An Ultrafinite Set Theory [Theory]

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From: Barb Knox on 6 Mar 2010 14:22

In article <87aaul213y.fsf(a)dialatheia.truth.invalid>,
Aatu Koskensilta <aatu.koskensilta(a)uta.fi> wrote:

> Barb Knox <see(a)sig.below> writes:
>
> > Just a pedantic point, but "natural numbers" should be reserved for
> > the structure that is the *standard* model for the Peano Postulates.
> > There are other (non-standard) structures which also conform.
>
> This depends on what we mean by "Peano Postulates". "Peano arithmetic"
> almost invariably refers to first-order arithmetic, a formal theory that
> does have non-standard models. "Peano Postulates" often, and usually
> always outside logic, refer to the (essentially second-order) axioms
> presented by Peano -- and these are categorical.

Only if one specifies the use of "the full second-order semantics",
which IMO is a bigger hammer than just stipulating "the standard model
of N" in the first place.

--
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| BBB b \ Barbara at LivingHistory stop co stop uk
| B B aa rrr b |
| BBB a a r bbb | Quidquid latine dictum sit,
| B B a a r b b | altum videtur.
| BBB aa a r bbb |
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From: FredJeffries on 7 Mar 2010 14:34

On Feb 24, 5:52 pm, RussellE <reaste...(a)gmail.com> wrote:
> I searched for "ultrafinite set theory" and all
> I found was a remark by Zermelo:
> "The 'ultrafinite antinomies of set theory',
> which the scientific reactionaries and
> anti-mathematicians eagerly and delightedly
> call on in their campaign ..."
>
> I get the impression Zermelo didn't like
> ultrafinitists.
>

You should not toss Zermelo off so lightly. If you would care to more
carefully examine the context of the above quote, you may find a clue
to a way of solving YOUR problem, i.e. finding a workable ultra-
finitistic set theory.

I took the liberty to search for the source of this quote and found
out that Zermelo used the term "Ultrafinite" to modify "Set", or
rather "non-set", and not "Set Theory" and seems to me to be refering
to what is known as Proper Classes.

The source is "Uber Grenzzahlen und Mengenbereiche: Neue
Untersuchungen uber die Grundlagen der Mengenlehre", Fundamenta
Mathematicae, 1930, 16: 29-47 which I won't even try owing to my high
school German classes being so long ago.

But I did find a translation of the relevant paragraph in Geoffrey
Hellman, "The Many Worlds Interpretation of Set Theory", Proceedings
of the Biennial Meeting of the Philosophy of Science Association, Vol.
1988, Volume Two: Symposia and Invited Papers (1988), pp. 445-455 on
page 447

<quote>
The "ultrafinite antinomies of set theory", which the scientific
reactionaries and anti-mathematicianse agerly and delightedly call on
in their campaign against set theory, these specious
"contradictions"a, rise solely from a confusion between the non-
categorical axioms of set theory and the various particular models of
them: What in one model appears as an "ultrafinite un- or super-set"
is in the next higher domain a perfectly good "set" with a cardinal
number and order type of its own, which serves as the foundation stone
for the construction of the new domain. The boundless series of
Cantor's ordinal numbers gives rise to an equally boundless series of
essentially different models of set theory, in each of which the whole
classical theory can be expressed. The polar opposite tendencies of
the thinking mind, creative progress and all-embracing completeness,
which lie at the root of Kant's "antinomies", find their symbolic
expression and resolution in the concept of the well-ordered
transfinite number-series, whose unrestricted progress comes to no
real conclusion, but only to relative stopping-points, the "boundary
numbers" that divide the lower from the higher models. And so the
"antinomies" of set theory, properly understood, lead not to a
restriction and mutilation, but rather to a further development (whose
scope cannot yet be taken in) and enrichment of mathematical science.
(Zermelo 1930, p. 47)
</quote>

See also José Ferreirós Domínguez, Labyrinth of thought: a history of
set theory and its role in modern mathematics, p.376
http://books.google.com/books?id=DITy0nsYQQoC&pg=PA376&lpg=PA376&dq=ultrafinite&source=bl&ots=na0S3ACwwJ&sig=ZEcJdhmKiHQ0_y81UnvlWPDwKX0&hl=en&ei=j-GTS6GwL4mgsgPo9an9Aw&sa=X&oi=book_result&ct=result&resnum=10&ved=0CC0Q6AEwCQ#v=onepage&q=ultrafinite&f=false

Take note particularly the explanation: The "ultrafinite non-sets" of
one model become legitimate sets in the next model.
and Zermelo's quoted above: What in one model appears as an
"ultrafinite un- or super-set" is in the next higher domain a
perfectly good "set" with a cardinal number and order type of its own,
which serves as the foundation stone for the construction of the new
domain.

Others have pointed out the seeming shifting sand paradox of using
numbers which are claimed not to exist or systems of which it is
claimed that they are inconsistent to try to create a solidly based
finitistic system.

Let us rather take to heart the insights of Nelson, Yessenin-Volpin
and , yes, even the notorious Cantorian Zermelo, and solidly build
upon a solid foundation.

Let us turn the Yessenin-Volpin method around: It has been conceded in
this thread that there is a consistent system with one object. Use
this one-object system to faithfully construct a two object system (a
la Zermelo) and submit it for certification-of-quality to the
inspectors and ask if it is a valid system. If you have done your work
well, they will surely (although after a bit of time for the
inspection added to the time spent constructing the extension) they
must concede the verity of your construction. I they didn't they would
be open to the criticism of unfaithfulness to the methods of Zermelo.

Having a two-object system, proceed to construct a four object system
and submit it for approval. Naturally, it will take a bit longer to
construct and have verified this system.

You may ask how to proceed in constructing these extensions without
running into the paradox of having to use numbers which don't yet
exist in the system.

Here is where to use Nelson's observation, with which anyone who has
any knowledge of computer system should have noticed: Tally numbers
(cardinals) are different sorts of numbers than positional numbers.
(Ed Nelson, Predicative Arithmetic p.173 available online at
http://www.math.princeton.edu/~nelson/books/pa.pdf)

<quote>
Originally, sequences of tally marks were used to count things. Then
positional notation -- the most powerful achievement of mathematics--
was invented. Decimals (i.e., numbers written in positional notation)
are simply canonical forms for variable-free terms of arithmetic. It
has been universally assumed, on the basis of scant evidence, that
decimals are the same kind of thing as sequences of tally marks, only
expressed in a more practical and efficient notation. This assumption
is based on the semantic view of mathematics, in which mathematical
expressions, such as decimals and tally marks, are regarded as
denoting abstract objects. But to one who takes a formalist view of
mathematics, the subject matter of mathematics is the expressions
themselves together with the rules for manipulating them-- nothing
more. From this point of view, the invention of positional notation
was the creation of a new kind of number.
</quote>

Instead of talking about tally numbers and decimals, I will follow
Rudy Rucker and use "Counting Numbers" and "Information Numbers". In
each system that you construct, consider the numbers therein as the
counting numbers. From them create that system's information numbers
using a positional notation scheme. Then when you create the next
system we use the previous system's information numbers as the new
system's counting numbers in a Zermeloization process.

I leave the details (and the glory) to others. I will just quote
Nelson again (Predicative Arithmetic p.75) as an answer to the "why
bother/so what" questions: There is a story of a bank employee who was
told to count a bundle of bills to verify that there was actually a
thousand of them. The employee began to count them: 1, 2, 3, ..., 61,
62, 63-- and then stopped, being convinced that since it had checked
perfectly all that way it must be correct.

From: FredJeffries on 7 Mar 2010 14:47

From: Transfer Principle on 7 Mar 2010 22:18

On Mar 4, 4:55 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote:
> Transfer Principle <lwal...(a)lausd.net> writes:
> > We're opponents, and so I expect nothing less.
> Newberry and I disagree on whether his intuitions are reasonable.
> I've never seen any real sense to his ideas, but I have no contempt
> for him. (This is not true for certain others, Andrew Usher, for
> example, who do behave in a truly contemptible manner.)

Ah yes, Newberry. I've noticed that thread back in its early
stages, but I didn't post there, since I didn't feel that
avoiding vacuous truth was worth defending.

But in that thread, Marshall Spight made a comment about
Newberry's idea and explicitly stated that it applied
equally to this thread as well. And so my next post in this
thread will be a response directed to Spight in which I
respond to that comment.

Also, that thread is now suddenly turning into another
discussion between Nam Nguyen and the anti-"cranks." I have
yet to decide whether I wish to make any comments there now.

From: Transfer Principle on 8 Mar 2010 01:04

On Mar 3, 10:18 pm, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote:
> Axiom of making sense: if x in y then Ur(x) and Set(y) or Set(x) and
> Class(y); and if x < y then Ur(x) and Ur(y).
> Axiom of well-ordering: < is a well-ordering of Ur; that is, < is a
> total order, and if Set(x) and there is a y in x, then there is a
> <-smallest y in x.
> Axiom of this or that: 0 is the <-least urelement; T is the <-greatest
> element.
> Comprehension for sets: if P(y) is a formula then all (universal
> closures of) formulas of the form
> (Ex)(Set(x) and (y)(y in x <--> Ur(x) & P))
> with the usual proviso on free variables in P.
> Comprehension for classes: if P(y) if a formula then all (universal
> closures of) formulas of the form
> (Ex)(Class(x) and (y)(y in x <--> Set(x) & P))
> with the usual proviso on free variables in P.
> There, a wonderful ultrafinite theory. (Those in the know may recognise
> this theory as a variant of third-order successor arithmetic with
> top. For arithmetic with top see e.g. Raatikainen's paper
> http://www.mv.helsinki.fi/home/praatika/finitetruth.pdf,

Interesting theory. Of course, since RE is the poster who
objected to standard theory, it's up to RE to make the final
decision as to whether this theory is acceptable or not.

According to the link given by Aatu in the post above, the
theory given by axioms A1-A10 has infinite models, including
the standard model of arithmetic. I'm not sure whether RE
will find that acceptable or whether he demands a theory all
of whose models are finite. But of course, as I myself found
out, it's far easier to give a theory with both finite and
infinite models than it is to give one with arbitrarily large
finite models but no infinite models.

> Now what? What use are we to put this theory to?

Presumably RE wants us to use this theory anywhere where we
normally use a standard theory, since he finds the standard
infinitary theory objectionable.

Thanks for the theory and link, Aatu!

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