Re: Moore on Skolem's Paradox


David C. Ullrich wrote:
> On 29 Sep 2005 11:26:04 -0700, "William of Ockham"
> <d3uckner@xxxxxxxxxxxxxx> wrote:
42. David C. Ullrich Sep 30, 11:36 am show options


> If you're not sure what some button in some program
> does the best thing is to just push the button and find out.
> Aargh.

What's wrong with this? This is my standard approach to software.


Ockham:
> An interpretation (as
>I understand) is the information which we must have about a sentence in
>order to understand it.

Ullrich:
> That's one of
> the reasons I say you simply don't understand what you're talking
> about - when we say things like "an interpretation of set theory"
> that is simply _not_ what the word "interpretation" means.

I got this from a textbook on model theory. It says "Sometimes we
write or speak a sentence S that expresses nothing either true or
false, because some crucial information is missing about what the words
mean. If we go on to add this information, so that S comes to express a
true or false statement, we are said to interpret S, and the added
information is called an interpretation of S." The last bit says "the
added information is called an interpretation of S".

Probably the textbook is wrong, but it makes me cautious of your
exhortation to read textbooks.

On the use of the word "reference", my dictionary gives two relevant
meanings: 1 the relation between a word and the object referred to, and
2 the object referred to. This agrees with standard philosophical
usage.

Ullrich:
> when you imagine a world where the proof of the existence of an
> uncountable set is invalid you're just not making any
> sense.

I didn't say this. I said, suppose a world where, because of the
meaning of the language people use, the reference of the expression
"the real numbers" is different from what it actually is, so that, in
their language, the proof of the uncountability of the reals is valid.


> In that imaginary world you're speaking of, where everything
> is countable, mathematics is simply nothing whatever like
> mathematics in our world. If you want to imagine a world
> where the sun is green and apples fall up, fine.
[...]
> Maybe what you have in mind is a world where the proof
> is valid but nonetheless everything is "really"
> countable. There is no such world.

Perhaps not, but could there be? And why indeed couldn't we imagine
such a world? First, it would look exactly the same, given the fact
that even a large but finite number of small things can appear
continuous, as digital images prove. Second, our mathematics would
seem exactly the same, and would even be true, so long as the model of
our mathematical theory was one of the countable models we were talking
about, and so long as any finite string that (in our world) defines a
real number like pi, or root 2, has a counterpart in this imaginary
world. Is it a physical impossibility? Why should it? Continuous
motion requires the truth of analysis, but analysis is true in the
imagined world. For example, the intermediate value theorem would be
true for definable continuous functions (i.e. what we call "definable
continuous functions").

So, the inhabitants of this world apparently speak exactly the same
language as us, but mean something slightly different by the phrase
"for all x", which for us would mean "for all definable x", whatever
sort of object x might be. In fact, they could even say that the reals
were uncountable. What they'd mean by this in our terms is that there
is no definable bijection between N and the definable reals, which
there isn't because then they could apply a diagonal argument and
define a real not in the image.

Then why would mathematics be "nothing whatever like mathematics in our
world."?

What we cannot do, of course, is to imagine that this possible world is
our world. For if so, there would be only countably many things, and
set theory would be true, but set theory says there are uncountably
many things. But then the inhabitants of this possible world would not
be able to imagine this either. Indeed, they could even say to
themselves, in their language "This possible world is not our world.
For if so, there would be only be countably many things, and set theory
would be true, but set theory says there are uncountably many things."
And this would be true, because "there would be only be countably many
things" expresses, in their language, the false idea that there would
be no power set of the naturals. But "power set of X" refers, in their
world, to all definable subsets of X! They simply cannot express what
we, as external observers can express.

On your argument that a relativity argument like this can "prove
anything", not so. The argument relies on the assumption that such a
world is possible (i.e. the supposition of its existence would not
result in contradiction) and that such a world would seem to its
inhabitants exactly as our world seems to us. Ockham, Leibniz,
Poincare and Einstein (and indeed Wittgenstein - the Private Language
Argument is a relativity argument) all used relativity arguments to
great effect.

>Nothing about the learning process seems to rule this possibility out.
> If the fixing of reference is
>performed solely by subjective mental acts, then there is nothing to
>prevent each of us fixing reference in a quite different way, which is
>another way of saying that (in respect of our public language),
>reference is not fixed at all.

> This seems extremely silly, at least in the present context. The
> fact that we can never be certain that what we mean by a term
> is the same as what someone else means by the same term has
> nothing to do specifically with the existence of uncountable
> sets or Skolem's "paradox" - again, it's the sort of thing
> you could use to refute anything anyone said about anything.

You don't seem to understand the argument, which is that the fixing of
reference cannot be performed solely by subjective mental acts, but
must include some sort of "public" baptism.

.