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From: Virgil on 5 Sep 2006 17:01
In article <44fda927(a)news2.lightlink.com>,
Tony Orlow <tony(a)lightlink.com> wrote:

> Actually, I am taking properties to be truth values applied to
> statements about a given object. "It is red" is a statement with a truth
> value close to 1 for most fire engines, and a truth value near 0 for
> most newspapers (except in old jokes). Fire engines have the property
> that that statement is usually true for them.

Actually many fire engines are not red. White and yellow are at least as
popular at least for newer engines, and are a god deal safer.
http://www.psychologymatters.org/solomon.html
From: Virgil on 5 Sep 2006 17:04
In article <44fdab7d(a)news2.lightlink.com>,
Tony Orlow <tony(a)lightlink.com> wrote:

> > That is again a clear demonstration of your ignorance. If you read just
> > the least bit on this subject, then you'd find how induction is sourced
> > in mathematical logic.
> >
> > MoeBlee
> >
>
> I find this last sentence vague because of the word "induction". If you
> refer to inductive proof, then I am familiar with the Peano axioms and
> the underlying logical construction which makes inductive proof valid.
> If you refer to inductive logic, the formulation of rules from instances
> of fact, then there are statistical methods coupled with feedback that
> make it possible. Which were you talking about?

The inductive methods of scientific investigation are no part of
mathematical logic. So that TO's question need not have been asked.
From: Virgil on 5 Sep 2006 17:16
In article <44fdac39(a)news2.lightlink.com>,
Tony Orlow <tony(a)lightlink.com> wrote:

> MoeBlee wrote:

> > You learned, as we all did, in about the fifth grade, that the naturals
> > are a subset of the integers are a subset of the rationals are a subset
> > of the reals. However, when we get into the real serious stuff, we see
> > that with certain constructions, that fifth grade notion of subset is
> > not precise and is sharpened into a rigorous formulation in set theory.
> >
> > MoeBlee
> >
>
> And yet, you are saying it's not technically correct. So, 1 is a natural
> but not a rational or real?

The 1 of the naturals and the 1/1 of the rationals and the 1.000... of
the reals are not identical, but there are isomorphic images of the set
of naturals with all its arithmetic within both the ordered field of
rationals and complete Archimedean ordered field of reals in which 1
maps to 1/1, or 1.000..., respectively.

So the rationals and reals each contain an image of the naturals in
which all of its essential properties are preserved.

When the distinction between the original naturals and these images
needs to be noted, one can refer to the images as the rational naturals
or real naturals, respectively. Otherwise, the distinction between
original and image is usually ignored.
From: Virgil on 5 Sep 2006 18:18
In article <44fdbf78(a)news2.lightlink.com>,
Tony Orlow <tony(a)lightlink.com> wrote:

> MoeBlee wrote:
> > Tony Orlow wrote:
> >> MoeBlee wrote:
> >>> Tony Orlow wrote:
> >>>> MoeBlee wrote:
> >>>>> Tony Orlow wrote:
> >>>>>> So, is it technically incorrect to say 1 = 3/3 = 1.000... = 0.999...? I
> >>>>>> never thought so.
> >>>>> Those are notations that can be understood precisely only in context of
> >>>>> the particular treatment in which they occur.
> >>>> BLAM!! That's exactly what I'm saying. If a particular treatment cannot
> >>>> distinguish between two elements, then they are the same,
> >>> By a 'treatment' of a formal system, I don't mean the formal system
> >>> itself, but rather an informal presentation of a formal system. For
> >>> example, a treatment would include the English text in a textbook that
> >>> discusses a formal system. So the treatment includes both the formal
> >>> system and the informal presentation of that formal system. For
> >>> example, a textbook, lecture, oral explanation, or even an Internet
> >>> post may be considered treatments or at least fragments of some
> >>> presumable overall treatment.
> >>>
> >>> And your "cannot distinguish" is too vague and broad. We prove equality
> >>> of objects in a theory in very specific ways. We don't just say, "Well,
> >>> no one seems to be able to distinguish them, so they must be the same."
> >>> For example, in set theory, x=y <-> Az(zex <-> zey).
> >
> >> That's assuming x and y are sets. If they are atomic objects,
> >> urelements, then A z not(zex v zey). In this case, neither x nor y has
> >> any elements, but does that mean they are the same?
> >
> > In such a theory, we have a primitive predicate for 'is a set'. Then
> > the axiom of extensionality is:
> >
> > (x is a set & y is a set) -> (x=y <-> Az(zex <-> zey)) and we would
> > have some kind of axiom saying there exists a unique set x such that Az
> > ~zex.
> >
> > That allows that there may be distinct urelements.
>
> Indeed, or we may NOT allow such things, if we adopt a rule of the form
> x = S: A P P(x) = P e S. That is, every object is a set of values which
> each property has when applied to that object. The truth value of each
> statement that may be made about x is EQUAL to the truth value of that
> property's membership in x. While some restrictions need to be made
> regarding what constitutes a property, this general notion seems sound, no?

TO presents bits and pieces, each of which, he claims, might be a part
of some sort of theory, but we cannot tell how those parts fit together
towards making a coherent theory without seeing all of them together.

is.
>
> That's nice. Is there a difference between an abstract object like a
> set, and its definition?

In ZF, et all, sets are not defined, they are among the undefined terms
which every mathematical theory must have.

Whenever one attempts to define everything, one ends, at some point,
with circularity.
From: Virgil on 5 Sep 2006 18:41
In article <44fdc460(a)news2.lightlink.com>,
Tony Orlow <tony(a)lightlink.com> wrote:

> MoeBlee wrote:

> >
> > Better would be a=b <-> AP(P(a) <-> P(b)).
>
> The difference between = and <-> disappears when logical truth values
> are quantities from 0 through 1, so I don't see that as any better, but
> equivalent.

"0 through 1"?

Does TO expect to find any truth values strictly between 0 and 1?

> I am aware that there are difficulties defining what constitutes a valid
> property in this sense, as Russell's Paradox demonstrates, but I think
> the kind of statement that produce such issues can be identified. That
> would be an interesting discussion....

In ZF, predicate definition of sets is limited to defining subsets of
sets which are otherwise known to exist, so that Russell's paradox is
vanquished.

Absent some mechanism of similar effectiveness, TO's system will crash.

>
> Ummm.... Isn't each isolated theory "subjective" in terms of the
> properties that it explores? If there is no universal system of cohesive
> mathematics, then this is surely the case. In the example of the
> staircase in the limit vs. the diagonal line, point-set topology cannot
> discern the two objects because it looks only at proximity of
> corresponding points. When one defines the objects using a
> segment-sequence topology, as I suggested, there is a very discernible
> difference between the two objects, namely, as I intuited from the
> beginning of that discussion, in direction of the curve.

But in TO's case, the objects are not mere sets of points but much more
complicated structures, about which TO asserts many properties he has
not proved which do not hold for mere sets of points.



> Thus, from the
> "subjective" perspective of sets of points, they are equal, but from the
> "subjective" perspective of sequences of segments, they are not. So, are
> they equal?

The issues are not subjective, but matters of definition. In standard
mathematics representing such objects by sets of points is standard, and
as TO wants them represented is not.

>
> Two objects are equal only if there exists no way to distinguish them.

If they are sets, they are indistinguishable if and only if every
member of either is a member of both and every non-member of either is a
non-member of both.

If they are TO's constructs and they are anything other than sets, it is
up to him to define what he means by equality.

> How do we know if this is the case? By enumerating all possible
> properties of each. Can we do that? No. We can only say that, given the
> set of properties under discussion in any given theory, the two are not
> distinguishable, within that theory. We cannot say that they are
> absolutely the same object.

That is the trouble with any such property definitions, you never can
tell whether you have considered all properties.

> > That's a confused view of the axiom of extensionality and the role of
> > variables.
>
> The perception of confusion would appear to be a subjective and rather
> relative phenomenon.

The only subject being confused by it is TO

> > For any PARTICULAR z, it's a property of y that z is or is not a member
> > of y. That doesn't entail that y IS the set of properties that y has.
>
> Consider each object in the universe to be a bit. Does each unique set
> correspond to a unique bit string

But as every "object" in TO's universe is a bit, every bitstring is a
bit also and one does not need any strings of bits separately.



> > Please just read a book on logic.
>
> Please just think hard. :)

TO rekes not his own rede.
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