items and
> must be usable by humans and other intellects. That's the way
> mathematics works: It is mainly a discussion with others or with
> oneself. Every item (including uncountable sets and inaccessible
> cardinals) must have finite definitions. Therefore there is no
> uncountable alphabet and there are not uncountably many languages.
>


Not in languages that humans actually use, no. But in the universe of
discourse of metamathematics, there are such languages.

> And all mathematics that has been done by Cantor can be discussed in
> any natural (i.e. countable) language. Further Cantor's diagonal
> argument works solely with digits (i.e., nodes). A proof that nobody
> can distinguish more than countable many paths in the Binary Tree is a
> contradiction of uncountability.
>


It's not.

> Regards, WM



Date Subject Author
12/14/12
Read The uncountability infinite binary tree.
Zaljohar@gmail.com
12/14/12
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Virgil
12/15/12
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Zaljohar@gmail.com
12/14/12
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William Elliot
12/15/12
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mueckenh@rz.fh-augsburg.de
12/15/12
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Rupert
12/15/12
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mueckenh@rz.fh-augsburg.de
12/15/12
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trj
12/15/12
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12/15/12
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mueckenh@rz.fh-augsburg.de
12/15/12
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Zaljohar@gmail.com
12/15/12
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mueckenh@rz.fh-augsburg.de
12/15/12
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Zaljohar@gmail.com
12/15/12
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Zaljohar@gmail.com
12/15/12
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mueckenh@rz.fh-augsburg.de
12/15/12
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mueckenh@rz.fh-augsburg.de
12/15/12
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Zaljohar@gmail.com
12/15/12
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mueckenh@rz.fh-augsburg.de
12/15/12
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Virgil
12/16/12
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Zaljohar@gmail.com
12/16/12
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mueckenh@rz.fh-augsburg.de
12/16/12
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Zaljohar@gmail.com
12/16/12
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mueckenh@rz.fh-augsburg.de
12/16/12
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Zaljohar@gmail.com
12/16/12
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mueckenh@rz.fh-augsburg.de
12/16/12
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Virgil
12/16/12
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ross.finlayson@gmail.com
12/16/12
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Virgil
12/16/12
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Zaljohar@gmail.com
12/16/12
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Zaljohar@gmail.com
12/16/12
Read Re: The uncountability infinite binary tree.
Virgil
12/17/12
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mueckenh@rz.fh-augsburg.de
12/17/12
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Virgil
12/15/12
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Zaljohar@gmail.com
12/15/12
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mueckenh@rz.fh-augsburg.de
12/15/12
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Virgil
12/15/12
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Tanu R.
12/15/12
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Virgil
12/16/12
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mueckenh@rz.fh-augsburg.de
12/16/12
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Virgil
12/17/12
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mueckenh@rz.fh-augsburg.de
12/17/12
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Virgil
12/15/12
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Virgil
12/16/12
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mueckenh@rz.fh-augsburg.de
12/16/12
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Virgil
12/17/12
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mueckenh@rz.fh-augsburg.de
12/17/12
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Virgil
12/15/12
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Virgil
12/15/12
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Rupert
12/15/12
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12/15/12
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netzweltler
12/15/12
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12/15/12
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Virgil
12/15/12
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Tanu R.
12/15/12
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Zaljohar@gmail.com
12/15/12
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Virgil
12/15/12
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Tanu R.
12/15/12
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namducnguyen
12/15/12
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Tanu R.
12/15/12
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namducnguyen
12/15/12
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Tanu R.
12/16/12
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Rupert
12/16/12
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mueckenh@rz.fh-augsburg.de
12/16/12
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Rupert
12/16/12
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mueckenh@rz.fh-augsburg.de
12/16/12
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Rupert
12/16/12
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ross.finlayson@gmail.com
12/16/12
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Rupert
12/16/12
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ross.finlayson@gmail.com
12/16/12
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Rupert
12/16/12
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ross.finlayson@gmail.com
12/16/12
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Rupert
12/16/12
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ross.finlayson@gmail.com
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mueckenh@rz.fh-augsburg.de
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Rupert
12/16/12
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mueckenh@rz.fh-augsburg.de
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Rupert
12/16/12
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mueckenh@rz.fh-augsburg.de
12/16/12
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Rupert
12/17/12
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mueckenh@rz.fh-augsburg.de
12/17/12
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Virgil
12/17/12
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mueckenh@rz.fh-augsburg.de
12/17/12
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Virgil
12/16/12
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mueckenh@rz.fh-augsburg.de
12/16/12
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Rupert
12/17/12
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mueckenh@rz.fh-augsburg.de
12/17/12
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fom
12/17/12
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Virgil
12/17/12
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Virgil
12/16/12
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Virgil
12/16/12
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Virgil
12/16/12
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m. m. m.
12/16/12
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Virgil
12/16/12
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mueckenh@rz.fh-augsburg.de
12/16/12
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Rupert
12/16/12
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mueckenh@rz.fh-augsburg.de
12/16/12
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Rupert
12/17/12
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mueckenh@rz.fh-augsburg.de
12/17/12
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Roland Franzius
12/17/12
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Virgil
12/17/12
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mueckenh@rz.fh-augsburg.de
12/17/12
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Virgil
12/17/12
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fom
12/16/12
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Virgil
12/16/12
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m. m. m.
12/16/12
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Virgil
12/16/12
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m. m. m.
12/16/12
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Virgil
12/16/12
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ross.finlayson@gmail.com
12/16/12
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trj
12/16/12
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Zaljohar@gmail.com
12/16/12
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mueckenh@rz.fh-augsburg.de
12/16/12
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Virgil
12/16/12
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Virgil
12/15/12
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Virgil
12/15/12
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Tanu R.
12/15/12
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Virgil
12/15/12
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Virgil
12/15/12
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Virgil

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