Re: Cantorian pseudomathematics

Tony Orlow wrote:
> MoeBlee said:
> > Tony Orlow wrote:
> > > MoeBlee said:
> > > > Tony Orlow wrote:
> > > > > MoeBlee said:
> > > > > > Tony Orlow wrote:
> > > > > > > Randy Poe said:
> > > > > > > > > There you say it ! But *I* don't think that "alternate decisions
> > > > > > > > > could be made from the same axioms". And *I* dont think
> > > > > > > > > that those theorems are "false", in the sense that they do
> > > > > > > > > not follow from the axioms. What *I* think is that the axioms
> > > > > > > > > themselves are false.
> > > > > > > >
> > > > > > > > What does it mean for an axiom to be false?
> > > > > > > >
> > > > > > >
> > > > > > > It means that they contradict other axioms. Now, you have some small set of
> > > > > > > axioms that don't contradict each other, but in the larger scope of mathemtics
> > > > > > > at large, the conclusions drawn from those axioms contradict conclusions drawn
> > > > > > > from other axioms of mathematics regarding sets such as Han is discussing. In
> > > > > > > the context of those other areas of math and their axioms, the axioms of set
> > > > > > > theory are flawed, as conflicts arise between them and the more solid
> > > > > > > computational axioms of the rest of mathemtics, and reality in general. The
> > > > > > > conclusions derived from the axioms are not correct.
> > > > > >
> > > > > > Please cite a theorem of set theory such that its negation is a theorem
> > > > > > of some other axiomatized theory that you hold to be important. And
> > > > > > what are these "other axioms of mathematics" that you have in mind?
> > > > > >
> > > > > > MoeBlee
> > > > > >
> > > > > > MoeBlee
> > > > > >
> > > > > >
> > > > > No one axiom needs to be the exact negation of another for contradictions to
> > > > > arise between them.
> > > >
> > > > I didn't ask for an example of an AXIOM and its negation. I asked for
> > > > an example of a THEOREM and its negation.
> > > >
> > > > > Certainly, the notion of an infinite number of increments
> > > > > never achieving an infinite value flies in the face of infinite series, where a
> > > > > sum can only converge to a finite value if the terms have a limit of 0 at oo.
> > > > > The constant increments of 1 have no such limit.
> > > >
> > > > That is gobbedlygook. I don't know of theorems of set theory that have
> > > > terms 'infinite number of increments', 'acheiving a value', and
> > > > 'constant increments'. I asked what THEOREM, that is, what provable
> > > > FORMULA of set theory is the negation of another THEOREM of another
> > > > theory that you hold to be important. Name the formula of set theory,
> > > > and name YOUR theory that has a theorem that negates the theorem of set
> > > > theory. Otherwise, your claim about "axioms" is just hot air.
> > >
> > > All countably infinite sets have the same size. Is this not a theorem of set
> > > theory with which you would say I disagree? According to my axioms of the
> > > inverse function and N=S^L, different countably infinite sets can have very
> > > different size, even with infinite ratios between sizes.
> >
> > You claimed that set theory contradicts "the more solid computational
> > axioms of the rest of mathemtics, and reality in general." Now you give
> > your own axioms (with undefined terms) that are not in an axiomatic
> > system and without a specified logistic system as part of the
> > "computational axioms of the rest of mathematics"? Please, what EXACTLY
> > is the set of "computational axioms of the rest of mathematics"? And
> > again, your "axioms" are not axioms unless you specify a logistic
> > system, and they cannot be evaluated for such things as consistency
> > unless you define your terms or say which terms are primitive and say
> > what OTHER axioms yours are supposed to go along with. And, again, your
> > "difference", your source of "contradicting set theory" is really about
> > definitions, not about formulas in the primitive language, which I've
> > explained to you too many times already.

> Look, the theory I am working on is not something I have a lot of personal time
> for, but something that has developed largely here over the last year. I have
> started putting together the definitions and axioms that I need, but it's not
> easy to lay new foundations for math. So, you are demanding a finished theory
> when it is not finished.

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".
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.

> I thought perhaps one could discuss mathemtical
> comcepts without having to publish a completed work, but I guess not, because
> "that's not the way mathematics works".

Strawman. No one has demanded publication. Just coherence.

> I wonder if you have ever tried to
> create your own axiomatic system, and what area it addressed? Just curious. How
> many non-mathematicians do you know that have attempted such a thing?

Yes I have. It addresses scales and chords used in jazz improvisation.
And no, you don't get a medal or brownie points just for being a
non-mathematician doing whatever you think it is you're doing.

> > > The infinitude of your omega as the size of the set of finite naturals is
> > > derived, is it not?

> > Depending on the particular treatment, usually one defines 'natural
> > number' ('finite' with 'natural number' is superfluous) as being a
> > member of omega. So, since the set of natural numbers IS omega, it
> > follows trivially that the set of natural numbers is equinumerous with
> > omega. Or, one can have a treatment in which we define 'natural number'
> > in a different way, then prove that omega is the set of natural
> > numbers. Either way comes to the same result.

> So, you can either define omega as the set and say that's the size of the set

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.

> or you can define natural numbers differently and prove omega is the size of
> the set?

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.

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

No. Definitions make it easier to work with axioms, but definitions are
dispensible and do not affect the theory (except in a technical sense
that the defined symbols then become part of a language in which the
theory is extended into). The theory, in its primitive form (which is
the only thing that really counts, since structures are given for the
language of primitives, not for the extended language of defined
symbols) is not affected by definitions. Please a read a book on
mathematical logic.

> > > Given the definiton of the naurals as the set with an
> > > identity relation between element count and value, this theorem is false, as
> > > count cannot exceed value.

> > Please, for the omega-th time, stop doing that. One does not refute
> > theorems by changing definitions (except in a very very trivial sense).

> You derivations depend, as you showed above, directly on the primitive
> definitions you use.

You see, you just don't know ANYTHING about this. There is no such
thing as a "primitive definition". 'primitive definition' makes about
as much sense as 'round square'.

> If one defines the set of naturals as the set where each
> member has its position as its value, then it follows trivially that the set of
> naturals cannot have infinite positions without infinite values. If you define
> an infinite set as one that includes infinite positions for elements in the
> set, then it follows that the set of finite naturals has a finite size.

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.

> > > > > Wherever the axioms for
> > > > > addition are stated, I am sure they are stated so as to avoid this issue, but
> > > > > addition should always mean making greater, and subtraction should mean making
> > > > > less.

> > > > Name your axioms, name your theory. Otherwise, your claim about
> > > > "axioms" is just hot air.

> > > N=S^L, where N is the size of the set of strings of length L and S is the size
> > > of the alphabet from which they are made.

> > That is not a theory. That is a bunch of undefined words and symbols.
> > To have an axiomatized theory, you need a logistic system, primitives,
> > and axioms that are stated (or can be stated) in the language with
> > those primitives.

> 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.

> > > The mapping function of a bijection describes the size of the infinite set
> > > being mapped in terms of the size of the set it's mapped to.

> > > Have you seen my axioms of finiteness? I've spewed a number of rules which
> > > qualify as axioms, and are justified outside of the axiomatic system.

> > Spewing rules with undefined terms irrespective of a logistic system
> > and primitives is not giving an axiomatization.

> Do you disagree with the rules as stated?

It's not a question of agreement. It's a question of understanding in a
CONTEXT, which is a logistic system, primitives, and axioms.

> I guess it is impossible for you to
> consider the merit of any given rule in isolation.

I at least need to know of at least a rough outline of what the rest of
the context might be.

> You need the entire theory
> in order to comment.

Not the entire theory; the entire set of axioms. But see comment above.

> 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.

> > > > > It may be by an infinitesimal fraction of the whole, but it should not be
> > > > > nonexistent. That aleph_0-1=aleph_0 makes no sense to me.

> > > > There is a bijection between Omega~{{0}} and Omega. In another post,
> > > > you recognized that such bijections exist. That is all that is at
> > > > stake. The rest is notation, which is NOT deductively creative (except
> > > > that a cardinality operation does use either the axiom of choice or the
> > > > axiom of regularity to formulate; but the 'equinumerosity' predicate
> > > > does not), as I've explained ad nauseum.

> > > That is all that is at stake for cardinality, but not for actual set size. If
> > > the definition of set size as cardinality doesn't change anything, then drop it
> > > and we'll be fine. Cling to it and insist that all other possible comparisons
> > > for infinite sets are invalid due to your axioms, and we'll continue. You're
> > > entitled to play with your system, but if you want to claim it's correct in any
> > > objective sense, you're going to get complaints until the end of time.

> > 1. "Actual set size". That's YOUR term. Set theory talks about
> > bijections, equinumerosity, and cardinality. 2. Who said that infinite
> > sets can't be compared in other ways than bijection? 3. No, I don't
> > want to claim that set theory is "correct in your objective sense".
> > That is a philosophical question, of which your term 'objective' is
> > undefined even in terms of a philosophical discourse.

> 1. Then equinumerous<>equibigulous

Fine. Then 'equibigulous' can be added to a stock of words already in
use: 'equinumerous', 'equipotent', 'equipollent', et. al.

> 2. those who claim that cardinality IS set size for infinite sets.

That is informal. You are free to make whatever formalizations you want
and then to give them whatever informal discussion you want. But since
the WORD 'size' is already used informally, it would be just as well
for you to use 'sizosity' or whatever. You still don't get it: WORDS
are not what is at stake here.

> 3. Objective: Independent of personal perspective.

Fine, independent of personal perspective, we can check whether
purported proofs in set theory are indeed proofs. Just put them in your
computer, and your computer will give you the answer. And you can even
check a purported proof that a certain model is a model of the axioms,
given whatever conditional assumptions are required. Just put that
purported proof in your computer, and your computer will give you an
answer. However, as to which models are models of what you call
"reality", then it is for YOU to demonstrate such an objective method.

> > > > So, name your axioms, name your theory. Othewise your claim about
> > > > "axioms" is just hot air.

> > > see above

> > See MY above.

> > > > > COncepts of set
> > > > > density have been developed, but not gotten far enough to really remedy the
> > > > > situation. Han seems to consider transifnite set theory to eb at odds with
> > > > > probability theory, and has given some quotes and diatribes on the subject,
> > > > > though I cannot say with great depth what foul conclusions set theory has
> > > > > postulated for infinite samples of values. My guess is they suck and he's
> > > > > right, even if he is a finitist. :)

> > > > "Concepts of density". Name your axioms, name your theory. Othewise
> > > > your claim about "axioms" is just hot air. And, by the way, set theory
> > > > posits axioms; every conclusion is either itself an axiom or provable
> > > > from the axioms in classical first order logic. Whatever "foul
> > > > conclusions" you think are "postulated" are ones that you can only
> > > > reject by rejecting either classical first order logic or an axiom of
> > > > set theory. So far, over thousands and thousands and thousands of
> > > > posts, you still haven't said what logic you do approve of and/or what
> > > > axiom of set theory you reject.

> > > I have, and claims to the contrary are bogus.

> > You have said what logic you approve and what axioms of set theory you
> > approve and reject? That's funny, because I've asked about a hundred
> > times already, and I didn't see a post in which you specified a
> > logistic system, primitives, and axioms. So, if I missed your answer,
> > then please forgive me and indulge me your answer again:

> > 1. What is your logisitc system?

> First order logic, I suppose. I haven't put it in symbols yet.

Classical, intuitionistic, or otherwise?

You suppose. Does this mean that if you break first order logic, then
we have agree to your conclusions anyway since you never committed to
first order logic? You don't get it. You keep asking me to comment on
certain particulars of your math. But I am quite disinclined to get
into that quagmire when you won't even tell me what LOGIC your math is
to be evaluated by.

> > 2. What are your primitives?

> Atoms include points, symbols, quantities. Collections include spaces, strings
> languages, etc. I am working on this, but it's not finished yet, so be patient.

As I understand, you've stated eight primitives and six axioms:

Ax(x is a point -> x is an atom)
Ax(x is a symbol -> x is an atom)
Ax(x is a quantity -> x is an atom)
Ax(x is a space -> x is a collection)
Ax(x is a string -> x is a collection)
Ax(x is a language -> x is a collection)

If that's not what you have in mind, then revise per what you do have
in mind.

> > 3. What are your axioms? (This requires stating them with primitives or
> > showing a sequence of definitions that revert to primitives.)

> Working on it.

> > 4. Subsumed by 3, but to reiterate: Which axioms of set theory do you
> > accept and reject?

> I have discussed that to some extent. Does your news reader allow you to see
> the responses to your posts, as opposed to just all the posts in the thread?

You're completely muddled on what axioms you accept and reject. Your
position seems to be that the axioms are okay, or some or okay but
others less okay, but that you don't like the way some of them are
used. But that is meaningless. As has been explained to you, there is
no such thing as disclaiming how an axiom is used.

> > 5. Please don't tell me that you accept all of the axiom of set theory
> > but that you reject the definitions and are replacing with different
> > definitions. That is not of any mathematical import. If you have
> > ADDITIONAL definitions to give, then fine, but you must give them so
> > that they revert to primitives.

> I never said that. I said the axiom of infinity needs a change. The axiom of
> choice doesn't even make sense as an axiom to me, but seems like an expression
> of dimensionality.

"Dimensionality". Oy vey, the axiom of choice can be stated as simply
as this: If you have a set S, then you can take an element from each
non-empty element of S to form a new set. If you don't like the
implications of that, then fine, but then you have to give up certain
results in analysis too.

> > > Does none of what I said above
> > > sound familiar? No, it just sounded "confused" to you. Well, it's not a perfect
> > > seamless system yet, but I am not at all confused about it.

> > The worst is not that you're confused, but that you don't KNOW (or
> > refuse to recognize) that you're confused.

> Again, all you can say is I am confused, but you are the one who can't even
> formulate a question to clarify what I am talking about. A declaration of
> confusion is not a refutation or even a valid repsonse.

You're repeating and we're going in circles. I've asked you the MOST
IMPORTANT question. And, as I said, there's no point in trying to clear
confusions one by one when they can only be cleared by addressing the
root. Aside from that, I don't claim to refute by declaring that you
are confused. Such declarations serve two purpopses: 1. Just to state
for the record that my response is in the context of my not assenting
to your confusions and 2. To remind you, yet again in vain alas, that
you are in fact confused.

> > > Don't pretend that
> > > I haven't responded to these questions, and then dismiss the responses I give
> > > without reason.

> > The reasons have been given by dozens of other posters. And I gave you
> > additional reasons regarding fundamentals. But you've not bothered to
> > try to understand what I wrote. I infer that since I assume that you
> > are not stupid. If you're not stupid and you bothered to try to
> > understand what I wrote, then you'd make some progress understanding
> > it.

> You ask for too much too fast, anxious to tell me where I'm wrong. I am not a
> professional mathematician, so if you want a full axiomatic system, then you
> have to be patient.

I'm not anxious about the progress of your system. I just recommend
that you read a book.

> > > Try reading it again, why dontcha, and see if you can muddle
> > > through it enough to formulate a question, how's about?

> > How's about you give the courtesy of just telling me by what logic
> > system I am to evaluate your mathematics, what are the primitive terms
> > in your system, and what is your set of axioms. There are libraries
> > full of mathematics I have never had the chance to work through. This
> > is mathematics written by mathematicians who give me the courtesy of
> > stating their logistic system, primitives, and axioms (or I can easily
> > surmise by context). And there is philosophy of mathematics written by
> > authors who I can tell know something about foundations and the history
> > of mathematics and the history of foundations. So I have no reason to
> > do YOUR work for you trying to untangle your confusions.

> I guess that's a "no", you can't formulate a question about what you didn't
> understand. I notice your previous response is snipped, where you told me I was
> confused, but then said nothing specific about the inverse function rule I
> proposed. I am not asking you to create an axiomatic system for me. I'll get to
> it, but life's busy.

You miss the point again. I'm not going to waste my time trying to
understand something that I have no reason to believe is coherent. What
you say about set theory is incorrect, time after time; your terms are
undefined or defined circularly, time after time; your logic is
incorrect, time after time; you miss MY points, given over and over,
time after time. So, no, I am not motiviated to try to understand what
shows no promise of even being coherent enough to be understood. It's
not my job to untangle your tangles.

> > > I get the feeling that
> > > mathemticians don't actually play with concepts, but only follow rules.

> > You're wrong again. Read some books written by mathematicians about
> > mathematics.

> > > I
> > > really was hoping someone with more experience might pick up on some of the
> > > ideas and formulate them to the satisfaction of their colleagues, but it seems
> > > the goal is to stifle any new ideas in an ego contest of refutation.

> > No it's not. New ideas are welcome. But welcoming new ideas does not
> > mean declaring a safe season for correcting basic errors of logic,
> > pointing to undefined terms, circular definitions, and misconstrual of
> > the mathematics that's available.

> And yet, I have been accused of all those things when they have not been true.
> People accuse me of quantifier dyslexia

You don't understand how to make sure you don't confuse the order of
your quantifiers.

> when I talk about the ramifications of
> the identity function between element count and value in the naturals. People
> claim my definitions are circular, when they are not, but they have misread. I
> am acccused of logical errors, and have had to correct the corrections on
> several occasions. Sometimes, someone actually DOES point out a problem with my
> thinking, and I am happy to admit it, such as when I was suggesting that log
> (log(log.....log(n)))))))) might map the naturals to the reals in [1,2] (or
> something, I don't remember exactly). When you complaint hat I haven't defined
> "alphabet" or "string" or "length", when I am using these terms in their
> commonmathematical senses, that's just a fillibuster tactic. Of the criticisms
> I have received here, about 95% are bull.

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.

> > And it's really ironic that you wish people would give your "new ideas"
> > a closer look when you won't even give the mathematics you purport to
> > improve, correct, and refute(!) ANY LOOK AT ALL! You go on about set
> > theory this and set theory that and how it's incorrect but you don't
> > even know what set theory IS. You've never even studied the subject.
> > Not even, and especially, at the most basic level. And you complain
> > that people won't give YOUR ruminations a chance? And you WONDER that
> > certain people take you for a pompous ignoramus? Sheesh.

> I was taught set theory, and introduced to the transfinite portions, some
> decades ago. It was wrong, so I turned my back on it and studied infinity
> through other avenues. I am not an expert, and have asked many questions
> regarding it in the last year, for instance, regarding well ordering of the
> reals, and many other topics. I've l;earned a lot, but what I have learned
> convinces me all the more that it's off the mark, and I'm obviously not alone
> in that.

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

> > > A sad
> > > state of things, really, but I guess that's "natural selection" for you, in the
> > > world of memes.

> > Memes are what you generate. As to mathematics, of course, for you, if
> > you're not selected, then there just HAS to be something wrong with the
> > selection

> Memes are passed from mind to mind, and sometimes from mutations in an
> individual given personal experience and novel connections. Everyone wants to
> reproduce their memes. When you think you have invincible memes, and other keep
> attacking them, then what do you do? You fight back. We'll see what memes
> survive and thrive.

They'll make a movie about it. How one man defied the mathematical
establishment and won. How his memes survived and he found the love of
a sexy cocktail waitress too.

MoeBlee.

.

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