Re: Skolem's 'Paradox'
- From: "george" <greeneg@xxxxxxxxxx>
- Date: 9 Sep 2005 14:45:19 -0700
William of Ockham wrote:
> What is this paradox?
> I have yet to find any explanation of it that
> makes sense.
Much as it galls me to agree with a contemptuous
1-liner from Torkel Franzen, a stopped clock is
right twice a day. Just find a decent logic book.
That is far smarter than trolling in THIS pond.
> My crude understanding is
Just that: crude, so why flaunt it here,
where all it will get you is insulted for being
barbarically uncouth? Do your f'ing HOMEwork for
a change!
> Here's an explanation of the paradox
> which I have adapted from a
> well-known writer.
If he was "well-known" then he probably
wasn't a logician. Popularized accounts are
NOT going to cut it here. I repeat (gawd,
this was painful enough the FIRST time),
TF was right: Go Find A Decent Textbook
on first-order logic.
> "If you are living (as an "internal observer") within
> the model, the set of sets of elements seems uncountable
> to you (because you cannot find a 1-1 function from sets
> of elements into elements in your world).
> Still, for me (an an "external observer") your
> uncountable set is countable - in my world
> I have a 1-1 function from
> sets of elements into elements!"
>
> This makes no sense to me at all.
Nor will it, UNTIL you
GO FIND A DECENT LOGIC TEXTBOOK.
"Model" here is a technical term.
Every model has a "domain" which is a
collection of objects. While every object
in this collection has to exist, other things
can be left out of it, and still go right
on existing. However, the model in some sense
does not "opine" or "believe" that these other
things exist.
> If "an uncountable set exists" is
> true, then "there is no mapping from
> elements to sets of elements" is
> surely true.
Your own use of logic here is really crappy.
That "there is no [surjection] on elements
ONto [all sub]sets of [THOSE SAME] elements"
is true ALL the time, PERIOD, COMPLETELY irrespective
of whether uncountable sets exist OR NOT.
IF there is going to be an if/then here, it is
going to be in the OPPOSITE direction, namely,
if there is no mapping on the naturals onto
all of their subsets, then "an uncountable set [namely, that
very class of all subsets of the naturals] exists" is true.
> So how can one exist at all,
> whether "outside the model"
> or not?
In the case of LST and the theory in question,
the mapping exists outside, and it is countable,
so the question of whether an uncountable set exists
does not even arise.
> The stuff about observers makes no sense to me.
It's a POPULAR account, dumbass.
If you want to get technical, replace the
"observer" with a set, namely, the domain of the model.
> If it merely "seems"
> that there is no mapping,
> then the sentence "there is no mapping" is
> false,
Wrong.
As I've told you before, AROUND HERE, we DON'T DO
"is true" or "is false". We INSTEAD do is true or false
IN A MODEL. You HAVE TO PICK a model, FIRST. AFTER that,
you can say "is true" or "is false". The fact that
"there is no mapping" is false in the bigger model
doesn't make it any less true in the smaller one.
> Apparently early writers
Why do you and Franz Fritsche have this fetish for the
opinions of early writers? Early means irrelevant.
As TF told you, GET A DECENT LOGIC BOOK. It will NOT
be early. It will NOT be popular. It WILL be CLEAR
about what is going on here. But the far more relevant
and surprising question is this: HOW DID YOU MANAGE to
find all these early quotes WITHOUT finding a decent
logic book? Decent logic books are ABUNDANT BY CONTRAST
with sources for these early quotes! What you are spouting
here implies that bagged elephants on safari but that
catching a few fireflies in your backyard is beyond your
hunting skills! THAT IS WHY you are getting such extreme
reactions!
> Von Neumann said "At present we can do no more than
> note that we have one more reason here to entertain
> reservations about set theory
Just HOW early was this?? The paradox was discovered
before 1920 so it would be VERY hard to call this one
MORE reason: it was one of the FIRST reasons: the OTHER
LATER reasons were "more".
> Zermelo declared the Skolem paradox a hoax.
That is overstating the case.
At best you could say he explained why it is not a paradox.
That does NOT suffice to make our continued insistence
on calling it one "hoaxical".
> In 1937 he wrote a small note entitled
> "Relativism in Set Theory and the
> So-Called Theorem of Skolem")
> in which he gives a refutation of
> "Skolem's paradox",
Wrong.
It is not possible to refute a correct proof.
Skolem did correctly prove that Zermelo's axioms had a
countable model, despite the fact that they prove the existence
of uncountable sets. That is all incontestable and it
is at least INformally, superficially, intuitively, paradoxical.
An explanation of why your intuition leads you astray here
and why this is not really a paradox is NOT a "refutation",
of either the proof OR the paradox.
> i.e., the fact that Zermelo-Fraenkel set
> theory--guaranteeing the existence of
> uncountably many sets--
No, PROVING the existence of ONE uncountable set.
The whole point of the Lowenheim-Skolem Theorem
(which ALSO has a proof) is that such a proof TURNS OUT
to be a GUARANTEE of basically NOTHING.
> has a countable model.
Indeed it has.
> Frankel said "Neither have the books yet been
> closed on the antinomy, nor has agreement on
> its significance and possible solution yet been
> reached.
That was overstating the case in the opposite
direction, since there IS NO antinomy here.
The issue is not about a possible solution.
The issue is about the existence of uncountable
sets (if you think that's where the burden of proof
lies). Skolem proved that proving the existence of
uncountable sets in a denumerable language in a first-
order theory is NOT sufficient to prove the existence
of uncountable sets, since you can model that whole
theory in a countable domain, since that theory's
models have VARYING opinions about "which" sets
are countable (or not).
> Skolem said "I believed that it was so clear
> that axiomatization in terms of sets was not
> a satisfactory ultimate foundation of mathematics
> that mathematicians would, for the most part,
> not be very much concerned with it. But in recent
> times I have seen to my surprise that
> so many mathematicians think that these axioms
> of set theory provide
> the ideal foundation for mathematics;
> therefore it seemed to me that
> the time had come for a critique.
OK, he critiqued. Mathematicians went right
on using first-order logic and set-theory ANYway.
But they also went far beyond those. In some ways,
his critique was therefore successful, to the extent
that it clarified that one would NEED to go beyond.
But that was already going to become clear for other
reasons. First-order ZFC's failure to prove the
existence of the set of natural numbers (at all,
in the first place) is a MUCH bigger reason for deciding to
go beyond it than its failure to exclude "small" models
might ever be.
.
- References:
- Skolem's 'Paradox'
- From: William of Ockham
- Skolem's 'Paradox'
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