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From: Transfer Principle on 5 Aug 2010 00:45
On Aug 2, 8:56 pm, Transfer Principle <lwal...(a)lausd.net> wrote:
> This post is already long enough, so I stop here. The next
> thing to consider would be the dual to Infinity. This axiom
> needs to tell us that all sets are infinitely divisible, so
> that the empty set 0 can't exist. One might call this axiom
> the "Axiom of Infinitesimal."

I've been thinking about how to come up with an axiom that
states the existence of infinitesimals.

At first, we might think that since according to RF,
infinitesimal and infinity are dual concepts, we could
take the dual of the Axiom of Infinity.

But let's see what happens:

ZF Axiom of Infinity:
Ex (0ex & Ay (yex -> yu{y}ex))

The dual of 0 is V, but then one has to wonder what the
dual of yu{y} is supposed to be. For that matter, we
wonder what the dual of {0} is. Since {0} is a set such
that the only element of {0} is 0, its dual ought to be
a set such that it is a structure of only V. But there
is already such a set -- namely V itself! So all of the
von Neumann naturals would collapse to V.

Instead, what if we tried something like:

Ex (Ay (xsy -> Ez (zsy & xsz & ~x=z)))

But this still fails -- we know that x is not equal to
V, but there need not exist any sets other than x and V.

The reason that we are having so much trouble trying to
define "infinitesimal" is that, as we found out back in
one of the IST (another infinitesimal theory) threads,
"infinitesimal" can't be defined in first-order theory
without introducing a new term called "standard," and
even then there is no set of infinitesimals.

So any formula I can write which appears to define
"infinitesimal" would hold for some set that's larger
than infinitesimal as well!

Let me think about this for a while longer...
From: R. Srinivasan on 5 Aug 2010 02:33
On Jul 29, 3:22 pm, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote:
> Transfer Principle <lwal...(a)lausd.net> writes:
> > If so, then perhaps this is one way to raise the reputations of the
> > sci.math finitists. We're to convince them just to leave the existence
> > of infinite sets open rather than assert their non-existence, just as
> > the mainstream leaves the existence of large cardinals open rather
> > than assert their non-existence.
>
What makes you think that it is necessary for sci.math finitists to
"raise their reputations" from the viewpoint of "the mainstream",
which basically consists of a bunch of people who not only believe in
infinitary reasoning, but have a large stake in perpetuating it
irrespective of any arguments against infinite sets? A logical
argument against infinitary reasoning can only win respectability
amongst people who believe in logic.

Isn't it obvious that finitism will become much more respectable if
finitists can give a sound argument for the non-existence of infinite
sets, rather than just assume it away? Of course such an argument
will displease the mainstream and cause its originator to lose
"respectability" from their viewpoint. But at least I, as a finitist,
am not interesting in winning friends and influencing people within
the mainstream, given the kind of people who fit that description. The
"respectability" that I desire is in an absolute sense, from honest,
unbiased people who are truly dedicated to their professions. If one
has to wait for such people to show up, then so be it.

It is far more important to stick to one's convictions and be honest
and professional, rather than give in to such political considerations
as you advocate.

>
> The obvious thing is just to do interesting finitist mathematics and
> leave all the silly harangues against infinity out of it. What formal
> theory we might, for this or that purpose, formalize such mathematics in
> is of not much consequence or interest -- and indeed we can't decide on
> such formal matters until we have a well developed body of mathematical
> results, principles, techniques, modes of reasoning, and so on, to
> formalize. The development of mathematics simply does not consist of
> people putting forth random formal theories, and fundamental questions
> about foundations are very rarely purely formal.
>
Fundamental questions about foundations are formal and logical in
nature. You first have to lay out a sound framework before you do any
mathematics. If the framework is dubious and questionable, be assured
that people will question it, irrespective of what you consider as
"interesting" and what Transfer Principle considers as "respectable"
within the mainstream.

RS

From: MoeBlee on 5 Aug 2010 12:54
On Aug 4, 11:45 pm, Transfer Principle <lwal...(a)lausd.net> wrote:
> On Aug 2, 8:56 pm, Transfer Principle <lwal...(a)lausd.net> wrote:
>
> > This post is already long enough, so I stop here. The next
> > thing to consider would be the dual to Infinity. This axiom
> > needs to tell us that all sets are infinitely divisible, so
> > that the empty set 0 can't exist. One might call this axiom
> > the "Axiom of Infinitesimal."
>
> I've been thinking about how to come up with an axiom that
> states the existence of infinitesimals.

An axiom added to what other axioms?

We can prove the existence of a number system with infinitesimals from
ZF by adding "every filter can be extended to an ultrafilter", which
is entailed by the axiom of choice but is weaker than the axiom of
choice.

It's not clear to me that you understand that non-standard analysis
can be done in ZFC and that, indeed, the non-standard analysis
introduced by Robinson was within ZFC.

MoeBlee
From: Transfer Principle on 6 Aug 2010 23:51
On Aug 4, 11:33 pm, "R. Srinivasan" <sradh...(a)in.ibm.com> wrote:
> > Transfer Principle <lwal...(a)lausd.net> writes:
> > > If so, then perhaps this is one way to raise the reputations of the
> > > sci.math finitists. We're to convince them just to leave the existence
> > > of infinite sets open rather than assert their non-existence, just as
> > > the mainstream leaves the existence of large cardinals open rather
> > > than assert their non-existence.
> What makes you think that it is necessary for sci.math finitists to
> "raise their reputations" from the viewpoint of "the mainstream",
> which basically consists of a bunch of people who not only believe in
> infinitary reasoning, but have a large stake in perpetuating it
> irrespective of any arguments against infinite sets? A logical
> argument against infinitary reasoning can only win respectability
> amongst people who believe in logic.

By "raise their reputations," I mean convince the mainstream
to stop using five-letter insults against them.

For example, right here in this thread, MoeBlee refers to RF
and TO as "cranks." Many sci.math finitists, especially
Herc, HdB, and WM, are regularly so insulted. I want to
make it so that these finitists are insulted as much, and the
only way to do so is to "raise their reputations." (Note that
HdB and Srinvasan have very similar opinions re: finitism.)

> Isn't it obvious that finitism will become much more respectable if
> finitists can give a sound argument for the non-existence of infinite
> sets, rather than just assume it away?  Of course such an argument
> will displease the mainstream and cause its originator to lose
> "respectability" from their viewpoint. But at least I, as a finitist,
> am not interesting in winning friends and influencing people  within
> the mainstream, given the kind of people who fit that description. The
> "respectability" that I desire is in an absolute sense, from honest,
> unbiased people who are truly dedicated to their professions. If one
> has to wait for such people to show up, then so be it.
> It is far more important to stick to one's convictions and be honest
> and professional, rather than give in to such political considerations
> as you advocate.

Good for you, Srinivasan!

If, like Srinivasan, one doesn't care about "respectability" from
the mainstream or being called a five-letter insult, then don't
worry about it. I know that I'm going to be so insulted as well,
but I'll continue to defend finitists, infinitesimalists, and other
non-mainsteam posters regardless of respectability.
From: Transfer Principle on 7 Aug 2010 00:14
On Aug 5, 9:54 am, MoeBlee <jazzm...(a)hotmail.com> wrote:
> > I've been thinking about how to come up with an axiom that
> > states the existence of infinitesimals.
> An axiom added to what other axioms?
> We can prove the existence of a number system with infinitesimals from
> ZF by adding "every filter can be extended to an ultrafilter", which
> is entailed by the axiom of choice but is weaker than the axiom of
> choice.
> It's not clear to me that you understand that non-standard analysis
> can be done in ZFC and that, indeed, the non-standard analysis
> introduced by Robinson was within ZFC.

Yes, I'm already aware of the construction of Robinson's
hyperreals from ultrafilters using AC.

But the infinitesimals as described by RF and TO can't
so be constructed. TO's infinitesimals directly contradict
ZFC, since, for example, to TO the set size of the set N
of natural numbers is invertible as an infinitesimal, but
one can't invert the cardinalities of ZFC to obtain any
sort of infinitesimal hyperreals. Furthermore, TO has
_expressly_ rejected hyperreals (and surreals) as
equivalent to his infinitesimals.

So let me correct what I wrote:

I've been thinking about how to come up with an axiom that
states the existence of _TO's_ infinitesimals.

And we need a theory other than ZF, and an axiom other
than the ultrafilter axiom, to obtain TO's infinitesimals.

One thing that I was wondering, though, is how to deal with
zero and negative values. According to RF, we want to
divide a real number into smaller and smaller parts (or
structures) corresponding to smaller infinitesimals, but at
how do we obtain zero or negative values?

Interestingly enough, in another thread, there is discussion
of a system of "infinitesimals" where the symbol "z" is an
infinitesimal, but there is no standard 0. Perhaps the
infinitesimals of TO will turn out the same way. (For that
matter, does TO ever discuss negative values?)
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