Re: Cantorian pseudomathematics

Tony Orlow wrote:
> > No, I'm not. You claimed that set theory contradicts "the more solid
> > computational
> > axioms of the rest of mathemtics, and reality in general." I asked what
> > are these "more solid computational axioms of the rest of mathematics".

> I already pointed to infintie series, where 1+1+1+1.... diverges without
> question, but for some reason Cantorian set theorists deny that the supposedly
> infinite number of incrementations involved in generatiing the naturals
> produces any infinite values.

That sequence of sums does not converge. You don't need to posit that
as an axiom. It's a theorem of set theory. It is only your sloppy
thinking and ignorance of mathematics that makes you conflate two
separate things. You have not shown a FORMULA AND ITS NEGATION, because
the theorem of set theory that that sequence does not converge does not
contradict the theorem of set theory that no natural number is
infinite. The only reason you THINK there is a contradiction is that
you think of set theory in terms of your OWN undefined informal picture
of "incrementations" and 'generating" while whatever you THINK these
are and whatever way you THINK they work in set theory has nothing to
do with what ACTUALLY goes on in set theory.

> There's one contradiction for you.

A contradiction is a formula and its negation. You haven't shown one.

> You want an
> axiom? If the terms of the series do not have a limit of 0 as n->oo, then the
> series diverges. You want it in first order logic? I am sure you can translate
> that much more easily than I, so i am not doing your homework for you.

That will be the day when you do my homework for me. I don't need you
to formalize the theorem non-convergence of the sequence, since I've
already done that. What YOU need to formalize is the theorem that no
natural number is infinite so that YOU will realize that one is not the
negation of the other and that you don't have a way to extract a
contradiction from the pair.

> Can we consider it an axiom that a^b is finite for finite a and b, but not for
> infinite a or b and nonzero a and b?

Forget about axioms here. We already have theorems. For x,y finite, x^y
is finite. For either x or y infinite and x not=0 and y not=0, x^y is
infinite.

> If so, then perhaps we can agree that the
> set of all strings of length L, with a finite alphabet of size S, is S^L, which
> cannot be infinite unless L is infinite?

Correct. We ALREADY agreed. It's a theorem of set theory.

> If we consider digital number systems
> in this context, and the fact that each digital number has an implicit infinite
> leading string of 0's,

What are we talking about? Real numbers with digital representation? If
so, forget about "implicit" leading strings. Set theory doesn't have
"implicit" things.

> can we not say that every set of strings of a given
> length L includes as a subset the strings of all lesser lengths, since the
> leading 0's don't change the value of the string?

Forget about "leading zeros". We're talking about strings of length L.
And no, the set of strings of length L does NOT have as a subset the
strings of length less than L. The set of strings of length L has BY
DEFINITION only strings of length L.

> If so, then for the set of
> digital numbers to be actually infinite, we have to allow each string to have
> an infinite number of bits, and not all leading 0's, or you haven't added any
> new values to the set.

I ALREADY showed you how, with just ONE symbol in the alphabet, we have
an infinite number of strings. That's because strings don't have to be
of just ONE length L, but of an UNLIMITED number of FINITE lengths, L1,
L2, L3, ... Suppose the one symbol is '1'. Then we have these strings:

1
11
111
......

Infinite number of strings on an alphabet of just one symbol.

> So, what does it mean, if the set of dgital numbers is
> not infinite unless the strings are infinitely long? Well, a nonzero bit in an
> infinite position represents an infinite value, so the requirement that we
> have infinitely long binary strings to achieve an infinite set is equivalent to
> the requirement that we have infinite values in the set, if the set is to be
> infinite. This is because the set is nowhere dense. A set can only contain an
> infinite number of elements within a finite range if it has at least one point
> of condensation, or infinite density.

With more and more undefined and conflated terminology, that just
builds on and adds to your initial confusion, which I just explained to
you and had explained before and other posters have explained for
thousands of posts. Please do me a favor by not even bothering to waste
your time typing paragraphs as above to me. If you want to start at
square one, then fine. We can keep trying to get you to understand
square one. But you won't get me to move past square one with you until
you understand it.

> > You can't point to a single one. Your own "axioms" (let alone hardly
> > being "more solid computational axioms of the rest of mathematics") are
> > not axioms since they use undefined terms that are not even delcared to
> > be primitives. So, when you say that set theory contradicts "the more
> > solid computational
> > axioms of the rest of mathemtics, and reality in general," you're just
> > talking out of your hat.

> You can look at it that way, but you are talking from behind your screen of
> symbolic logic, unable to translate English into any mathemtical ideas, as far
> as I can tell.

I have no "screen". I can formalize or not at will. That I don't find
your mumbo jumbo enlightening does not entail a limitation of my
flexibility. It's not a matter of "looking at it that way". When you
said that that set theory contradicts "the more solid computational
axioms of the rest of mathematics, and reality in general," you're just
talking out of your hat.

> Everything we say here is published, in the sense that it's communicated
> publicly.

Fine, then the key word was 'completed'. The point still stands.
Completion is not demanded. Coherence is.

> You have an axiomatic system concerning jazz? That sounds interesting, but
> perhaps a little too rigorous for something like jazz. What kind of theorems
> can you prove with your axioms?

The point was not to axiomatize all of jazz improvisation. That would
be ridiculous. All I did was codify some relations between scales and
chords used in jazz. And it's only a small project and not even
completed. And hardly worth discussing its details here. I just
answered your question to let you know that I do know something about
starting a system from scratch.

> By the way, I don't expect a medal for trying, but I wouldn't mind a little
> patience while I develop things. Of course, you don't owe me anything.

The impatience is not with the development of your system but rather
with your stubborn refusal to even LOOK AT the system you deride and
instead to MISconstrue it and MISrepresent it virtually every time you
open your yap about it.

>> > No. I said that the set of natural number is equinumerous with omega,
> > trivially, because the set of natural numbers IS omega. To prove that
> > the cardinality of omega is omega, you prove that omega is a cardinal.
> > In most treatments that means it is the least ordinal (in the ordering
> > of ordinals) of all those ordinals with which it is equinumerous. Then
> > you can say that the cardinality of omega is omega itself.
>
> That sounds rather circular, but okay. The monkey's fist is tightly tied.

I have no idea what "the monkey's fist is tightly tied" means. But no,
there's nothing circular in what I described.

> > No. I said that you can define 'is a natural number' before defining
> > 'omega'. Then you prove that omega is the set of natural numbers. Then,
> > again, we prove that omega is a cardinal, from which it follows that
> > the cardinality of the set of natural numbers is itself, which is to
> > say, omega.

> So, in other words, you can define one thing and derive the other

What is 'thing'? You can define a predicate symbol, then prove a
theorem that mentions that predicate symbol; but the point I've been
saying over and over and over is that the theorem does not have to
mention the defined symbol. The theorem (with more primitive symbols)
is derivable even withOUT the definition. The definition just
ABBREVIATES symbolism so that the statement of the theorem is shorter,
but NOT more powerful. Shorter but NOT more expressive and NOT more
powerful. You'd understand this if you'd just READ A BOOK. The kinds of
definitions we're talking about do NOT add to meaning or deductive
power; but rather definitions are just ways of ABBREVIATING.

>, or vice
> versa,

Proving certain things facilitates making certain definitions. But
again, for the 2^omega-th time, definitions do not materially add to
the theory.

> which is to say that the definitions one starts with DO affect what
> axioms can be stated and hence what theorems can be derived.

You've got it COMPLETELY mixed up. What is it with you that you just
won't read a book rather than have people keep pounding their heads
against the thick wall of your skull to make you understand what you
REFUSE to understand anyway?

One does not START with definitions. One starts with primitives and
axioms. In the presentation one can give some definitions before giving
axioms, but the definitions do NOT affect the expressive or deductive
power of the axioms. Axioms do NOT depend on definitions. Definitions
can be used to make it easier, shorter to state axioms; but, one more
time: this is not required and does NOT affect the expressive or
deductive power of the axioms.

> > > It sounds like definitions are intimately intertwined with axioms
> > > regarding them, no?

"Intimately intertwined". Please read a book on mathematical logic so
that you can speak MEANINGFULLY about this.

> I understand what you're saying I think, that the mechanics of the logic are
> not affected by what the symbols mean,

The symbols don't mean onto themselves, but only by an interpretation
of the language. The mechanics of the logic work hand in hand with the
method of interpretation. Definitions are ways of abbreviating strings
of symbols that are given meaning per interpretations. Definitions do
not affect the meanings of the primitives nor the deductive power of
the axioms.

> but only by the operations defined by
> the axioms,

Axioms (not definitional axioms) "define" in a much different sense of
'define' than do definitions. The sense of 'define' (here 'define' is
informal) for axioms is to narrow the class of models of that are
models of the theory. And 'operations' is something else again. If you
read a book, then your remarks, even if criticisms of set theory, would
be coherent.

> but in the end, the conslusions deduced are interpreted, as the
> meaning is read back into the symbols, or the conclusion remains just a bunch
> of symbols, and not a statement.

The conclusions (theorems) are not interpreted directly. What is
interpreted are the primitive symbols. The interpretation then dictates
whether the theorems are true in that interpretation (actually,
structure). If the theorems are not all true in that interpretation
(structure, really), then that interpretation (structure, really) is
not a model of the theory.

> While the logical operations may all be
> correct in the proof, both the axioms that state the operations possible, and
> the definition of what the terms mean, combine to produce the meaning of the
> final statement. Am I so far off?

The meaning is given by a structure (an interpretation) of the
primitives. This does not reference the axioms except that the
primitives are what are used in the axioms. A structure is for a
LANGUAGE of a theory, not for a theory. A language for a theory can
have many different structures no matter and many different theories
can use the same language. The axioms don't dictate meaning per se, but
rather the axioms dictate WHICH interpretations will make all of the
theorems true. And definitions play NO part in this. Definitions do not
impart meaning to a theory. Definitions provide ABBREVIATIONS for
expressions built with primitives; and the primitives are given meaning
by an interpretation; and then the axioms dictate which interpretations
make the theorems from the axioms true.

> Didn't you ask me to define my primitives? Were you asking me to square the
> circle? Wold it have helped to say "the primitives you define" rather than "the
> primitive definitions"?

No, I didn't ask you to DEFINE primitives. I'm not a crazy person. One
doesn't DEFINE primitives. One STATES what the primitives are.
(Actually, technically, one defines the set of primitives, but that's
not what we're talking about.) You just need to say: "These are my
primitives: 'point', 'collection', 'quantity'..." (or whatever they
are). Then you give your axioms, using JUST those primitives and the
built-in logical symbols such as variables, quantifiers, and sentential
connectives. You can also give definitions before or after giving
axioms; but the definitions must be of a certain forms so that your
defined symbols can always be eliminated to revert to primitives and so
that your definitions do not increase the deductive power of your
theory.

> > Define 'position', 'value', 'infinite'. Define them back to whatever
> > you declare to be your primitives. Otherwise, at least as far as
> > getting a response from me is concerned, you can save yourself the
> > trouble of typing paragraphs such as the above.
>
> This is part of what I have been working on. Position may be defined as a whole
> number equal to the number of predecessors to an element's successor in a
> sequence.

STOP!

'whole number', 'equal', 'predecessors', 'successor', 'sequence'.

Are they primitives? If so, what are the axioms? Are they defined? If
so, what are the definitions?

Start with primitives and work up to defined terms. Or, if you work
back from defined terms to primitives, then you have to work ALL the
way back.

> Value may be defined as the property of an element which uniquely
> identifies it regardless of position.

'property', 'uniquely identfied'

Define 'em or state them as primitives.

> Infinite is defined thus, arithmetic
> operations being assumed to have been defined elsehwere:
>
> 1: x e (0,1] -> finite(x)
> 2: Finite(x) -> finite(0-x)
> 3: Finite(x) -> finite(1/x)
> 4: not(finite(x) or zero(x)) -> infinite(x)

What is that? A definition of 'finite' for real numbers? What about
sets in general? Anyway, this is not in definitional form for a
predicate. The definitional form for a predicate is a biconditional.
But I guess you could use it. I'd put it this way:

The set of finite reals is the intersection of the set of sets S such
that
(0 1] subset S
S is closed under changing sign
S is closed under the operation of dividing into 1

x is a finite real <-> (x is a real & x e the set of finite reals)

x is an infinite real <-> (x is a real & ~(x=0 v x is a finite real))

Theorem: x is a real -> exactly one: x is a finite real, x is an
infinite real, x=0

> There are other statements regarding finites vs infinites and infintiesimals,
> but this gets us to some definition of infinite. Does it seem at all circular,
> assuming 0, 1, -, and / have been defined?

0, 1, -, / defined. Indeed, and all the way back to two primitives.

This seems okay as a way of partitioning the real numbers, though you'd
do a lot better to choose a word other than 'finite'. 't-finite' or
something so that people know you're talking about something different.

EXCEPT one big thing: You say you reject the axiom of infinity. So what
IS undefined is '(0 1]'. What interval is that supposed to be if not a
subset of the set of reals? Without the axiom of infinity, I don't know
where you think the set of real numbers comes from.

> > > Certainly those are not rock-bottom primitives, and N=S^L is derivable as a
> > > combinatoric theorem, but may be used as an axiom within the system. Are you
> > > saying that you don't understand what strings, alphabets, and length mean? Do I
> > > really need to define these terms for you in this discussion? This is the kind
> > > of lawyerism that Han refers to as a trick. If you didn't understand the
> > > paragraph, as for clarification. If you did, then don't waste my time
> > > complaining. Did you did you not understand it?
> >
> > You miss the point. I understand the theorem just fine, AS A THEOREM
> > FROM AXIOMS. But in your style, it is just free-floating verbiage that
> > HAPPENS to be correct only because it is a theorem of SET THEORY, not
> > because you've made it a THEOREM or even an AXIOM of whatever your
> > system is supposed to be.
>
> So, you agree it's correct. Good. So, what do you have to say about the
> implications of that fact for natural numbers and languages?

First, let's get a grip on the initial point. It's not an axiom that,
as you incorrectly claimed, contradicts set theory.

Second, your questions about implications is too open-ended. What am I
supposed to do, start listing every theorem I can think of that follows
from the aforementioned theorem?

> COnsider the universe to be the context. There is no universe in ZFC, as Ross
> often says. There is a universe, and in the universe, that fact regarding
> symbolic systems is universally true, with or without tranfinite tomfoolery.

First, it is a theorem of ZFC that there is no set of all sets. For any
interpretation of the language of the ZFC, there is a universe of
discourse. Second, what exactly what is it that you claim to be a fact
regarding symbolic systems that is universally true?

> Then you need to ask more specific questions.

No, I don't need anything from you. YOU need to read a book. Your
ignorance of mathematics trumps my disinterest in the ramifications of
your ignorance of mathematics.

> Assume the logistic system is
> first order logic. Assume definitions are as normally considered, unless stated
> otherwise. If you want to understand what my universe is, when it comes to the
> naturals, the universe is the real number line which includes all quantities,
> and the naturals are embedded in it.

I'm not asking about your UNIVERSE. I'm asking about your theory. Even
if we were talking about a universe, then you'd have to say what the
real number line is. There is the set of real numbers. And there is the
standard ordering on the set of real numbers. And there is <R
less-than> which is the ordered pair of the set of real numbers and the
standard ordering on the set of real numbers. You may add whatever
definition of 'line' you want, but without a definition, your remarks
are just metaphorical language.

> If that's not what you mean, please ask
> specifically what part of the context eludes you, because I don't understand
> what it is you don't understand, if you agree with the facts, but aren't
> satisfied with the statement of them.

You missed the point again. I agree that the sentence is a theorem of
set theory. But since you don't have a theory, it is meaningless to say
how the sentence impacts your system, since you don't have one.

> What else does the theory consist of but the definitions and axioms which
> comprise its foundation?

A theory is a set of formulas closed under entailment. Definitions are
auxillary and serve only to extend the language of a theory but not the
class of models of the original theory. To specify a theory, one must
at least say what set of formulas the theory is closed from. You have
not given such a specification.

> There are derivable theorems int he theory, but
> derivation of theorems is the easy part. Once all the axioms are set up in
> symbolic format, any machine can do the rest, right?

A machine can enumerate the theorems. A machine cannot decide the
theorems (for consitent theories of sufficient expressive capability).

> As I've said, the entire
> set of primitives and axioms is not yet complete,

You mean 'not finished', since 'complete' is something else (I'm saying
that not as a correction, but just to be clear).

> but in progress. I'd like to
> make sure it's as right as possible, so patience, please.

Patience is exhausted not with your formalization but with your
self-ratifying ignorance.

> > > If this is so, then perhaps you shouldn't think about it.
> > > When I have what might be a completed axiomatized pre-digested theory, I'll let
> > > you know.
> >
> > I'm sure the world will hear about it.
>
> Maybe through usenet. Is that a bad thing? Maybe the world will accept it.
> Wouldn't that be annoying? :)

Maybe the world will treat your treatise like you treat set theory:
Reject it without even bothering to the read the first chapter.

> Yes, Bigulosity has a good ring to it, and we can have equibigulous sets,
> relatively semibigulous sets (like the evens), micro-, macro-, and (thanks
> Ross) meso-bigulous sets and values. I think Bigulosity sounds silly and yet
> pompous enough to add a little olor to mathemtical temrinology. :D

I'll leave the christenings to you. You're certainly enamored with your
creations.

> Concepts of universal truth are what is at stake here, and the desire to delve
> deeper into infinite realms, untethered by "contability", omega, and other
> standard concepts. You are challenging me, and I appreciate it, but I don't
> have ready what you seem to need. It's in the works.

Forget about what I need. You need to read a book.

> There is no argument regarding the objectively correct application of logical
> operations on logical statements. The question is whether those starting
> assumptions which define the models correspond to actual facts of reality. When
> they are not derived or justified, but seem arbitrary, then they cannot be at
> all guaranteed to be true at all. I believe it is up to thse claiming axioms to
> justify them in a non-axiomatic context.

Three points of view (and blends of them): (1) Axioms embody sure
statements as starting points. (2) Axioms are chosen for their
deductive power to prove all (or as many as we can get of) what we
consider to be sure statements. (3) Axioms need only be consistent.

But "correspond to actual facts of reality" is a whopper of undefined
verbiage (and I don't mean formally defined, but rather even defined
for a philosophical discussion). As to arbitrariness, one might not be
so concerned with arbitrariness so much as deductive power (as in point
of view (2)). Moreover, one might not consider the axioms of ZFC to be
so arbitrary. Each one of them "corresponds" to a pretty basic "sense"
of how "sets" work. And, the point for someone who claims arbitrariness
is to show a set of axoms that is not arbitrary but that has sufficient
deductive power.

> You see, while the "foundations" of
> math today are considered symbolic and logical, the beginnings of math were
> more geometrical, or about tallies, the first number systems. Developing
> symbolic systems itself is a matter of geometry. So, in my view, a
> geometrically-based foundation is equally valid.

Foundations concern both formal axiomatization and philosophical
matters. One should not take a particular formalization as itself a
philosophy.

As to your "geometrtically-based" foundation, no one can opine until
you produce one. That you personally have a geometric bent does not
entail that a geometric approach is required, though if you produce
one, and is coherent, then people should evaluate it on the bases any
system is evaluated.

> Classical. I have looked a little at intuitionistic logic, but I don't see the
> point to it. Classical logic works fine as far as I am concerned.

Okay, that's a start.

> Ax(x is a point -> x is an atom of space)
> Ax(x is a symbol -> x is an atom of language)
> Ax(x is a number -> x is an atom of quantity)
> Ax(x is a space -> x is an order-distinguished collection of points)
> Ax(x is an alphabet -> x is an order- and value-distinguished collection of
> symbols)
> Ax(x is a string -> x is an order-distinguished collection of symbols which are
> elements of an alphabet)
> Ax(x is a language -> x is an order- and value-distinguished collection of
> strings)

This is not clear now. Is 'atom of space' one predicate or are 'atom'
and 'space' two predicates? And how does 'of' work?

I suggest you don't use English words. For the first axiom, perhaps:

Let 'P', intuitively for 'is a point', 'A' for 'is an atom'; 'S' for
'is a space'; 'e' for 'is an element of'.

Ax(Px -> (Ax & Ey(Sy & xey)))

Or whatever; but you have to be able to reduce to predicate letters and
not rely on the connotations of the English words. The same for the
rest of what you've added: 'order-distinguished',
'value-distinguished'.

> That is partly because your theory is not a single theory, but a collection of
> models depending on what axioms you include or don't.

What are you talking about with "collection of models"? Please stay
away from technical verbiage you don't know the definitions of. Given a
fixed treatment, there is a single theory.

Here's a single theory Z:

Z is the set of consequences of the following axioms (in a language of
just two primitives: '=' and 'e') in first order logic with identity:

Extensionality
Separation (schema)
Power Set
Union
Pairing
Infinity

That is one definite theory.

Here's another, ZF is just like Z except the axioms are:

Extensionality
Replacement (schema)
Power Set
Union
Infinity

Others:

ZR is the same as Z except add Regularity

ZC is the same as Z except add Choice

ZRC is the same as ZR except at Choice

ZFC is the same as ZF except add Choice

ZFR is the same as ZR except add Regularity

ZFRC is the same as ZFR except add Choice

And we can have other theories, some allowing urelements and some
allowing proper classes by having 'is a set' as primitive predicate.

In any case, if it is required to be specific as to exactly which
theory we're talking about, then we can simply stipulate.

> Like I said, I don't
> really understand the use of the axiom of choice. It seems like a dimensional
> statement to me.

In other words, "I don't want to pay attention to the very simple thing
you just said and I don't want to look at a book that talks about the
subject. I just want to keep saying 'It looks dimensional' to me."

> The axiom of infinity is incorrectly stated, and I reject it.

It's not incorrectly stated. You just reject it. Fine. But can we at
least be clear (on a case by case basis if necessary) on the point that
you are not relying in any way on the axiom of infinity for any
conclusion?

> At some point I'll get back to analyzing the lot of them and writing down
> exactly where I agree and disagree with the premises. The conclusions in many
> instances are roundly rejected, not because the deduction is incorrect. It's
> not. It's because of the axioms, and how they are used.

STOP THAT! Stop that nasty habit of saying "how they are used". Just
say you don't accept conclusions from the axioms. There's no MEANING to
"how they are used". They're used in proofs. That's the only sense of
how they are used. If you don't want to use them in proofs, then don't.
But this "how they are used" is just juvenile language.

> It just sounds like creating a tuple of coordinates, choosing a value from each
> of a dimensional set.

Do you want to know what the axiom of choice is, or do want to just
keep meaninglessly mumbling, "it seems like a dimensional set"?

Ab(beT -> ~b=0) -> Ef(f is a function & domain(f)=T & Ac(ceT ->
f(c)ec))

I.e., if T is a set of non-empty sets, the there is a function f on T
such that, for all c in T, f(c) is a member of c. So if you have T, a
set of non-empty sets, then f is function that picks a member from each
member of T. Take the range of f to be the "new" set formed from T by
choosing a member of each member of T.

There are equivalents that mention Cartesian products and other things,
but you just need to get it through your thick meme-brains that the
axiom of choice, whether you agree with it or not, is not difficult to
fathom, does not need your "dimensional sets" and is as simple as that
there is a function that chooses one member of each non-empty member of
a given set.

> Why is this significant to well-ordering the real
> numbers, for instance?

Why don't you read a damn book on set theory? The signficance is that
with the axiom of choice we prove that there exists a well ordering of
ANY set. The basic idea is simple: Roughly put, we don't know that a
set has a well ordering since, for any ordering relation, we don't know
that each subset of a set has a least element. But the axiom of choice
says that we can just CHOOSE, arbitrarily choose, a least element. The
axiom of choice is EQUIVALENT to the well ordering principle. With the
choice function, we chose an element, which we call the least element.
That power of choosing "blindly" makes the axiom of choice
controversial, but the axiom is not at all hard to understand.

> And what results in analysis would I have to give up,
> that you don't already think I have rejected?

I have no idea what you accept and reject in analysis. The question is
not even meaningful at this point, given the rest of our discussion
about your complete fog about what a theorem even IS.

> Well, when you say that, instead of commenting on any of the subject matter of
> what I've said, I see that, as I have seen before here many times, as a way of
> sloughing off the ideas. Maybe English isn't your primary language or
> something, but if something doesn;t make sense, can you at least formulate a
> question enough to address it?

I've answered this already, in plain English. I must promise myself not
to keep answering questions I've already answered, so I hope this is
the last time: I don't care to ask you questions about muddles you've
made when you won't address your more FUNDAMENTAL confusions.

> By the way, I am not an expert in transfinite proofs by any means, but I am
> also not confused about my assertions, despite your repetitions to that effect.

Like I said.

> Instead of doing something original, I should learn the standard method.

False dichotomy.

> Hmmmm.. I need to brush up on my mathmetical foundation requirements, but
> formulating the best primtiives and axioms to tie them together isn't going to
> be helped by reading many books on other subjects.

Yes it will.

> Go read a book=Stop being an ignoramus.
>
> Thanks for the advice.

Please take it.

> Do you have specific examples of circular deifnitions on my part, logical
> errors? Some statements on standard set theory are not stated as rigorously as
> you would like, but my different conclusions are not wrong when they conflict
> with standard theory. This is exactly what I am talking about. I am not wrong
> to say the set of finite naturals is quantitatively finite. You have a Dedeking
> set-theoretic definition, so fine, it's Dedekid infinite. But, that doesn't
> make me wrong to say it's Orlow-finite. It's statements like that that make
> guys like me say your theory is trash.What goes around comes around.

You're obviously way too emotional about this if you think personal
spite is a basis to reject a mathematical theory or call it 'trash'. By
the way, I don't usually use Dedekind infinity but rather plain
infinity (not finite, i.e., not equinumerous with a natural number),
since the equivalence of plain infinity and Dedekind infinity requires
the axiom of choice. And I do not hold that what you say about your own
system in progress is incorrect merely for not being set theory. And
how I ironic of you: You say the axiom of infinity is "incorrectly
stated" but protest that anyone finds YOUR ruminations to be ill
founded. You don't even know what set theory IS but you can say what is
incorrect about it in context of YOUR ruminations, while anyone who
points out the confusions in your ruminations has not sufficient
grounds since YOUR ruminations are not to be judged in context of set
theory.

> If you had followed the simple exampels I gave of the inverse function rule,
> you might have verified that it actually does seem to work. But, even with a
> simple explanation and two examples, it is all just gobbledygook to you. And
> I'M the confused one? If you say so. At least I can parse English.

Parse mathematics, not English.

> > You don't understand how to make sure you don't confuse the order of
> > your quantifiers.
>
> On what do you base that falsehood? Your declarations are bull. Back 'em up.

Start with your confusion about the set of finite strings. It is NOT
the set of strings for a given length L.

> > Yes, you're right. And all of your critics are wrong (except 5% of the
> > time). You just convinced me. I don't know why I didn't see it before.
> > I feel like a great weight has been lifted from me. I see clearly know.
> > You have the answers; it cannot be doubted.
>
> Ho hum. Look at the record, and back up your statements, or drop it.

You just claimed that your critics are wrong 95% of the time. That's
for YOU to substantiate.

> I can get
> nasty too. That you folks can't fathom what Han is trying to say is
> disgraceful. You're so microsopically educated that you can't understand
> anything but symbolic logic any more, or accept any possible conclusions except
> for the tenets of the Cantorian religion.

Of course, if I disagree and point out your confusions and lack of
understanding of the subject, then I am ignorant of anything but
symbolic logic and I have adopted a religion.

> > You can't learn it except systematically. Bits and pieces cannot give
> > an understanding. Start at the beginning. Get a book.

> Get a book on English, and one on scientific method, while you're at it.

You mean, while YOU'RE at it. I'm always in the market for good books
on those subjects. But does this mean that you really are going to
study a book on mathematical logic?

MoeBlee

.

Relevant Pages

  • Re: infinity
    ... >> Not while discussing the consequences of the axioms as they ... I commented that they were irrelevant to, and contradictory ... existed for over 2 millennia without needing any infinite naturals. ... A natural number is finite if the set of naturals less than or equal to ...
    (sci.math)
  • Re: Infinity
    ... >> system and axioms, ... >> making your proofs in set theory. ... > infinite set, and I apologize, the original question had nothing to ... >> equivalent with that of the definition of 'Dedekind infinite'. ...
    (sci.math)
  • Re: Infinity
    ... > and axioms, ... > about 'Dedekind infinite' we're talking about set theory, ... since you don't know what set theory IS. ... While you're right that I misstated the Dedekind definition of an infinite set, ...
    (sci.math)
  • Re: Set Theory: Should You Believe
    ... Why, in your opinion, is the orthodoxy in set theory ... mathematics, ignoring the misgivings of geniuses like Poincare, Weyl, ... When NW said that "You don't need axioms", ... NAFL theories (but infinite proper classes, ...
    (sci.logic)
  • Re: Calculus XOR Probability
    ... The same axioms that give us transfinite set theory are needed to ... give us the naturals, and from them the other number systems on ... The invertible functions of IFR map reals to reals. ... infinite number. ...
    (sci.math)