Re: Anti-diagonalist page

On 8 Sep 2005 20:20:30 -0700, slaterbh@xxxxxxxxxxxxxxxxxx wrote:

>It seems possible that there are other points of contention,
>I'm not sure, because of my uncertainty regarding exactly
>what Slater means by various things (he doesn't seem too
>interested in clarifying):
>
>It seems to me you are wanting the impossible. If you are not already
>into the discussion about Abstraction Principles and their ontological
>presumptions, then no-one can digest the issues for you in a paragraph.
> Likewise with the content of Wright's piece on Skolem; and Lavine's
>book on infinity. All this requires quite a bit of reading. Then
>there's the full text of my electronic paper, which was mentioned right
>at the start of this string, and which, one might assume, would be some
>help in 'understanding Slater'. Of course, if the idea is just to have
>a brief chat about this and that, then nothing too lengthy or difficult
>will be of interest.

My first reaction was that this is very convenient - it's just
too hard to explain.

Then I downloaded your paper and read through it quickly.
I didn't see anything that appeared to me to address any
of the objections I've made to what's in that brief
excerpt that has appeared here. In detail:

(i) Most of the paper seems to be about paradoxes associated
with self-reference. By sheer coincidence I've already stated
quite explicitly, before my comments on your comments on
"Skolem's paradox", that if you were to claim that there was
something to be resolved regarding, say, Russel's paradox
that axiomatic set theory just ignores I wouldn't find
anything wacky about that. It seems you claim to have a
uniform solution to paradoxes of self-reference; I have
no quarrel with that.

But regarding "numbers":

(ii) First, we should note that mathematicians simply
do _not_ say that the number of reals is larger than
the number of naturals, except when they're speaking
very informally - instead they say that the cardinality
of the reals is greater than the cardinality of the
naturals. This is a good thing, because it makes it
clear that what they're talking about is not the
common notion of "number". I'm going to use the
word "number" instead of "cardinality" below, simply
because the word "number" appears in quotations from
your paper, but we need to bear in mind that what I'm
talking about is _cardinality_. (If cardinality is
not what you're talking about then you're attacking
a strawman, refuting claims that no mathematician
actually makes!)

(iii) You claim to be refuting the statement

(*) There exist infinite sets with different numbers
of elements.

But nowhere in the paper do I see a definition of
exactly what you _mean_ when you say (*). Without
a definition (*) is simply meaningless. You really
do need to clarify what you _mean_ by (*): If you
mean one thing then you're simply wrong, while if
you mean something else then my reaction would be
"ok, you're right about that, so what? That has
no relevance to what we mean by (*) in mathematics."

You really should explain what you mean by (*).
I've asked what you mean by various things -
when you say that I need to read your paper I
assume that I'll find definitions there. I
didn't. Maybe I missed it. Whether the definition
appears in (*) or not: What _do_ you mean by (*)?

(iii) In the paper I see this:

" The presupposition in Set Theory is that

(P)(En)(nx)Px.

i.e. that, for each P, there is a determinate number, such that there
is that number of P's."

Good of you to include the "i.e." or I might not have
known what the notation (nx) meant.

This statement is simply _false_. Exactly why it's
false depends on exactly what we mean by "a determinate
number", which is alas still not clear to me, but:

(a) of course if "a determinate number" means "a finite
number" a priori then (P)(En)(nx)Px is false, but
that's a totally straw man, nobody ever claimed such
a silly thing.

(b) if "a determinate number" means a cardinal, then
what you say is false as well - there is no such
"presupposition" in set theory. For some P, as you
point out, (En)(nx)Px is not even a theorem of set
theory (for example if Px means x=x.) For other
Px, for example "x is a natural number" or
"x is a real number", (En)(nx)Px is a theorem
of set theory, but it is not a "presupposition",
it is a _theorem_. You haven't said anything
about why the proof of that theorem is wrong.

(iv) Similarly, you say

"for no justification for saying the natural numbers have a number can
be given. For Dedekind defined infinite sets as those that could be
put into one-one correlation with proper subsets of themselves, so the
criteria for 'same number' bifurcate: if any two such infinite sets
were numerable, then while, because of the correlation, their numbers
would be the same, still, because there are items in the one not in
the other, their numbers would be different."

But without a definition the final "their numbers would be
different" is not susceptible to proof.

Reading the paper really didn't answer any of my objections.
Instead I find many examples of errors with the propery
that I've already explained _why_ they're errors, even
before knowing that they were errors you were actually
making. For example:

(v) I read this:

(**) "The key question therefore is: if there is a determinate number
of natural numbers, then by what process is it determined? Replacing
'the number of natural numbers' with 'Aleph zero' does not make its
reference any more determinate."

What you say is the key question may well be the key question.
And it is true that the definition of Aleph_0 is in fact
"the number of natural numbers", or (much) more properly
"the cardinality of the set of natural numbers".

And it is true that _if_ that were the _only_ definition
in the universe then we would be left without any determination
of what Aleph_0 actually was. But that is a straw man, because
there _are_ other definitions in set theory. Defining Aleph_0
to be the cardinality of the set of natural numebers _does_
make the reference of "Aleph_0" determinate, because we
have _previously_ given a definition of the cardinality
of any set.

There are at least two definitions that have been used
at various times:

Definition 1: If S is a set then the cardinality of S is
the (proper) class of all sets which are in 1-1 correspondence
with S.

That's a simple intuitive definition, but it has the
(_aesthetic_) flaw that a cardinal is now a huge
ungainly sort of object.

Also I suppose that Definition 1 uses a certain amount
of "abstraction" - if we insist there are problems
with abstraction although we can't even give a hint
what the problems are in a short space then we may have
problems with that. Luckily that's not the standard
definition these days - the more standard definition
doesn't depend on abstraction this way.

The more standard definition these days relies on the
Axiom of Choice: Using AC we prove that if S is a set
then there is a smallest ordinal which can be put
into 1-1 correspondence with S ("smallest" in the sense
of the natura order on the ordinals, not in the sense
of cardinality.) Having proved that we then define

Definition 2: If S is a set then the cardinality of S
is the smallest ordinal which can be put into 1-1
correspondence with S.

With Definition 2 it's easy to say exactly what
Aleph_0 is: If we define Aleph_0 to be the cardinality
of the natural numbers then Aleph_0 turns out to be
exactly {0,1,2,...} (the intersection of all the
infinite ordinals).

*********************

Of course when I say this or that is deteminate I'm using
the word "determinate" in a somewhat abstract sense, meaning
precisely determined by the axioms of set theory. Regarding
this see Jiri's post: That's really the only sort of
determinateness we can expect. You talk a lot about 1
and 2, but you can't tell me exactly what 1 _is_ (or
if you can I'd be interested to hear about it.)
Where I come from 0 = {}, the empty set, and then
1 = {0}. That's "determinate" _given_ some basic
assumptions about set theory, not in any absolute
sense - Definition 2 is determinate in the same sense.

You should really tell me what you mean by (*),
and maybe also answer some of the points above.
Just saying you're right but it takes too long
to explain is not very convincing - a simple
definition of the _meaning_ of (*) should not
take that much space. (Hmm, if the definition
of (*) really does take too much space to include
in a usenet post then we have proof of the
strawmanosity of all this, since the definition
used in mathematics is very simple, and if you're
using some other definition then your comments
have no relevance to the math.)

************************

David C. Ullrich
.

Relevant Pages

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