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From: David R Tribble on 2 Jul 2010 23:35
David R Tribble wrote:
>> But no one (except perhaps Zuhair) has offered anything resembling
>> a cogent, coherent, consistent theory.
>

Transfer Principle (Walker) wrote:
> I'd argue that Srinivasan, in the current thread that he shares
> with Charlie-Boo, offers the theory NBG-Infinity+"D=0."

That's great. I have not read those posts. If they really
have something interesting, and most importantly, consistent,
to offer, then more power to them.


David R Tribble wrote:
>> Tell us, in all the time that you've been re-stating their theories,
>> have you ever come up with anything that really would stand
>> as an interesting alternative to standard theories?
>

Transfer Principle (Walker) wrote:
> It depends on what one means by "interesting alternative to
> standard theories."
> If one demands that such alternative, say, provide for an
> axiomatization for the sciences, then not yet. But keep in
> mind that ZFC has already had a century's head start. One
> should not expect a post that contains all the axioms and
> schemata of a proposed alternative, evidence that the theory
> isn't trivially inconsistent, _and_ evidence that the theory
> provides for the sciences _all_ in one concise post that
> Tribble or anyone can read in 15 minutes.

True, but it should be obvious with a few minutes' worth
of text whether the ideas are worth pursuing or not. Illogical
arguments, undefined and vague terminology, and blatant
misunderstandings of basic concepts are large, bright clues
about the worth of "new" ideas.


> Theories take time to develop. But most posters who already
> accept full ZFC have neither the patience nor any incentive
> to wait as we develop the theory. They're much more likely
> to ignore such developmental posts. On the other hand, it's
> easy for them to jump on theories that are inconsistent,
> incoherent, and so on, with five-letter insults.

I don't read as many sci.math posts as you, but my observation
leads me to the opposite conclusion. Anyone with a reasonable
idea is given lots of free advice for improving their concepts.

The problem is that most "new idea" posters are cranks, and
behave like cranks.

Look at the exchanges in the last half of this thread, for
example. Sometimes it's like pulling teeth to get a straight
answer out of Tony on even simple ideas of his.


> But I don't believe that a poster who objects to ZFC for one
> reason or another should be forced to use it merely because
> there is no real alternative

You say that a lot, but who is forcing anyone to use ZFC?
All we're asking is for logically sound arguments.


> Srinivasan is a finitist, so
> he opposes ZFC. TO wants sets to be strictly larger than
> their proper subsets, so he opposes ZFC. Since nowhere in
> the physical world can one prove that an infinite object
> exists, it can't be _ruled_out_ that there exists a theory
> satisfying most of Srinivasan's, or TO's, or AP's, or RF's,
> desiderata and still axiomatize the sciences.

The important difference between Tony and the other cranks,
and someone like Srinivasan (whose posts I have not read), is
that the former despise and reject standard theory. They are
not trying to create a theory or model above and beyond
ZFC, but rather they are trying to condemn ZFC, discard it,
and replace it with something "better" of their own invention.
That attitude is what makes them cranks. That, and being
completely illogical in most of what they write.

Assuming that Srinivasan does not take that approach
(and I'm giving him the benefit of the doubt here), I'd say
that he acts like a mathematician instead of a crank,
doing actual mathematics instead of condemning standard
ideas without justification. I take it that his "rejection" of
ZFC is of a totally different character than the rejection given
by the cranks.

Oh, and by the way, when you talk about things like "finite
within the model" and "outside the model", you are talking
way over Tony's head.
From: FredJeffries on 3 Jul 2010 10:08
On Jun 28, 12:47 am, Transfer Principle <lwal...(a)lausd.net> wrote:

> Let's state this formally. We start with ZF and add to its language a
> new primitive symbol -- but not "tav" or "size" or anything like that,
> but the symbol "<=" a two-place infix predicate symbol. We also define
> another symbol "~=" also a two-place infix predicate symbol:
>
> x~=y <->def (x<=y & y<=x)
>
> Here are some axioms for <= and ~=:
>
> 1. <= is a preorder.
> 2. Axy (x<=y v y<=x)
> 3. Axy ((card(x) = card(y) & card(x) < omega) -> x~=y)
>
> (That is, x~=y agrees with cardinality for finite sets.)
>
> 4. Axy ((card(x) < card(y) & card(x) < omega) -> (x<=y) & ~(y<=x))
>
> (That is, x<=y agrees with cardinality for finite sets.)
>
> 5. Axyz ((x,y,z mutually disjoint & x<=y) -> xuz<=yuz)
>
> (Here "u" denotes Boolean union.)
>
> So far, this theory is consistent (assuming ZF is), since
> standard cardinality has all of the properies ascribed to
> the relations <= and ~=. (Notice that if "x<=y" denotes
> "there exists an injection from x to y," then "x~=y"
> denotes "there exists a bijection from x to y" because of
> the Schroeder-Bernstein Theorem.) But we ask, is there
> another set size (such as Bigulosity) that also has these
> properties as well as guaranteeing that sets don't have
> the same set size as any of their proper subsets, or can
> we prove from these properties that there _must_ exist a
> set and its proper subset with the same size.
>
> In other words, can we prove the following:
>
> Eab (b proper subset a & a<=b)

I may be missing something completely obvious, but wouldn't something
along the following show that your 1) - 5) and

> 6. Aab (b proper subset a -> ~(a<=b))

is not inconsistent?:

Suppose the universe can be well ordered.

Define a<=b by

i) if a and b are both finite then a<=b agrees with cardinality

ii) if a is finite and b is infinite then a<=b

iii) if both a and b are infinite, use the characteristic function
with respect to the universal well ordering to compare them
lexicographically.

Thus, just as the (infinite) subsets of the Natural Numbers can be
mapped to (base-2 representations of) real numbers between 0 and 1 and
compared via the Real Number natural order we can compare any two sets
(in this well-orderable universe)

Of course this order may have other objectionable features, For
instance, if 1 is the first element of the well order then any
infinite set containing 1 will be larger than any infinite set not
containing 1.
From: MoeBlee on 3 Jul 2010 17:46
On Jul 2, 6:43 pm, Transfer Principle <lwal...(a)lausd.net> wrote:

> But most posters who already
> accept full ZFC

Whatever it means to "accept" ZFC,

> have neither the patience nor any incentive
> to wait as we develop the theory.

I don't have patience (actually, I have too MUCH patience) with the
posting tactics of certain people who might have ideas toward some new
theory or another. I don't have patience with their ignorance and
disinformative postings about mathematics, their incoherence and
various forms of irrationality, as well as their arrogance throughout.

I'm fine with someone who just says they're working on some pre-
formalized notions toward making a theory and gives commentary on what
they'd like to achieve and related matters involved.

> But I don't believe that a poster who objects to ZFC for one
> reason or another should be forced to use it merely because
> there is no real alternative.

Okay, fair enough. Who's forcing anyone?

MoeBlee

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