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| giovanni lagnese |
 Posted: Sat Jan 10, 2004 6:37 pm |
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the smallest non-definable ordinal is definable.
why is this argument wrong? |
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| William Elliot |
| Posted: Sat Jan 10, 2004 10:00 pm |
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On Sat, 10 Jan 2004, giovanni lagnese wrote:
If there is a non-definable ordinal, then
Quote: the smallest non-definable ordinal is definable.
which is a contradiction. Therefore....
Quote: why is this argument wrong?
It isn't. You didn't include the complete argument. |
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| Charlie-Boo |
| Posted: Sat Jan 10, 2004 10:03 pm |
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"giovanni lagnese" <ti.oiligriv@rotnac_groeg> wrote
Quote: the smallest non-definable ordinal is definable.
why is this argument wrong?
Are there any ordinals that are not definable?
Charlie Volkstorf |
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| giovanni lagnese |
| Posted: Sat Jan 10, 2004 10:43 pm |
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"Charlie-Boo" <chvol@aol.com> ha scritto nel messaggio
news:3df1e59f.0401101903.760bc98f@posting.google.com...
Quote: Are there any ordinals that are not definable?
sure! |
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| giovanni lagnese |
| Posted: Sun Jan 11, 2004 1:19 am |
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Now I have just solved the problem, I have understood.
Thank you. |
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| David C. Ullrich |
| Posted: Sun Jan 11, 2004 7:08 am |
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On Sat, 10 Jan 2004 23:37:51 GMT, "giovanni lagnese"
<ti.oiligriv@rotnac_groeg> wrote:
Quote:
the smallest non-definable ordinal is definable.
why is this argument wrong?
Because of fuzziness about the definition of "definable".
There must be undefinable ordinals, because there are only
countably many formulas of ZF that can be used to define
an ordinal. But "the smallest non-definable ordinal" is _not_
a definition of a ordinal _by_ a formula of ZF.
************************
David C. Ullrich |
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| David C. Ullrich |
| Posted: Sun Jan 11, 2004 7:09 am |
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On 10 Jan 2004 19:03:42 -0800, chvol@aol.com (Charlie-Boo) wrote:
Quote: "giovanni lagnese" <ti.oiligriv@rotnac_groeg> wrote
the smallest non-definable ordinal is definable.
why is this argument wrong?
Are there any ordinals that are not definable?
There are certainly ordinals that are not definable by
formulas of ZF, because there are only countably many
such formulas.
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David C. Ullrich |
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| David C. Ullrich |
| Posted: Sun Jan 11, 2004 7:11 am |
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On Sat, 10 Jan 2004 19:00:41 -0800, William Elliot <marsh@xyzt.org>
wrote:
Quote: On Sat, 10 Jan 2004, giovanni lagnese wrote:
If there is a non-definable ordinal, then
the smallest non-definable ordinal is definable.
which is a contradiction. Therefore....
why is this argument wrong?
It isn't. You didn't include the complete argument.
The resolution of the apparent paradox is not what
you seem to be hinting at - there certainly do exist
non-definable ordinals.
(Well, first we need to give a precise definition of
"definable ordinal" (and in fact it's fuzziness over
this definition that leads to the "paradox"). But
whatever definition we give is going to involve
a definition in some sense in the language of
ZF; hence there are only countably many
definitions, hence only countably many
definable ordinals.)
************************
David C. Ullrich |
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| Antonio Espejo |
| Posted: Sun Jan 11, 2004 3:06 pm |
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David C. Ullrich <ullrich@math.okstate.edu> wrote in message news:<k0f200h8545mv9jr693cijub5n2kdknhme@4ax.com>...
Quote: But "the smallest non-definable ordinal" is _not_
a definition of a ordinal _by_ a formula of ZF.
Yes, it is. Why not? |
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| Charlie-Boo |
| Posted: Mon Jan 12, 2004 12:25 am |
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David C. Ullrich <ullrich@math.okstate.edu> wrote
Quote: On 10 Jan 2004 19:03:42 -0800, chvol@aol.com (Charlie-Boo) wrote:
"giovanni lagnese" <ti.oiligriv@rotnac_groeg> wrote
the smallest non-definable ordinal is definable.
why is this argument wrong?
Are there any ordinals that are not definable?
There are certainly ordinals that are not definable by
formulas of ZF, because there are only countably many
such formulas.
If you require that the definition be expressed in a particular
formalism (e.g. ZFC) whose set of definitions has cardinality less
than the cardinality of the set of ordinals, then sure, but (1) who
says it has to be ZFC, and (2) might there be a formalism with a set
of definitions with cardinality that is not less than that (there's
one obvious - trivial - possibility, although I'm not sure if it'd
work!)?
C-B
Quote: Charlie Volkstorf
************************
David C. Ullrich |
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| David C. Ullrich |
| Posted: Mon Jan 12, 2004 7:34 am |
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On 11 Jan 2004 21:25:28 -0800, chvol@aol.com (Charlie-Boo) wrote:
Quote: David C. Ullrich <ullrich@math.okstate.edu> wrote
On 10 Jan 2004 19:03:42 -0800, chvol@aol.com (Charlie-Boo) wrote:
"giovanni lagnese" <ti.oiligriv@rotnac_groeg> wrote
the smallest non-definable ordinal is definable.
why is this argument wrong?
Are there any ordinals that are not definable?
There are certainly ordinals that are not definable by
formulas of ZF, because there are only countably many
such formulas.
If you require that the definition be expressed in a particular
formalism (e.g. ZFC) whose set of definitions has cardinality less
than the cardinality of the set of ordinals, then sure, but (1) who
says it has to be ZFC, and (2) might there be a formalism with a set
of definitions with cardinality that is not less than that (there's
one obvious - trivial - possibility, although I'm not sure if it'd
work!)?
The short version: I was assuming that "definable" means
something like the _standard_ meaning. That doesn't
mean it has to be ZFC, but it does restrict us to a countable
language.
The long version:
For example, one could fix a model of set theory, and then
for each ordinal 0 in the language add a new constant symbol
c_o. Then we have a large language in which every ordinal
is "definable".
This sort of "large language" has uses, for example in
setting up non-standard analysis. But when one is talking
about "definable" whatever, with no qualification on what
"definable" means, one usually assumes a countable
language. The reason is that in a countable language
we can actually write down a symbol in English for
each formal symbol in the language - the "constant
symbols" mentioned in the previous paragraph are
a somewhat curious sort of "symbol", since we don't
and can't encode the symbols in English - most of
them are "symbols" that we can't write down, hence
not the sort of thing one is usually referring to when
one talks about definability.
A shorter version of the long version: If we allow
that sort of "large language" then everything is
definable, hence the word "definable" ceases to
mean anything.
Quote: C-B
Charlie Volkstorf
************************
David C. Ullrich
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David C. Ullrich |
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| David C. Ullrich |
| Posted: Mon Jan 12, 2004 7:35 am |
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On 11 Jan 2004 12:06:19 -0800, CASAFARFARA@terra.es (Antonio Espejo)
wrote:
Quote: David C. Ullrich <ullrich@math.okstate.edu> wrote in message news:<k0f200h8545mv9jr693cijub5n2kdknhme@4ax.com>...
But "the smallest non-definable ordinal" is _not_
a definition of a ordinal _by_ a formula of ZF.
Yes, it is.
What formula defines it?
************************
David C. Ullrich |
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| Aatu Koskensilta |
| Posted: Mon Jan 12, 2004 7:56 am |
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David C. Ullrich wrote:
Quote: A shorter version of the long version: If we allow
that sort of "large language" then everything is
definable, hence the word "definable" ceases to
mean anything.
While every ordinal is obviously ordinal definable, i.e. definable with
ordinal parameters, everything need not be.
--
Aatu Koskensilta (aatu.koskensilta@xortec.fi)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus |
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| David C. Ullrich |
| Posted: Mon Jan 12, 2004 9:44 am |
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On Mon, 12 Jan 2004 14:56:11 +0200, Aatu Koskensilta
<aatu.koskensilta@xortec.fi> wrote:
Quote: David C. Ullrich wrote:
A shorter version of the long version: If we allow
that sort of "large language" then everything is
definable, hence the word "definable" ceases to
mean anything.
While every ordinal is obviously ordinal definable, i.e. definable with
ordinal parameters, everything need not be.
I said _that sort of large language_, not "that large language"...
************************
David C. Ullrich |
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| Aatu Koskensilta |
| Posted: Tue Jan 13, 2004 2:35 am |
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David C. Ullrich wrote:
Quote: On Mon, 12 Jan 2004 14:56:11 +0200, Aatu Koskensilta
aatu.koskensilta@xortec.fi> wrote:
David C. Ullrich wrote:
A shorter version of the long version: If we allow
that sort of "large language" then everything is
definable, hence the word "definable" ceases to
mean anything.
While every ordinal is obviously ordinal definable, i.e. definable with
ordinal parameters, everything need not be.
I said _that sort of large language_, not "that large language"...
I misinterpreted you, then. Sorry.
--
Aatu Koskensilta (aatu.koskensilta@xortec.fi)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus |
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