Re: Cantorian pseudomathematics

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. 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". 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?

>
> > 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,
or you can define natural numbers differently and prove omega is the size of
the set? It sounds like definitions are intimately intertwined with axioms
regarding them, no?

>
> > 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. 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.

>
> > > > 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?

>
> > 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? I guess it is impossible for you to
consider the merit of any given rule in isolation. You need the entire theory
in order to comment. 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.

>
> > > > 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
2. those who claim that cardinality IS set size for infinite sets.
3. Objective: Independent of personal perspective.

>
> > > 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.

>
> 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.

>
> 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?

>
> 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.

>
> > 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.

>
> > 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.

>
> > 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.

>
> > 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 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.

>
> 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.

>
> > 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.

>
> MoeBlee
>
>

--
Smiles,

Tony
.

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