Is it possible to formally define a "potential" infinity?

[Dan Christensen]

DEFINING ACTUAL INFINITY

A set S is said to be actually infinite iff there exists an injective function f: S --> S such that f is not surjective. (There are other, equivalent definitions.)


DEFINING POTENTIAL INFINITY

Is it possible to formally define the potentially infinite?

A set S is said to be potentially infinite iff .....????


Dan

Download my DC Proof 2.0 software at http://www.dcproof.com
Visit my Math Blog at http://www.dcproof.wordpress.com

[FredJeffries]

On Wednesday, July 1, 2015 at 6:07:12 AM UTC-7, Dan Christensen wrote:
> DEFINING ACTUAL INFINITY
>
> A set S is said to be actually infinite iff there exists an injective function f: S --> S such that f is not surjective. (There are other, equivalent definitions.)

There are also other non-equivalent definitions

>
>
> DEFINING POTENTIAL INFINITY
>
> Is it possible to formally define the potentially infinite?

Certainly. It has been done in functional programming
https://en.wikipedia.org/wiki/Lazy_evaluation#Working_with_infinite_data_structures
http://www.cs.nott.ac.uk/~txa/mgs.2014/inf.html
https://www.haskell.org/tutorial/functions.html#tut-infinite
http://courses.cs.washington.edu/courses/cse341/08au/haskell/infinite/

https://www.google.com/search?q=infinite+data-structure

>
> A set S is said to be potentially infinite iff .....????

Sets (in the strict mathematical sense of the word) are not said to be "potentially infinite". You need some sort of structure or order to talk about "potentially infinite". Sets have no structure, no order. That's what makes them so useful.

[Dan Christensen]

On Wednesday, July 1, 2015 at 6:44:39 PM UTC-4, FredJeffries wrote:
> On Wednesday, July 1, 2015 at 6:07:12 AM UTC-7, Dan Christensen wrote:
> > DEFINING ACTUAL INFINITY
> >
> > A set S is said to be actually infinite iff there exists an injective function f: S --> S such that f is not surjective. (There are other, equivalent definitions.)
>
> There are also other non-equivalent definitions
>
> >
> >
> > DEFINING POTENTIAL INFINITY
> >
> > Is it possible to formally define the potentially infinite?
>
> Certainly. It has been done in functional programming
>

Not interested in functional programming as such. But thanks for the links anyway.

>https://en.wikipedia.org/wiki/Lazy_evaluation#Working_with_infinite_data_structures
> http://www.cs.nott.ac.uk/~txa/mgs.2014/inf.html
> https://www.haskell.org/tutorial/functions.html#tut-infinite
> http://courses.cs.washington.edu/courses/cse341/08au/haskell/infinite/
>
> https://www.google.com/search?q=infinite+data-structure
>
> >
> > A set S is said to be potentially infinite iff .....????
>
> Sets (in the strict mathematical sense of the word) are not said to be "potentially infinite". You need some sort of structure or order to talk about "potentially infinite". Sets have no structure, no order. That's what makes them so useful.

The set of natural numbers has structure and an ordering can be defined on it. It is an actual infinity by the formal definition above. Is there no formal definition of "potential infinity?" I suspect that it cannot be formally defined, and that it is realm of philosophers, not mathematicians.

[George Greene]

> On Wednesday, July 1, 2015 at 6:07:12 AM UTC-7, Dan Christensen wrote:
> > DEFINING POTENTIAL INFINITY
> >
> > Is it possible to formally define the potentially infinite?

On Wednesday, July 1, 2015 at 6:44:39 PM UTC-4, FredJeffries wrote:

> Certainly. It has been done in functional programming

No, it hasn't.

> https://en.wikipedia.org/wiki/Lazy_evaluation#Working_with_infinite_data_structures
> http://www.cs.nott.ac.uk/~txa/mgs.2014/inf.html
> https://www.haskell.org/tutorial/functions.html#tut-infinite
> http://courses.cs.washington.edu/courses/cse341/08au/haskell/infinite/

All of those are finitary representations OF ACTUALLY infinite data-types.

> > A set S is said to be potentially infinite iff .....????
>
> Sets (in the strict mathematical sense of the word)
> are not said to be "potentially infinite".

Exactly. In any set theory worthy of the name, there are going
to exist actually infinitely many sets. No matter how big a set is,
you can "always" add another element to it, in principle, so all talk
of potential infinity in THAT context is just bullshit. You can have
a set theory in which all the sets are finite and all the infinite collections
are proper classes, but they are still ACTUALLY infinite proper classes.
In that context, all sets are actually finite and all proper classes are
actually infinite. NOTHING IS EVER potentially infinite.

> You need some sort of structure or order to talk about
> "potentially infinite".

You need a context in which variables rather than their values are
important. In that context, you could have a variable and allege that
it is of type "potentially infinite" if it in fact IS NOT potentially
infinite, but, even though it is always guaranteed to be finite, it has
no uppoer bound on HOW LARGE finite.

> Sets have no structure, no order.

Oh, horseshit. The fact that sets don't have inherent order is a living
nightmare. People are inventing set-representations of ordered things
ALL THE TIME (ordinals, the Kuratowski definition of ordered pair, ad nauseam)
to get around it.

> That's what makes them so useful.

For some people, some of the time. The fact that no particular order is
inherent means you are free to define your own. But you are sort of NOT
free to revel in the non-ordered-ness of everything.

[Julio Di Egidio]

On Thursday, July 2, 2015 at 6:33:06 PM UTC+1, George Greene wrote:
> > On Wednesday, July 1, 2015 at 6:07:12 AM UTC-7, Dan Christensen wrote:

> > Sets (in the strict mathematical sense of the word)
> > are not said to be "potentially infinite".
>
> Exactly.

Exactly wrong! Potential vs. actual infinity is first and foremost a philosophical notion, and it just means what it means: one is the never ending but ever finite, the other is accomplished infinity. But those are just qualifications with no absolute meaning in mathematics: that, and the fact that mostly everybody is confused (and echoing it), is the only reason why things get confused. In a system with sequences and limits, one can say that a sequence is a potential infinity while the limit of a sequence is an actual infinity. Just informal qualifications. In a set theory, one can say that N is potentially infinite while N* (its compactification) is actually infinite. Finitary mathematics only deals with the potentially infinite. The transfinite is the realm of actual infinities. Just qualifications: they may belong to the logic, but, strictly speaking, they do not belong to the mathematics.

Yes, N is a *potentially infinite* set, of course. And the fact that "set" and "unfinished" are logically incompatible is rather yet another instance of the fundamental issues of standard set theory. Not per chance, in strict finitism, which is a proper formalisation of finitary mathematics, N is *not* a set.

Julio

[Julio Di Egidio]

On Thursday, July 2, 2015 at 7:07:33 PM UTC+1, Julio Di Egidio wrote:
> On Thursday, July 2, 2015 at 6:33:06 PM UTC+1, George Greene wrote:
> > > On Wednesday, July 1, 2015 at 6:07:12 AM UTC-7, Dan Christensen wrote:
>
> > > Sets (in the strict mathematical sense of the word)
> > > are not said to be "potentially infinite".
> >
> > Exactly.
>
> Exactly wrong!

I have snipped too much. Anyway you continued with "In any set theory worthy of the name, there are going to exist actually infinitely many sets." Then, since by other clues I must maintain that you are surely not in agreement with me, you are rather (not even) wrong. Follows the rest of the answer:

[George Greene]

On Thursday, July 2, 2015 at 2:07:33 PM UTC-4, Julio Di Egidio wrote:
> On Thursday, July 2, 2015 at 6:33:06 PM UTC+1, George Greene wrote:
> > > On Wednesday, July 1, 2015 at 6:07:12 AM UTC-7, Dan Christensen wrote:
>
> > > Sets (in the strict mathematical sense of the word)
> > > are not said to be "potentially infinite".
> >
> > Exactly.
>
> Exactly wrong!

No, he's right, and to say you suck a philosophy would be an insult to suckers eerywhere.

> Potential vs. actual infinity is first and foremost a philosophical notion,

This is sci.logic.
If you can't coherently say what you mean, then you don't get a get out of jail free card just by calling it "philosophical". NOTHING is "first and foremost a philosophical notion". That's just something that fools like you say when you are too stupid to say what you mean.

> and it just means what it means:

it doesn't mean anything.

> one is the never ending but ever finite,

This IS IMPOSSIBLE, IDIOT.
If it doesn't end then IT IS ACTUALLY infinite.
BY DEFINITION.

There is A TYPE conflict here.
A *VARIABLE* is one thing.
A CONSTANT is another!
A VARIABLE can be what YOU call "potentially infinite"
IF its values ARE ALWAYS ACTUALLY finite but there is no upper
bound on them and the NUMBER of its possible values is therefore
ACTUALLY INFINITE.
But there IS NO "thing" that is "potentially infinite" and there is
no "amount" that is "potential infinity"!!

> the other is accomplished infinity.
> But those are just qualifications with
> no absolute meaning in mathematics:

Oh, again, HORSESHIT. EVERYthing gets to have an absolute meaning in mathematics. In mathematics, you can just STIPULATE AXIOMS *REQUIRING* things to have the properties you want them to have(as long as the desiderata are not mutually inconsistent). If you KNOW what you mean then you CAN SAY so.
YOU *DON'T*GET* to just wave your hands on go "I'm being a philosopher".
You're being an ignoramus.
I JUST IN FACT GAVE YOU a way to get mathematically precise about it.
There are tons of places where it is just standard linguistic practice
to write, in an expression, the variable, when the purpose of doing that
is to get the expression EVALUATED by REPLACING the variable with its VALUE.
But despite the fact that we routinely write one to mean the other, the
variable and its values are INHERENTLY DIFFERENT TYPES of things.
"Potentially infinite" is not merely "philosophical" -- rather, it applies in contexts where you are thinking about A *VARIABLE* AS OPPOSED to its (generally finite; actually absolutely finite, all ACTUALLY INFNITELY MANY of them are finite) VALUE.

[George Greene]

On Thursday, July 2, 2015 at 2:07:33 PM UTC-4, Julio Di Egidio wrote:

> Yes, N is a *potentially infinite* set, of course.

N is an ACTUALLY infinite set.
IT ACTUALLY HAS INFINITELY many members. And it IS "finished".
Nevertheless, it is true that every one of those infinitely many elements is finite. Rational human beings simply do not have any PROBLEM with this.
It is not any kind of tension or contradiction.
It does not NEED to be explained.
The set is actually infinite.
The elements are actually finite.
NOTHING is "potential".

IN ORDER TO MAKE "potential infinity" relevant, you have to invoke TYPE.
You have to be aware that some mathematical entities are different KINDS
of things than others. In particular, A VARIABLE is a different TYPE of thing
from A CONSTANT -- EVEN when the type-of-the-variable is or includes that
selfsame constant.

It is reasonable and meaningful to say, of a VARIABLE of TYPE N -- i.e.,
a variable with a set-of-possible-values that IS N, i.e., a variable whose
possible values are natural numbers, a variable whose value is ONE natural
number -- that that VARIABLE is "potentially infinite". That is a gross
abuse of language because the point is that the variable is NOT infinite,
not even potentially -- its value is always guaranteed to be actually finite.
But because THERE IS NO UPPER BOUND on said value, infinity becomes relevant to this variable in the following 2 ways: way#1) it has an actually infinite number of different possible values, and way#2) if YOU ACTUALLY WANTED TO STORE this variable in some computer, then you would need an actually infinite number of bits to do it, because even though, every time you changed its value, you would only be using a finite initial segment of those bits and infinitely many of them would remain unused, all infinitely many of them could POTENTIALLY NEED to be used, at SOME VARIOUS times, even though they could never be used all at once.

[WM]

On Friday, 3 July 2015 11:37:46 UTC+2, George Greene wrote:
> On Thursday, July 2, 2015 at 2:07:33 PM UTC-4, Julio Di Egidio wrote:
> > Yes, N is a *potentially infinite* set, of course.
>
> N is an ACTUALLY infinite set.

That is your insane belief - nothing else.

Regards, WM

[Julio Di Egidio]

On Friday, July 3, 2015 at 10:31:17 AM UTC+1, George Greene wrote:

> But there IS NO "thing" that is "potentially infinite" and there is
> no "amount" that is "potential infinity"!!

You are just a fucking idiot, that's all. *Plonk*

Julio

[Virgil]

In article <0c1c77be-824e-4a79...@googlegroups.com>,

WM, who believes in a set F for which
F \ F = |N and who believes that a mapping carrying the mth power of the
nth prime in |N to m/n in Q+ does NOT surject |N onto Q+, is hardly in a
position to claim that anyone else's beliefs are insane!
--
Virgil
"Mit der Dummheit kampfen Gotter selbst vergebens." (Schiller)

[George Greene]

On Friday, July 3, 2015 at 9:07:46 AM UTC-4, WM wrote:
> > N is an ACTUALLY infinite set.
>
> That is your insane belief - nothing else.

Bitch, please.
It's IN A THEORY. The theory is consistent.
All the models of the theory believe it as well.
It's not just me.
Your problem is that you think you can define it OUTside some theory.
You don't even GET TO TALK about mathematical things in non-mathematical
terms. No one does. Well, maybe, with N, they do, because N is special.
But you CAN'T just say that something in math is "insane" -- you HAVE TO *DERIVE*A*CONTRADICTION* from it. You have to actually refute it.
Nobody can help it if you don't understand the sign on the door to THIS forum.

[George Greene]

On Friday, July 3, 2015 at 12:41:13 PM UTC-4, Julio Di Egidio wrote:
> You are just a fucking idiot, that's all. *Plonk*

Your plonking me won't stop anyone from reading my refutations of you.

[Julio Di Egidio]

Read your nonsense and incompetence... You keep violently making an ass of yourself on things you know shit-all about (you narrow-minded red neck), while you won't answer questions that you are asked on topics you are supposed to know... Hence I have simply decided you are not worth any more effort from my side: you are useless, and you are in my kill-file under all respects from now on, together with the many monsters also pestering these and all neighbourhoods. EOD.

Julio

[George Greene]

On Friday, July 3, 2015 at 4:11:14 PM UTC-4, Julio Di Egidio wrote:
> You keep violently making an ass of yourself on things you
> know shit-all about (you narrow-minded red neck),

Liar.

> while you won't answer questions
> that you are asked on topics you are supposed to know...

I can't help it if I haven't seen all your questions.
Can you be more specific? I don't respect people who are
asking questions, most of the time. You haven't asked any questions
about Cantor's arguments. You have been TELLING, NOT asking.
I have as well. I don't HAVE any QUESTIONS about this.
And I gave you your relevant answer about the difference between a constant
and a variable. That really is what is happening here.

[George Greene]

On Friday, July 3, 2015 at 4:11:14 PM UTC-4, Julio Di Egidio wrote:
> Read your nonsense and incompetence...

Says the fool who thinks he can refute Cantor's theorem, or that it has a type error. Jeezus.

[Virgil]

In article <33e7ffb3-dffb-44a4...@googlegroups.com>,

Julio's "arguments" above are all strictly ad hominem and thus totally
irrelevant to either honest mathematics or honest logic!

[Zeit Geist]

Then, YOU please give a formalized definition of "potential infinity". If you cannot, then you have no business being here.

> Regards, WM

ZG

[George Greene]

On Friday, July 3, 2015 at 6:35:05 PM UTC-4, Zeit Geist wrote:
> Then, YOU please give a formalized definition of "potential infinity".
> If you cannot, then you have no business being here.

Lots of people about whom that could be said. That it is unmoderated
is a strength of this forum but for better or for worse, the smart people
have given up.

Clue about "business being here": the sign above the door SAYS
sci.logic.
If you are not going to take a logical approach then.....

[George Greene]

On Wednesday, July 1, 2015 at 9:07:12 AM UTC-4, Dan Christensen wrote:
> DEFINING ACTUAL INFINITY
>
> A set S is said to be actually infinite iff there exists an injective function f: S --> S such that f is not surjective. (There are other, equivalent definitions.)
>
>
> DEFINING POTENTIAL INFINITY
>
> Is it possible to formally define the potentially infinite?
>
> A set S is said to be potentially infinite iff .....????

That is way oversimplified. "Their" point is sort of that BECAUSE infinity in general IS potential, and sets are completed finished objects of mathematical inquiry, there is no such thing as an actually infinite set.
That is sort of defensible; you can define a theory where all the sets
are finite and all the infinite classes are proper. But it's only SORT OF defensible, the point being that the infinite classes, despite being too big
to be sets, are still "finished", "completed", well-understood entities (proper
classes) that ARE ACTUALLY infinite.

It is truly a shame that WM and Julio are both commenting on this.
Julio actually had something to say but there is all that WM noise....

[George Greene]

On Wednesday, July 1, 2015 at 9:07:12 AM UTC-4, Dan Christensen wrote:

> DEFINING ACTUAL INFINITY
>
> A set S is said to be actually infinite iff there exists an injective function f: S --> S such that f is not surjective. (There are other, equivalent definitions.)
>
>
> DEFINING POTENTIAL INFINITY
>
> Is it possible to formally define the potentially infinite?

Getting formal is just a mistake.
Something is infinite if and only if it is not finite. All the NUMBERS you KNOW are finite. I don't know how old you were before you were taught about
aleph_0 or omega+omega but all that stuff is above&beyond. Every natural number and every real number YOU KNOW is finite. It doesn't make sense to call something infinite if there is not SOME aspect of NUMBER being used to indicate its SIZE. And if that size is BIGGER THAN any number, well, again, that's INformal.

This is a lot more intuitive if you allow 0 to count as a natural number; you need it both in PA and for founding ordinals in ZF.

WithOUT getting formal, you can handwave that for any natural number, there are more natural numbers than that, so no natural number is the number OF natural numbers.

Therefore the class of all&only-the natural numbers is infinite (in size, in cardinality). Invoking Dedekind is a mistake. Formally, though, you might have to invoke something slightly more intuitive even if it is logically equivalent; i.e., all natural numbers are finite (BY DEFINITION) and any finite
set of them has a largest element.

[George Greene]

On Thursday, July 2, 2015 at 2:07:33 PM UTC-4, Julio Di Egidio wrote:

> Exactly wrong! Potential vs. actual infinity
> is first and foremost a philosophical notion,

No, it isn't. It's an utter delusion by medieval cretins who had never heard of computer science and didn't know the difference between a constant and a variable.
The MERE NOTION of that difference either elucidates OR ELIMINATES the QUESTION.

[Virgil]

On Friday, July 3, 2015 at 6:07:46 AM UTC-7, WM wrote:
> On Friday, 3 July 2015 11:37:46 UTC+2, George Greene wrote:
> > On Thursday, July 2, 2015 at 2:07:33 PM UTC-4, Julio Di Egidio wrote:
> > > Yes, N is a *potentially infinite* set, of course.
> >
> > N is an ACTUALLY infinite set.
>
> That is your insane belief - nothing else.

A fairly standard distinction between finite sets and infinite sets, at
least everywhere outside of WM's worthless world of WMytheology,
is that a FINITE set can never be injected into any proper subset of
itself but an INFINITE, by being non-finite, set CAN be injected into
proper subsets of itself.

The mapping n -> n+1 injects |N to a proper subset of |N.

Pray tell, WM, how does that FAIL to show that |N is NOT FINITE
and thus IN-FINITE according to that standard definition?

And if WM refuses to allow the word "infinite" to be used for sets which
can be injected to finite subset of themselves, how about "non-finite"?

[Virgil]

On Friday, July 3, 2015 at 6:07:46 AM UTC-7, WM wrote:
> On Friday, 3 July 2015 11:37:46 UTC+2, George Greene wrote:

> > On Thursday, July 2, 2015 at 2:07:33 PM UTC-4, Julio Di Egidio wrote:

> > > Yes, N is a *potentially infinite* set, of course.
> >

> > N is an ACTUALLY infinite set.
>
> That is your insane belief - nothing else.

Depends on one's definition of a set being infinite or only finite.

The standard distinguishing issue for set finiteness is whether a given
set has any injections to any of its proper subsets. Sets which don't
are finite, sets which do are not finite.

|N, for example, has n -> n+1 mapping |N to a proper suset of |N.

At least for any |N satisfying the Peano axioms.

So unless WM can produce an example of a potentially infinite set which
does not allow injection to any of its proper but presumably also
potentially infinite subsets, he is shit out of luck!

[Virgil]

In article <76df7187-13e8-45ef...@googlegroups.com>,

George Greene <gre...@email.unc.edu> wrote:

> On Wednesday, July 1, 2015 at 9:07:12 AM UTC-4, Dan Christensen wrote:
> > DEFINING ACTUAL INFINITY
> >
> > A set S is said to be actually infinite iff there exists an injective
> > function f: S --> S such that f is not surjective. (There are other,
> > equivalent definitions.)
> >
> >
> > DEFINING POTENTIAL INFINITY
> >
> > Is it possible to formally define the potentially infinite?
> >
> > A set S is said to be potentially infinite iff .....????
>
>
> That is way oversimplified. "Their" point is sort of that BECAUSE infinity
> in general IS potential, and sets are completed finished objects of
> mathematical inquiry, there is no such thing as an actually infinite set.

That depends both on one's definition of a set being infinite and
whether one accepts that there are sets satisfying the Peano axioms.
A set for which there are no injections from it to any of its proper
subsets is, by a fairly standard definition, finite, but that is false
for any set satisfying the Peano axioms, like |N.

So in any set theory allowing Peano sets there is a deinition of a set
being infinite (non-finite) that Peano sets all satisfy!

[WM]

On Friday, 3 July 2015 21:27:53 UTC+2, George Greene wrote:
> On Friday, July 3, 2015 at 9:07:46 AM UTC-4, WM wrote:
> > > N is an ACTUALLY infinite set.
> >
> > That is your insane belief - nothing else.
>
>

> It's IN A THEORY. The theory is consistent.

Only if you deny rational thinking.



0
1,0
2,1,0
...

If this set is actually infinite, then there are more finite sequences than any sequence has terms. This been contradicted by the nested arithmogeometrical triangle:

1

1
22

3
31
322

3
31
322
4444

5
53
531
5322
54444

...

which cannot have aleph_0, i.e., more than every finite number of symbols, neither in height nor in width.

Regards, WM

[WM]

On Saturday, 4 July 2015 00:35:05 UTC+2, Zeit Geist wrote:


> Then, YOU please give a formalized definition of "potential infinity".

Peano axioms and ZF axiom of infinity are such formalizations. Actual infinity comes in by unfounded belief only.

Regards, WM

[Virgil]

In article <3e57f2e3-7634-4afc...@googlegroups.com>,

A set is finite when it cannot be injected into any proper subset of
itself, and it is non-finite or actually infinite when it can be
injected into a proper subset (like n -> n+1 maps |N to |N\{1}).

When is a set potentially infinite but neither finite nor actually
infinite?

[Virgil]

In article <90c4b5cc-fea0-4e66...@googlegroups.com>,

WM <muec...@rz.fh-augsburg.de> wrote:

> On Friday, 3 July 2015 21:27:53 UTC+2, George Greene wrote:
> > On Friday, July 3, 2015 at 9:07:46 AM UTC-4, WM wrote:
> > > > N is an ACTUALLY infinite set.
> > >
> > > That is your insane belief - nothing else.
> >
> >
> > It's IN A THEORY. The theory is consistent.
>
> Only if you deny rational thinking.

WM knows a lot about denying rational thinking as it is his
standard modus operandi

>
> 0
> 1,0
> 2,1,0
> ...
>
> If this set is actually infinite

WHat are the members of that set?
If each line is a member, one can map each line to the next, thus
mapping the set to a proper subset of itself and thus satisfying the
definition of actual infiniteness!

So WM is
WRONG !
AGAIN ! !
AS USUAL ! ! !

[Nam Nguyen]

On 04/07/2015 9:12 AM, Virgil wrote:
> In article <3e57f2e3-7634-4afc...@googlegroups.com>,
> WM <muec...@rz.fh-augsburg.de> wrote:
>
>> On Saturday, 4 July 2015 00:35:05 UTC+2, Zeit Geist wrote:
>>
>>> Then, YOU please give a formalized definition of "potential infinity".
>>
>> Peano axioms and ZF axiom of infinity are such formalizations. Actual
>> infinity comes in by unfounded belief only.
>
> A set is finite when it cannot be injected into any proper subset of
> itself, and it is non-finite or actually infinite when it can be
> injected into a proper subset (like n -> n+1 maps |N to |N\{1}).
>
> When is a set potentially infinite but neither finite nor actually
> infinite?

Fwiw, the set of counter examples of Goldbach Conjecture is both
potentially infinite and potentially finite.

--
-----------------------------------------------------
There is no remainder in the mathematics of infinity.

NYOGEN SENZAKI

[Nam Nguyen]

On 04/07/2015 9:30 AM, Nam Nguyen wrote:
> On 04/07/2015 9:12 AM, Virgil wrote:
>> In article <3e57f2e3-7634-4afc...@googlegroups.com>,
>> WM <muec...@rz.fh-augsburg.de> wrote:
>>
>>> On Saturday, 4 July 2015 00:35:05 UTC+2, Zeit Geist wrote:
>>>
>>>> Then, YOU please give a formalized definition of "potential infinity".
>>>
>>> Peano axioms and ZF axiom of infinity are such formalizations. Actual
>>> infinity comes in by unfounded belief only.
>>
>> A set is finite when it cannot be injected into any proper subset of
>> itself, and it is non-finite or actually infinite when it can be
>> injected into a proper subset (like n -> n+1 maps |N to |N\{1}).
>>
>> When is a set potentially infinite but neither finite nor actually
>> infinite?
>
> Fwiw, the set of counter examples of Goldbach Conjecture is both
> potentially infinite and potentially finite.

Goldbach Conjecture is about the natural numbers.

Now, let nGC = The number of counter examples of Goldbach Conjecture.

Obviously nGC, which could still be a valid natural number, is about
something (the Conjecture) that is about the natural numbers: sort
of like "THIS statement is true".

_A natural number about the natural numbers_ ? Incompleteness - indeed -
goes beyond formal system provability!

Think about it.

[WM]

On Saturday, 4 July 2015 17:12:40 UTC+2, Virgil wrote:
> In article <3e57f2e3-7634-4afc...@googlegroups.com>,
> WM <muec...@rz.fh-augsburg.de> wrote:
>
> > On Saturday, 4 July 2015 00:35:05 UTC+2, Zeit Geist wrote:
> >
> >
> > > Then, YOU please give a formalized definition of "potential infinity".
> >
> > Peano axioms and ZF axiom of infinity are such formalizations. Actual
> > infinity comes in by unfounded belief only.
>
> A set is finite when it cannot be injected into any proper subset of
> itself, and it is non-finite or actually infinite when it can be
> injected into a proper subset (like n -> n+1 maps |N to |N\{1}).
>
> When is a set potentially infinite but neither finite nor actually
> infinite?

When it has no cardinal number aleph_0.

Regards, WM

[Virgil]

In article <47a22977-1d09-437a...@googlegroups.com>,

WM <muec...@rz.fh-augsburg.de> wrote:

> On Saturday, 4 July 2015 17:12:40 UTC+2, Virgil wrote:
> > In article <3e57f2e3-7634-4afc...@googlegroups.com>,
> > WM <muec...@rz.fh-augsburg.de> wrote:
> >
> > > On Saturday, 4 July 2015 00:35:05 UTC+2, Zeit Geist wrote:
> > >
> > >
> > > > Then, YOU please give a formalized definition of "potential infinity".
> > >
> > > Peano axioms and ZF axiom of infinity are such formalizations. Actual
> > > infinity comes in by unfounded belief only.
> >
> > A set is finite when it cannot be injected into any proper subset of
> > itself, and it is non-finite or actually infinite when it can be
> > injected into a proper subset (like n -> n+1 maps |N to |N\{1}).
> >
> > When is a set potentially infinite but neither finite nor actually
> > infinite?
>
> When it has no cardinal number aleph_0.

The |N, by having cardinality aleph_0, must be actually infinite , as it
clearly is not potentially infinite according to WM's own definition!

[graham...@gmail.com]

On Sunday, July 5, 2015 at 1:12:40 AM UTC+10, Virgil wrote:
>
> A set is finite when it cannot be injected into any proper subset of
> itself, and it is non-finite or actually infinite when it can be
> injected into a proper subset (like n -> n+1 maps |N to |N\{1}).

If someone were to walk up the infinite corridor of Hilbert's Hotel he wouldn't know if it was potentially infinite or absolute infinity

ALL(x):[x in N => ... ] where N is the set of natural numbers

>
> When is a set potentially infinite but neither finite nor actually
> infinite?

You can use


N C S


OR


ALL(x) x e N -> x e S


OR


(0 e S)
^
(X e S) -> (s(X) e S)



If a finite induction formula exists then absolute infinity is merely academic.

[Virgil]

In article <f9930916-aaf8-48d0...@googlegroups.com>,

graham...@gmail.com wrote:

> On Sunday, July 5, 2015 at 1:12:40 AM UTC+10, Virgil wrote:
> >
> > A set is finite when it cannot be injected into any proper subset of
> > itself, and it is non-finite or actually infinite when it can be
> > injected into a proper subset (like n -> n+1 maps |N to |N\{1}).
>
>
> If someone were to walk up the infinite corridor of Hilbert's Hotel he
> wouldn't know if it was potentially infinite or absolute infinity

He would know it was not finite, which is all that infinite means!

>
> ALL(x):[x in N => ... ] where N is the set of natural numbers

For one thing:
ALL(x):[x in N => x+1 in N ] where N is the set of natural numbers

[George Greene]

> On Friday, 3 July 2015 21:27:53 UTC+2, George Greene wrote:
> > It's IN A THEORY. The theory is consistent.
>
> Only if you deny rational thinking.

Oh, horse-shit.
The theoretical paradigm we are talking about DEFINES rational thinking,
on the left-brain side, anyway. There are spatial/geometrical forms of
reasoning that are inherently different, but a lot of them can also be
translated into the string-based paradigm we are talking about.

>
> 0
> 1,0
> 2,1,0
> ...
>
> If this set is actually infinite, then there are more finite sequences than any sequence has terms.

OF COURSE THERE ARE!!

> This been contradicted by the nested arithmogeometrical triangle:

No, it hasn't, and it can't be; the rules for that geometry ARE JUST DIFFERENT.
A contradiction looks like A SYNTACTIC contradiction in this theory.
But you would have to define some axioms for describing these sequences, first.
OBVIOUSLY, if you have an infinite sequence of finite sequences, then the number
of sequences is greater than the number of terms in any one sequence.
Equally obviously, if you have an infinite set of finite numbers, then the
number of numbers in the set is bigger than any one number IN the set.
This is NOT a problem or a contradiction and you don't get to draw
"arithmgeometrical triangles" to make it so.

>
> 1
>
> 1
> 22
>
> 3
> 31
> 322
>
> 3
> 31
> 322
> 4444
>
> 5
> 53
> 531
> 5322
> 54444
>
> ...

You hve not followed your own convention here, apparently.
I can't discern your alleged pattern. It was not necessary for you
to even try this with triangles.

> which cannot have aleph_0, i.e.,
> more than every finite number of symbols, neither in height nor in width.

Each INDIVIDUAL triangle is finite in height and width, BUT OBVIOUSLY,
if you have INFINITELY many triangles, they can have, in toto, infinitely
many symbols. There is also,equally obviously, an infinite version of this
triangle-- just like there is an infinite binary tree-- that IS THE LIMIT OF
this infinite sequence of finite triangles.
Said triangle IS ACTUALLY infinite and DOES NOT HAVE a bottom row,
just as surely as N does not have a biggest natnum.

[WM]

On Tuesday, 7 July 2015 05:56:59 UTC+2, George Greene wrote:


> > 0
> > 1,0
> > 2,1,0
> > ...
> >
> > If this set is actually infinite, then there are more finite sequences than any sequence has terms.
>
> OF COURSE THERE ARE!!

So this triangle has a height aleph_0 according to set theory.

>
> > This been contradicted by the nested arithmogeometrical triangle:
>
> No, it hasn't, and it can't be; the rules for that geometry ARE JUST DIFFERENT.
> A contradiction looks like A SYNTACTIC contradiction in this theory.
> But you would have to define some axioms for describing these sequences, first.

The sequences are taken fromn the set of natural numbers by Axiom of Separation. Simply drop the commas:

0
10
210
...

And if you have problems with the succesor of 9 use this sequence

1
11
111
...

> OBVIOUSLY, if you have an infinite sequence of finite sequences, then the number
> of sequences is greater than the number of terms in any one sequence.

Contradicted by the arithmogeometrical triangle below.

> Equally obviously, if you have an infinite set of finite numbers, then the
> number of numbers in the set is bigger than any one number IN the set.

That is not obvious but it is nonsense, since there is no number of numbers in an infinite set!

> This is NOT a problem or a contradiction and you don't get to draw
> "arithmgeometrical triangles" to make it so.
>
> >
> > 1
> >
> > 1
> > 22
> >
> > 3
> > 31
> > 322
> >
> > 3
> > 31
> > 322
> > 4444
> >
> > 5
> > 53
> > 531
> > 5322
> > 54444
> >
> > ...
>

> I can't discern your alleged pattern.

Sorry, then my proof is not digestible for you.

> It was not necessary for you
> to even try this with triangles.

If you have problems with the successor of 9, use 1, 22, 111, 2222, ..., all these are natural numbers obtained by restricted comprehension from the set |N.

>
> > which cannot have aleph_0, i.e.,
> > more than every finite number of symbols, neither in height nor in width.
>
> Each INDIVIDUAL triangle is finite in height and width, BUT OBVIOUSLY,
> if you have INFINITELY many triangles,
> they can have, in toto, infinitely
> many symbols.

There is a complete triangle, according to set theory. It cannot have infinitely many symbols in any horizontal line and in any vertical line.

> There is also,equally obviously, an infinite version of this
> triangle-- just like there is an infinite binary tree-- that IS THE LIMIT OF
> this infinite sequence of finite triangles.

Yes, it is the limit. But your claim is wrong. According to set theory aleph_0 is the cardinal number of all *finite* numbers. And for all n in |N: aleph_0 > n. This is contradicted by the simple fact that all sides of the limit triangle are finite. Even in the limit of all natural numbers we have no infinite number among them.

> Said triangle IS ACTUALLY infinite and DOES NOT HAVE a bottom row,
> just as surely as N does not have a biggest natnum.

It does not have a bottom row. In particular it does not have an infinite bottom row. All it has is finite. Just like the set of natural numbers which has not a "bottom row" or any aleph_0 either.

Regards, WM

[WM]

On Tuesday, 7 July 2015 05:32:50 UTC+2, Virgil wrote:
> In article <f9930916-aaf8-48d0...@googlegroups.com>,
> graham...@gmail.com wrote:


> > If someone were to walk up the infinite corridor of Hilbert's Hotel he
> > wouldn't know if it was potentially infinite or absolute infinity
>
> He would know it was not finite, which is all that infinite means!

Not in any argument identifying a real number with an infinbite digit sequence.
Further the walker would never know whether it was finite or not until he had gone the complete way.

Regards, WM

[George Greene]

On Tuesday, July 7, 2015 at 8:59:16 AM UTC-4, WM wrote:
> > I can't discern your alleged pattern.
>
> Sorry, then my proof is not digestible for you.

No, you are either being unclear, or failing to follow your own intended pattern properly. The precise nature of the pattern is not really relevant
to the "proof" in any case -- lots of different patterns would serve the purpose
equally well.

> If you have problems with the successor of 9,

I don't; that is not relevant.

> There is a complete triangle, according to set theory.
> It cannot have infinitely many symbols in any horizontal
> line and in any vertical line.

IDiot, you have NOT DRAWN a RIGHT triangle! IF YOU DO then
in the USUAL orientation, YES IT DOES have infinitely many symbols ON TWO
of its lines, and the 3rd side DOES NOT EVEN EXIST!

>
> > There is also,equally obviously, an infinite version of this
> > triangle-- just like there is an infinite binary tree-- that IS THE LIMIT OF
> > this infinite sequence of finite triangles.
>
> Yes, it is the limit. But your claim is wrong. According to set theory aleph_0 is the cardinal number of all *finite* numbers.

Of THE CLASS of all finite numbers, yes.

> And for all n in |N: aleph_0 > n.

Yes.

> This is contradicted by the simple
> fact that all sides of the limit triangle are finite.

FOOL, *NO* side of the limit triangle is finite. ALL THREE of its sides are of length aleph_0.

[George Greene]

On Tuesday, July 7, 2015 at 9:01:56 AM UTC-4, WM wrote:

> Not in any argument identifying a real number with an infinbite digit sequence.
> Further the walker would never know whether
> it was finite or not until he had gone the complete way.

It DOESN'T MATTER what THE WALKER would or wouldn't know. WE are NOT the walker and WE know.

[George Greene]

On Tuesday, July 7, 2015 at 8:59:16 AM UTC-4, WM wrote:

> If you have problems with the successor of 9, use 1, 22, 111, 2222, ...,
> all these are natural numbers obtained by restricted comprehension
> from the set |N.

You're really over-killing trying to talk about ZFC. You don't even believe
ZFC. The rest of us understand it better than you do, so you don't have to
explain to us how to use it to construct stuff. That's what WE do for YOU.

The example would've been clearer if you had just said that the nth row
should be n copies of n. It doesn't matter whether they are or aren't
single digits. They could be big numbers with large numbers of digits.
It would look more like a trumpet than a triangle in a fixed font, but
there's no arbitrary lower limit on font-size; you could've still made
it look like a triangle.

> There is a complete triangle,
> according to set theory.

Of course.

> It cannot have infinitely many symbols in any horizontal line

OBVIOUSLY -- every horizontal line has n numbers so of course it's
finite.

> and in any vertical line.

Wrong. ALL the "vertical" lines are infinitely long. There are
infinitely many rows, therefore, BOTH SIDES of the triangle are infinitely
long. IF YOU DO IT RIGHT,
1
22
333
4444
55555
666666
7777777
88888888
999999999
(continue as you will; I personally would choose
10101010101010101010
because WHO CARES if the sides are parabolic??)
then the left side of the triangle IS N and the right side of the triangle IS N and the "bottom side" of the "triangle" DOES NOT EXIST.

[Virgil]

In article <737dc02e-69da-4eb3...@googlegroups.com>,

WM <muec...@rz.fh-augsburg.de> wrote:

> On Tuesday, 7 July 2015 05:32:50 UTC+2, Virgil wrote:
> > In article <f9930916-aaf8-48d0...@googlegroups.com>,
> > graham...@gmail.com wrote:
>
>
> > > If someone were to walk up the infinite corridor of Hilbert's Hotel he
> > > wouldn't know if it was potentially infinite or absolute infinity
> >
> > He would know it was not finite, which is all that infinite means!
>
> Not in any argument identifying a real number with an infinbite digit
> sequence.

Hilbert's Hotel does not need any real numbers other than natural
numbers, none of which requires any infinite digit sequences,

[Virgil]

In article <6a1796ce-d9ec-4683...@googlegroups.com>,

WM <muec...@rz.fh-augsburg.de> wrote:

> On Tuesday, 7 July 2015 05:56:59 UTC+2, George Greene wrote:
>
>
> > > 0
> > > 1,0
> > > 2,1,0
> > > ...
> > >
> > > If this set is actually infinite, then there are more finite sequences
> > > than any sequence has terms.
> >
> > OF COURSE THERE ARE!!
>
> So this triangle has a height aleph_0 according to set theory.

What "triangle"?

Where in WM's alleged limit are there three identifiable sides or three
identifiable vertices? Without exactly 3 of each, whatever else WM's
alleged limit sets might be they are not triangles!

[Julio Di Egidio]

On Tuesday, July 7, 2015 at 4:48:41 PM UTC+1, Virgil wrote:

> Hilbert's Hotel does not need any real numbers other than natural
> numbers, none of which requires any infinite digit sequences,

Hilbert's Hotel is nonsense: a potentially infinite hotel is never full.

For a little bit more explication:

<http://seprogrammo.blogspot.co.uk/2014/02/hilberts-impossible-hotel.html>

Julio

[Charlie-Boo]

On Wednesday, July 1, 2015 at 9:07:12 AM UTC-4, Dan Christensen wrote:
> DEFINING ACTUAL INFINITY
>
> A set S is said to be actually infinite iff there exists an injective function f: S --> S such that f is not surjective. (There are other, equivalent definitions.)
>
>
> DEFINING POTENTIAL INFINITY
>
> Is it possible to formally define the potentially infinite?
>
> A set S is said to be potentially infinite iff .....????
>
>

> Dan
>
> Download my DC Proof 2.0 software at http://www.dcproof.com
> Visit my Math Blog at http://www.dcproof.wordpress.com

Yes. To be able to ask the question it must be defined. For the question to be meaningful, it must be formal.

C-B

[WM]

Am Dienstag, 7. Juli 2015 15:18:27 UTC+2 schrieb George Greene:
> On Tuesday, July 7, 2015 at 8:59:16 AM UTC-4, WM wrote:
> > > I can't discern your alleged pattern.
> >
> > Sorry, then my proof is not digestible for you.
>
> No, you are either being unclear, or failing to follow your own intended pattern properly. The precise nature of the pattern is not really relevant
> to the "proof" in any case -- lots of different patterns would serve the purpose
> equally well.

Of course. But the existence of many proofs does not undermine mine.

>

>
> > There is a complete triangle, according to set theory.
> > It cannot have infinitely many symbols in any horizontal
> > line and in any vertical line.
>
> IDiot, you have NOT DRAWN a RIGHT triangle!

I did not say so. The triangle has horizontal and vertical lines.

> IF YOU DO then
> in the USUAL orientation, YES IT DOES have infinitely many symbols ON TWO
> of its lines,

Not if all natural numbers are finite.

> and the 3rd side DOES NOT EVEN EXIST!

The third side is irrelevant, but it exists as the separation into two different triangles

5
53
531
53
5

and

22
4444

shows.

> > Yes, it is the limit. But your claim is wrong. According to set theory aleph_0 is the cardinal number of all *finite* numbers.
>
> Of THE CLASS of all finite numbers, yes.

Class or set or not does not change the facts, if they are facts.

>
> > And for all n in |N: aleph_0 > n.
>
> Yes.
>
> > This is contradicted by the simple
> > fact that all sides of the limit triangle are finite.
>
> FOOL, *NO* side of the limit triangle is finite. ALL THREE of its sides are of length aleph_0.

Then aleph_0 is not the cardinal number of the set of finite numbers. Then infinite natural numbers are required to complete the set and set theory is even a worse mess.

Regards, WM

[WM]

You believe. You believe to know. Better you should know to believe.

Regards, WM

[WM]

Am Dienstag, 7. Juli 2015 15:28:23 UTC+2 schrieb George Greene:
> On Tuesday, July 7, 2015 at 8:59:16 AM UTC-4, WM wrote:
> > If you have problems with the successor of 9, use 1, 22, 111, 2222, ...,
> > all these are natural numbers obtained by restricted comprehension
> > from the set |N.
>
> You're really over-killing trying to talk about ZFC. You don't even believe
> ZFC.

I believe in ZFC. I do not believe in the common interpretation of the axiom of infinity.

> The rest of us understand it better than you do, so you don't have to
> explain to us how to use it to construct stuff. That's what WE do for YOU.

I have done it first. But fine if you understand that it is correct.

Fine that you understand and share my view. Now return to my version. I did it right. There is no final edge existing. But everything that is existing, is finite --- nowhere is aleph_0.

Regards, WM

[Virgil]

In article <7454f9fe-8798-435d...@googlegroups.com>,

WM <wolfgang.m...@hs-augsburg.de> wrote:

> Am Dienstag, 7. Juli 2015 15:28:23 UTC+2 schrieb George Greene:
> > On Tuesday, July 7, 2015 at 8:59:16 AM UTC-4, WM wrote:
> > > If you have problems with the successor of 9, use 1, 22, 111, 2222, ...,
> > > all these are natural numbers obtained by restricted comprehension
> > > from the set |N.
> >
> > You're really over-killing trying to talk about ZFC. You don't even
> > believe
> > ZFC.
>
> I believe in ZFC. I do not believe in the common interpretation of the axiom
> of infinity.

Since believing in the common or standard interpretation of the axiom
of infinity in ZFC is a necessary part of believing in ZFC, at least one
of WM's last two statements above is necessarily FALSE.

>
>
> > The rest of us understand it better than you do, so you don't have to
> > explain to us how to use it to construct stuff. That's what WE do for YOU.
>
> I have done it first. But fine if you understand that it is correct.

What we understand is that WM does not understand, and his repeaed
nonsense claims reeatedly prove he does not understand!

> > > There is a complete triangle,
> > > according to set theory.

What axioms of ZFC say THAT WM'S alleged limit set is even a triangle,
much less a complete one?

> > > It cannot have infinitely many symbols in any horizontal line
> >
> > OBVIOUSLY -- every horizontal line has n numbers so of course it's
> > finite.
> >
> > > and in any vertical line.
> >
> > Wrong. ALL the "vertical" lines are infinitely long.

Where in WM's mystical mythical triangular sequence does one find any
vertical line that is not matched exactly in length by a horizontal line?

[Virgil]

In article <875713d5-dbc9-4313...@googlegroups.com>,

WM <wolfgang.m...@hs-augsburg.de> wrote:

> Am Dienstag, 7. Juli 2015 15:21:08 UTC+2 schrieb George Greene:
> > On Tuesday, July 7, 2015 at 9:01:56 AM UTC-4, WM wrote:
> >
> > > Not in any argument identifying a real number with an infinbite digit
> > > sequence.
> > > Further the walker would never know whether
> > > it was finite or not until he had gone the complete way.
> >
> > It DOESN'T MATTER what THE WALKER would or wouldn't know. WE are NOT the
> > walker and WE know.
>
> You believe.

We KNOW:
That every positive real is representable uniquely by a strictly
increasing sequence of ever longer decimals,each being a finite initial
segment of all it successors and of its unique infinite limit sequence.

We KNOW:
That WM, even within his own wild weird wacky worthless world of
WMytheology, has no acceptable definition for "potentially infinite" to
match up with the proper and usable definition of a set being infinite
if it allows an injection of itself to a proper subset of itself!

[Virgil]

In article <59d6f9dd-9c9a-41e9...@googlegroups.com>,

WM <wolfgang.m...@hs-augsburg.de> wrote:

> Am Dienstag, 7. Juli 2015 15:18:27 UTC+2 schrieb George Greene:
> > On Tuesday, July 7, 2015 at 8:59:16 AM UTC-4, WM wrote:
> > > > I can't discern your alleged pattern.
> > >
> > > Sorry, then my proof is not digestible for you.
> >
> > No, you are either being unclear, or failing to follow your own intended
> > pattern properly. The precise nature of the pattern is not really relevant
> > to the "proof" in any case -- lots of different patterns would serve the
> > purpose
> > equally well.
>
> Of course. But the existence of many proofs does not undermine mine.

So far we have yet to see any VALID proofs by WM.

> >
>
> >
> > > There is a complete triangle, according to set theory.

While there may be a limit set for a sequence of trianlges, nothing is
set theory requires that limit also to be a triangle, unless WM can
identify it as having three vertices and three lines as its sides
connecting those vertices.

Note that the limit of a sequence of rational numbers need not be, and
often is not, a rational number.


!

>
> The triangle has horizontal and vertical lines.

Unless WM can prove that HIS limit set forms a triangle, by showing
that, like each member of the sequence of which it is the alleged limit,
it has three vertices pairwise connecting three boundary sides, or three
such sides pairwise intersecting in three vertices, there is no reason
to suppose his alleged limit set is anything at all like a triangle.

[Julio Di Egidio]

On Tuesday, July 7, 2015 at 10:15:25 PM UTC+1, Virgil wrote:

> there is no reason
> to suppose his alleged limit set is anything at all like a triangle.

Except plain logic: if there is limit, a that may be not finite but certainly is a triangle, because what we are talking about is a limit-*triangle*. IOW, a *sequence of triangles* is a sequence of *triangles*. Indeed you surreptitiously slip sets into the picture, but, while you can use sets to answer, what you use does not change the nature of the *question*.

Julio

[Virgil]

In article <57720dc4-130b-4e2f...@googlegroups.com>,

Julio Di Egidio <ju...@diegidio.name> wrote:

> On Tuesday, July 7, 2015 at 10:15:25 PM UTC+1, Virgil wrote:
>
> > there is no reason
> > to suppose his alleged limit set is anything at all like a triangle.
>
> Except plain logic: if there is limit, a that may be not finite but certainly
> is a triangle, because what we are talking about is a limit-*triangle*. IOW,
> a *sequence of triangles* is a sequence of *triangles*.

But there is no way that any reasonable meaning for a limit of WM's
sequence can be a figure having three vertices connected by three
vertices which is what a triangle would be if there were anything like
such a limit triangle.

> Indeed you
> surreptitiously slip sets into the picture

Since WM is the one claiming something about sets himself, I am not the
one introducing them!

[Dan Christensen]

On Friday, July 3, 2015 at 9:07:46 AM UTC-4, WM wrote:

> On Friday, 3 July 2015 11:37:46 UTC+2, George Greene wrote:
> > On Thursday, July 2, 2015 at 2:07:33 PM UTC-4, Julio Di Egidio wrote:
> > > Yes, N is a *potentially infinite* set, of course.

> >
> > N is an ACTUALLY infinite set.
>
> That is your insane belief - nothing else.
>

Do you claim that the set of natural numbers is only a potential infinity? If so, let us see your formal definition of a potential infinity. Fill in the blank: A set is a potential infinity if and only if _______________________.

Put up or shut up, troll.

Dan

Download my DC Proof 2.0 software at http://www.dcproof.com
Visit my Math Blog at http://www.dcproof.wordpress.com

**********

Really stupid quotes from WM here at sci.math:

"In my system, two different numbers can have the same value."
-- Wolfgang Mueckenheim (WM), sci.math, 2014/10/16

"1+2 and 2+1 are different numbers."
-- Wolfgang Mueckenheim (WM), sci.math, 2014/10/20

"Axioms are rubbish!"
-- Wolfgang Mueckenheim (WM), sci.math, 2014/11/19

[WM]

Am Mittwoch, 8. Juli 2015 07:39:19 UTC+2 schrieb Dan Christensen:
> On Friday, July 3, 2015 at 9:07:46 AM UTC-4, WM wrote:
> > On Friday, 3 July 2015 11:37:46 UTC+2, George Greene wrote:
> > > On Thursday, July 2, 2015 at 2:07:33 PM UTC-4, Julio Di Egidio wrote:
> > > > Yes, N is a *potentially infinite* set, of course.
> > >
> > > N is an ACTUALLY infinite set.
> >
> > That is your insane belief - nothing else.
> >
>
> Do you claim that the set of natural numbers is only a potential infinity?

Since there is no other infinity (remember I have contradicted it) and since it is not finite, the claim stands.

Regards, WM

[WM]

Am Mittwoch, 8. Juli 2015 06:27:43 UTC+2 schrieb Virgil:
> In article <57720dc4-130b-4e2f...@googlegroups.com>,
> Julio Di Egidio <ju...@diegidio.name> wrote:
>
> > On Tuesday, July 7, 2015 at 10:15:25 PM UTC+1, Virgil wrote:
> >
> > > there is no reason
> > > to suppose his alleged limit set is anything at all like a triangle.
> >
> > Except plain logic: if there is limit, a that may be not finite but certainly
> > is a triangle, because what we are talking about is a limit-*triangle*. IOW,
> > a *sequence of triangles* is a sequence of *triangles*.
>
> But there is no way that any reasonable meaning for a limit of WM's
> sequence can be a figure having three vertices connected by three
> vertices which is what a triangle would be if there were anything like
> such a limit triangle.

Set theory claims that there are aleph_0 terms of the sequence 1, 22, 111, 2222, ...

Whether you visualize it in one line or in a triangle like

1
22
111
2222
...

or in another geometric configuration does not change the facts.

Regards, WM

[Dan Christensen]

So, you do not distinguish a potential infinity from any other sort of infinity. Do you agree that a set S is infinite if there exists an injection f: S --> S that is not a surjection?

[Julio Di Egidio]

On Wednesday, July 8, 2015 at 5:27:43 AM UTC+1, Virgil wrote:
> In article <57720dc4-130b-4e2f...@googlegroups.com>,

> Julio Di Egidio <j***@diegidio.name> wrote:
> > On Tuesday, July 7, 2015 at 10:15:25 PM UTC+1, Virgil wrote:
> >
> > > there is no reason
> > > to suppose his alleged limit set is anything at all like a triangle.
> >
> > Except plain logic: if there is limit, a that may be not finite but certainly
> > is a triangle, because what we are talking about is a limit-*triangle*. IOW,
> > a *sequence of triangles* is a sequence of *triangles*.
>
> But there is no way that any reasonable meaning for a limit of WM's

That is upside down: you should say "formalisation", not "meaning".

> sequence can be a figure having three vertices connected by three
> vertices which is what a triangle would be if there were anything like
> such a limit triangle.

*You*, i.e. standard set theory, have no way: and I am not surprised.

Julio

[Virgil]

In article <54682681-ff3d-445f...@googlegroups.com>,

Considering WM's trach record here, the fact that WM has contradicted
the idea of proper infiniteness is rather evidence of its propriety,

[Virgil]

In article <3fe833f6-f343-4b74...@googlegroups.com>,

WM <wolfgang.m...@hs-augsburg.de> wrote:

> Am Mittwoch, 8. Juli 2015 06:27:43 UTC+2 schrieb Virgil:
> > In article <57720dc4-130b-4e2f...@googlegroups.com>,
> > Julio Di Egidio <ju...@diegidio.name> wrote:
> >
> > > On Tuesday, July 7, 2015 at 10:15:25 PM UTC+1, Virgil wrote:
> > >
> > > > there is no reason
> > > > to suppose his alleged limit set is anything at all like a triangle.
> > >
> > > Except plain logic: if there is limit, a that may be not finite but
> > > certainly
> > > is a triangle, because what we are talking about is a limit-*triangle*.
> > > IOW,
> > > a *sequence of triangles* is a sequence of *triangles*.
> >
> > But there is no way that any reasonable meaning for a limit of WM's
> > sequence can be a figure having three vertices connected by three
> > vertices which is what a triangle would be if there were anything like
> > such a limit triangle.
>
> Set theory claims that there are aleph_0 terms of the sequence 1, 22, 111,
> 2222, ...

Set theory only claims that there is one such term for each member of |N

>
> Whether you visualize it in one line or in a triangle like
>
> 1
> 22
> 111
> 2222
> ...
>
> or in another geometric configuration does not change the facts.

When WM visualizes it all on one line does WM still visualize its limit
as a triangle! If not then at least one of WM;s alleged facts is false!

[Virgil]

In article <22915a00-a470-4b07...@googlegroups.com>,

Julio Di Egidio <ju...@diegidio.name> wrote:

> On Wednesday, July 8, 2015 at 5:27:43 AM UTC+1, Virgil wrote:
> > In article <57720dc4-130b-4e2f...@googlegroups.com>,
> > Julio Di Egidio <j***@diegidio.name> wrote:
> > > On Tuesday, July 7, 2015 at 10:15:25 PM UTC+1, Virgil wrote:
> > >
> > > > there is no reason
> > > > to suppose his alleged limit set is anything at all like a triangle.
> > >
> > > Except plain logic: if there is limit, a that may be not finite but
> > > certainly
> > > is a triangle, because what we are talking about is a limit-*triangle*.
> > > IOW,
> > > a *sequence of triangles* is a sequence of *triangles*.
> >
> > But there is no way that any reasonable meaning for a limit of WM's
>
> That is upside down: you should say "formalisation", not "meaning".
>
> > sequence can be a figure having three vertices connected by three

> > edges which is what a triangle would be if there were anything like

> > such a limit triangle.
>
> *You*, i.e. standard set theory, have no way: and I am not surprised.
>
> Julio

Consider the one dimensional analog, the sequence of FISONs (Finite
Initial Sequences Of Naturals), each having both a first and a last (but
not always different) member, but the limit |N has no last member, so a
sequence of segments (having two endpoints) has a limit which is no
longer a segment.

Thus in two dimensions the limit of a sequence of segmentws of ever
longer lengths need not be a segment.
Thus the limit of WM's triangles, which are necessarily bounded by three
segments and have three vertices, need not be a triangle.

[Julio Di Egidio]

On Wednesday, July 8, 2015 at 5:05:44 PM UTC+1, Virgil wrote:
> In article <22915a00-a470-4b07...@googlegroups.com>,

> Julio Di Egidio <j***@diegidio.name> wrote:
> > On Wednesday, July 8, 2015 at 5:27:43 AM UTC+1, Virgil wrote:

<snip>

> > > sequence can be a figure having three vertices connected by three
> > > edges which is what a triangle would be if there were anything like
> > > such a limit triangle.
> >
> > *You*, i.e. standard set theory, have no way: and I am not surprised.
>

> Consider the one dimensional analog, the sequence of FISONs (Finite
> Initial Sequences Of Naturals), each having both a first and a last (but
> not always different) member, but the limit |N has no last member, so a
> sequence of segments (having two endpoints) has a limit which is no
> longer a segment.

That logic is indeed broken: the limit is simply not N! Conversely, the limit-segment is still a segment, and if your mathematics cannot reach it, that is an issue with your mathematics.

(Triangles do not add anything significant to this discussion, I agree on that.)

Julio

[Virgil]

In article <5715653f-844b-4eeb...@googlegroups.com>,

Julio Di Egidio <ju...@diegidio.name> wrote:

> On Wednesday, July 8, 2015 at 5:05:44 PM UTC+1, Virgil wrote:
> > In article <22915a00-a470-4b07...@googlegroups.com>,
> > Julio Di Egidio <j***@diegidio.name> wrote:
> > > On Wednesday, July 8, 2015 at 5:27:43 AM UTC+1, Virgil wrote:
> <snip>
> > > > sequence can be a figure having three vertices connected by three
> > > > edges which is what a triangle would be if there were anything like
> > > > such a limit triangle.
> > >
> > > *You*, i.e. standard set theory, have no way: and I am not surprised.
> >
> > Consider the one dimensional analog, the sequence of FISONs (Finite
> > Initial Sequences Of Naturals), each having both a first and a last (but
> > not always different) member, but the limit |N has no last member, so a
> > sequence of segments (having two endpoints) has a limit which is no
> > longer a segment.
>
> That logic is indeed broken: the limit is simply not N!

When one has any sequence of sets, each of which is a proper subset of
its all its successors, in any sane set theory their limit set clearly
exists and is their union set, which in this case is |N! ANyone who
disputes that is the one with fractured logic!

> Conversely, the
> limit-segment is still a segment

A segment, by definition, ALWAYS has 2 endpoints.
The union of all FISONs can have only one endpoint.
At least in the true mathematics free of such corruptions as WM'S
WORTHLESS WORLD OF WMYTHEOLOGY.

> and if your mathematics cannot reach it,
> that is an issue with your mathematics.

NO uncorrupted mathematics can find two endpoints for the union of all
FISONs (Finite Initial Sets Of Naturals)

>
> (Triangles do not add anything significant to this discussion, I agree on
> that.)
>
> Julio

[WM]

Am Mittwoch, 8. Juli 2015 22:47:05 UTC+2 schrieb Virgil:


>
> A segment, by definition, ALWAYS has 2 endpoints.
> The union of all FISONs can have only one endpoint.

The triangle does not contain the union of all FISONs but only all FISONs. There are aleph_0 FISONs.

Regards, WM

[WM]

Am Mittwoch, 8. Juli 2015 18:05:44 UTC+2 schrieb Virgil:


> Consider the one dimensional analog, the sequence of FISONs (Finite
> Initial Sequences Of Naturals), each having both a first and a last (but
> not always different) member, but the limit |N has no last member, so a
> sequence of segments (having two endpoints) has a limit which is no
> longer a segment.

There are aleph_0 finite sequences, according to set theory.

>
> Thus in two dimensions the limit of a sequence of segmentws of ever
> longer lengths need not be a segment.

No, but the number of sequences is a fixed quantity larger than any finite number.

> Thus the limit of WM's triangles, which are necessarily bounded by three
> segments and have three vertices, need not be a triangle.

Whatever it may be, according to set theory it has aleph_0 lines and no line that has aleph_0 elements.

Regards, WM

[WM]

Am Mittwoch, 8. Juli 2015 15:50:27 UTC+2 schrieb Dan Christensen:
> On Wednesday, July 8, 2015 at 6:15:57 AM UTC-4, WM wrote:
> > Am Mittwoch, 8. Juli 2015 07:39:19 UTC+2 schrieb Dan Christensen:
> > > On Friday, July 3, 2015 at 9:07:46 AM UTC-4, WM wrote:
> > > > On Friday, 3 July 2015 11:37:46 UTC+2, George Greene wrote:
> > > > > On Thursday, July 2, 2015 at 2:07:33 PM UTC-4, Julio Di Egidio wrote:
> > > > > > Yes, N is a *potentially infinite* set, of course.
> > > > >
> > > > > N is an ACTUALLY infinite set.
> > > >
> > > > That is your insane belief - nothing else.
> > > >
> > >
> > > Do you claim that the set of natural numbers is only a potential infinity?
> >
> > Since there is no other infinity (remember I have contradicted it) and since it is not finite, the claim stands.
> >
>
> So, you do not distinguish a potential infinity from any other sort of infinity.

Since there is no other sort.

Regards, WM

[Dan Christensen]

What WM snipped:

Do you agree that a set S is infinite iff there exists an injection f: S --> S that is not a surjection?

> Since there is no other sort.
>

Now, please answer the question. Or alternatively...

Do you agree that a set S is FINITE iff for all injections f: S --> S, f is also a surjection?

Or maybe you have your own favourite definition of finite and infinite? In that case, please state them here giving the necessary and sufficient conditions for a set being finite (or infinite).

I hope you are not going to say that definitions, like axioms, are also "rubbish."

Dan

Download my DC Proof 2.0 software at http://www.dcproof.com
Visit my Math Blog at http://www.dcproof.wordpress.com

[Virgil]

In article <4847bc7f-1f17-4661...@googlegroups.com>,

WM <wolfgang.m...@hs-augsburg.de> wrote:

> Am Mittwoch, 8. Juli 2015 18:05:44 UTC+2 schrieb Virgil:
>
>
> > Consider the one dimensional analog, the sequence of FISONs (Finite
> > Initial Sequences Of Naturals), each having both a first and a last (but
> > not always different) member, but the limit |N has no last member, so a
> > sequence of segments (having two endpoints) has a limit which is no
> > longer a segment.
>
> There are aleph_0 finite sequences, according to set theory.

According to which set theory?
Accordint to the best of my revcollection, neither ZFC nor NBG has any
axiom or required definition even memntioning any aleph_0.

> >
> > Thus in two dimensions the limit of a sequence of segmentws of ever
> > longer lengths need not be a segment.
>
> No, but the number of sequences is a fixed quantity larger than any finite
> number.

My point exactly!

>
> > Thus the limit of WM's triangles, which are necessarily bounded by three
> > segments and have three vertices, need not be a triangle.
>
> Whatever it may be, according to set theory

Whatever it many be according to any actual set theory,
it is NOT what WM claimed it must be according to his set theory!

[Virgil]

In article <6490f49c-7c2d-48d1...@googlegroups.com>,

WM <wolfgang.m...@hs-augsburg.de> wrote:

> Am Mittwoch, 8. Juli 2015 22:47:05 UTC+2 schrieb Virgil:
>
>
> >
> > A segment, by definition, ALWAYS has 2 endpoints.
> > The union of all FISONs can have only one endpoint.
>
> The triangle does not contain the union of all FISONs

A triangle, at least everywhere outside of WM's worthless world of
WMytheology, must have three vertex points pairwise connected by three
line segments.

Until WM can show us that the limits of his sequences all have three
vertex points pairwise connected by three line segments, none of those
limits are triangles, despite his false claims!

[Virgil]

In article <0788bf27-48e8-4784...@googlegroups.com>,

And there can be no such thing a merely potential infiniteness for sets
until it can be and has been properly defined, which as yet it has not
been and apparently cannot be!

A set, S, is, by standard definition, finite if and only if every
injection from S to S is a surjection, and a S is thus non-finite or
in-finite if there is some injection from S to S which is not a
surjection.

These are standard definitions accepted everywhere outside of WM's
worthless world of WMytheology.

When WM maunders on about potential infiniteness, he is talking about
wwhat has not been and apparently cannot be properly defines at all, and
is thus nonsense!

Proper mathematics does not speculate on the properties of the
undefineable!

[George Greene]

On Tuesday, July 7, 2015 at 3:41:48 PM UTC-4, WM wrote:
> I believe in ZFC. I do not believe in the common
> interpretation of the axiom of infinity.

There still has to be SOME *set* that is closed under successor if you believe that axiom AT ALL. Nobody is even asking you to "interpret" it. Whatever set YOU pick will BE ACTUALLY infinite AS WELL. If you have an uncommon interpretation, BY ALL MEANS USE it!! It will NOT impact the fact that N IS CONSTRUCTIBLE FROM the axiom of infinity, BY PROOFS in ZFC, so IF you believe ZFC....

[George Greene]

On Tuesday, July 7, 2015 at 3:41:48 PM UTC-4, WM wrote:

> Fine that you understand and share my view.

YOU do not understand your view, which makes it hard for anyone to share it.

> Now return to my version.

No, we are NOT going to return to your version, because your version is not
coherently stated. Mine is.

> I did it right.

No, you didn't.

> There is no final edge existing.

Right.

> But everything that is existing, is finite --- nowhere is aleph_0.

Aleph_0 IS THE LEFT SLANT *SIDE* AND THE RIGHT SLANT *SIDE*, BOTH of which
exist and NEITHER of which ENDS, and BOTH of which are actually infinite.

You can't say that something doesn't exist just because it's infinite.
By your argument, THE WHOLE TRIANGLE ITSELF would fail to exist (because it's infinite), even though the ROWS DO exist (because they're finite).
You don't get to say that the members exist but the collection doesn't!

[George Greene]

On Wednesday, July 8, 2015 at 4:47:05 PM UTC-4, Virgil wrote:
> A segment, by definition, ALWAYS has 2 endpoints.
> The union of all FISONs can have only one endpoint.

It's a mistake to even ADMIT WM's FISONs.
In Z, we start WITH *ZERO*, NOT 1.
Every natural is represented by the set of all (and only) SMALLER naturals,
INCLUDING zero.
WM likes to start with 1 because then can claim he is sharing ZFC's opinion
that the set representing n has n elements, but then go on to claim that n
itself is the biggest element of the set. Thus N can't be a set because it
doesn't share this property with all these things of which it is the alleged limit. That last part is a complete non-sequitur but WM treats it as though it were self-evident, even though he can't axiomatize it.

[Virgil]

In article <7d2ca3dc-de5a-45c6...@googlegroups.com>,

George Greene <gre...@email.unc.edu> wrote:

> On Wednesday, July 8, 2015 at 4:47:05 PM UTC-4, Virgil wrote:
> > A segment, by definition, ALWAYS has 2 endpoints.
> > The union of all FISONs can have only one endpoint.
>
> It's a mistake to even ADMIT WM's FISONs.
> In Z, we start WITH *ZERO*, NOT 1.
> Every natural is represented by the set of all (and only) SMALLER naturals,
> INCLUDING zero.

If one starts, as one should , with the Peano axioms, the unique
non-successor of a Peano defined inductive set can serve equally well as
either a zero or a one depending on how one defines the arithmetical
structure to be imposed on the basic inductive sets produced by the
Peano's axioms. It is easily done either way!

Until the advent of electronic computers and computer science in the mid
twentieth century, most mathematicians presumed that "the set of natural
numbers" was the same as "the set of counting numbers" and started with
1 rather than 0, and a significant proportion of them still do.

> WM likes to start with 1 because then can claim he is sharing ZFC's opinion
> that the set representing n has n elements, but then go on to claim that n
> itself is the biggest element of the set. Thus N can't be a set because it
> doesn't share this property with all these things of which it is the alleged
> limit. That last part is a complete non-sequitur but WM treats it as though
> it were self-evident, even though he can't axiomatize it.

[Virgil]

In article <1d7526ad-11c7-4a3c...@googlegroups.com>,

WM only believes, unjustifiably, in his own infallibility!

[WM]

Am Donnerstag, 9. Juli 2015 19:29:43 UTC+2 schrieb George Greene:


> You can't say that something doesn't exist just because it's infinite.
> By your argument, THE WHOLE TRIANGLE ITSELF would fail to exist (because it's infinite), even though the ROWS DO exist (because they're finite).
> You don't get to say that the members exist but the collection doesn't!

Wrong. The whole triangle is infinite. There is no fixed quantity aleph_0 describing the number of lines. That is only your wrong belief. It is easy to get a contradiction from the triangle I posted.

Regards, WM

[WM]

Am Donnerstag, 9. Juli 2015 19:26:55 UTC+2 schrieb George Greene:
> On Tuesday, July 7, 2015 at 3:41:48 PM UTC-4, WM wrote:
> > I believe in ZFC. I do not believe in the common
> > interpretation of the axiom of infinity.
>

> Whatever set YOU pick will BE ACTUALLY infinite AS WELL.

Prove it.

> If you have an uncommon interpretation, BY ALL MEANS USE it!! It will NOT impact the fact that N IS CONSTRUCTIBLE FROM the axiom of infinity, BY PROOFS in ZFC, so IF you believe ZFC....

|N is constructible but |N is not finished. There is no fixed quantity aleph_0 larger than every natural number and giving the number of numbers, to be precise.

Regards, WM

[Virgil]

In article <b80a79f4-50ad-40af...@googlegroups.com>,

WM <wolfgang.m...@hs-augsburg.de> wrote:

> Am Donnerstag, 9. Juli 2015 19:26:55 UTC+2 schrieb George Greene:
> > On Tuesday, July 7, 2015 at 3:41:48 PM UTC-4, WM wrote:
> > > I believe in ZFC. I do not believe in the common
> > > interpretation of the axiom of infinity.
> >
> > Whatever set YOU pick will BE ACTUALLY infinite AS WELL.
>
> Prove it.
>
> > If you have an uncommon interpretation, BY ALL MEANS USE it!! It will
> > NOT impact the fact that N IS CONSTRUCTIBLE FROM the axiom of infinity, BY
> > PROOFS in ZFC, so IF you believe ZFC....
>
> |N is constructible but |N is not finished.

|N being constructible means that it can be constructed,
and when it has been constructed there is no part of ir left unconstruced

> There is no fixed quantity aleph_0 larger than every natural number
> and giving the number of numbers

Maybe not in WM's worthless world of WMytheology, but no one but WM
lives there!

[Virgil]

In article <b80a79f4-50ad-40af...@googlegroups.com>,
WM <wolfgang.m...@hs-augsburg.de> wrote:

> Am Donnerstag, 9. Juli 2015 19:26:55 UTC+2 schrieb George Greene:
> > On Tuesday, July 7, 2015 at 3:41:48 PM UTC-4, WM wrote:
> > > I believe in ZFC. I do not believe in the common
> > > interpretation of the axiom of infinity.
> >
> > Whatever set YOU pick will BE ACTUALLY infinite AS WELL.
>
> Prove it.

ANY non-empty set in in ZFC which is closed under s -> s \/ {s}, and
there are such sets, will satisfy the definition of not being finite!
And that is all that is needed!

>
> > If you have an uncommon interpretation, BY ALL MEANS USE it!! It will
> > NOT impact the fact that N IS CONSTRUCTIBLE FROM the axiom of infinity, BY
> > PROOFS in ZFC, so IF you believe ZFC....
>
> |N is constructible but |N is not finished.

If |N is a set at all, it is a finished set, as no set theory outside of
WM's worthless world of WMytheology, allows unfinished objects to be
sets.

> There is no fixed quantity
> aleph_0 larger than every natural number and giving the number of numbers,

There is fixed quantity, often denoted by aleph_0, larger than every
natural number and giving the number of natural numbers, though not the
number of numbers.

[Virgil]

In article <f8d9948e-3f18-4b37...@googlegroups.com>,

WM <wolfgang.m...@hs-augsburg.de> wrote:

> Am Donnerstag, 9. Juli 2015 19:29:43 UTC+2 schrieb George Greene:
>
>
> > You can't say that something doesn't exist just because it's infinite.
> > By your argument, THE WHOLE TRIANGLE ITSELF would fail to exist (because
> > it's infinite), even though the ROWS DO exist (because they're finite).
> > You don't get to say that the members exist but the collection doesn't!
>
> Wrong. The whole triangle is infinite. There is no fixed quantity aleph_0
> describing the number of lines.

Everywhere outside of WM's worthless world of WMytheology that allegedly
unnamable quantity is named aleph_0

> That is only your wrong belief. It is easy to
> get a contradiction from the triangle I posted.

Every WM argument is inherently self-contradictory!

[Julio Di Egidio]

On Wednesday, July 8, 2015 at 9:47:05 PM UTC+1, Virgil wrote:
> In article <5715653f-844b-4eeb...@googlegroups.com>,

> Julio Di Egidio <j***@diegidio.name> wrote:
> > On Wednesday, July 8, 2015 at 5:05:44 PM UTC+1, Virgil wrote:
> > > In article <22915a00-a470-4b07...@googlegroups.com>,
> > > Julio Di Egidio <j***@diegidio.name> wrote:
> > > > On Wednesday, July 8, 2015 at 5:27:43 AM UTC+1, Virgil wrote:
> > <snip>
> > > > > sequence can be a figure having three vertices connected by three
> > > > > edges which is what a triangle would be if there were anything like
> > > > > such a limit triangle.
> > > >
> > > > *You*, i.e. standard set theory, have no way: and I am not surprised.
> > >
> > > Consider the one dimensional analog, the sequence of FISONs (Finite
> > > Initial Sequences Of Naturals), each having both a first and a last (but
> > > not always different) member, but the limit |N has no last member, so a
> > > sequence of segments (having two endpoints) has a limit which is no
> > > longer a segment.
> >
> > That logic is indeed broken: the limit is simply not N!
>
> When one has any sequence of sets, each of which is a proper subset of
> its all its successors, in any sane set theory their limit set clearly
> exists and is their union set,

When the sequence in question is *suitably* defined, its limit of course exists.

> which in this case is |N!

The limit set cannot be N, because:

> > Conversely, the
> > limit-segment is still a segment
>
> A segment, by definition, ALWAYS has 2 endpoints.

Exactly, the second endpoint being the limit point. Hence the limit set *must* contain the limit point. QED.

Julio

[Julio Di Egidio]

On Thursday, July 9, 2015 at 6:29:43 PM UTC+1, George Greene wrote:
> On Tuesday, July 7, 2015 at 3:41:48 PM UTC-4, WM wrote:

<snip>

> > There is no final edge existing.
>
> Right.

It is right if you are describing standard theory and its approach, but, and as I have already tried to explain up-thread, it is already wrong by logic, the logic of the question: a triangle has three edges, period. Hence one rather concludes that standard theory is useless as an infinitary mathematics. To begin with.

Julio

[Virgil]

In article <6eaedd81-a005-4b6d...@googlegroups.com>,

The sequence of FISONs is suitably defined and has |N as its limit set

> Exactly, the second endpoint being the limit point. Hence the limit set
> *must* contain the limit point. QED.

What forces the limit of a sequence of segments to be a segment!
The sequence of segments [0,n] has limit [0,oo) which is not a segment!

[Virgil]

In article <4980af3e-c998-4cfc...@googlegroups.com>,

Julio Di Egidio <ju...@diegidio.name> wrote:

> On Thursday, July 9, 2015 at 6:29:43 PM UTC+1, George Greene wrote:
> > On Tuesday, July 7, 2015 at 3:41:48 PM UTC-4, WM wrote:
> <snip>
> > > There is no final edge existing.
> >
> > Right.
>
> It is right if you are describing standard theory and its approach, but, and
> as I have already tried to explain up-thread, it is already wrong by logic,
> the logic of the question: a triangle has three edges, period.

The sequence of real compact real intervals, each containing its upper
bound, {[0,n], n in |N}, has limit [0,oo) which is not a compact
interval and does not contain its upper bound.

Thus the limit, if any exists, of a sequence of triangles need not be a
triangle.

[Julio Di Egidio]

On Friday, July 10, 2015 at 6:13:36 PM UTC+1, Virgil wrote:
> In article <4980af3e-c998-4cfc...@googlegroups.com>,

> Julio Di Egidio <j***@diegidio.name> wrote:
> > On Thursday, July 9, 2015 at 6:29:43 PM UTC+1, George Greene wrote:
> > > On Tuesday, July 7, 2015 at 3:41:48 PM UTC-4, WM wrote:
> > <snip>
> > > > There is no final edge existing.
> > >
> > > Right.
> >
> > It is right if you are describing standard theory and its approach, but, and
> > as I have already tried to explain up-thread, it is already wrong by logic,
> > the logic of the question: a triangle has three edges, period.
>
> The sequence of real compact real intervals, each containing its upper
> bound, {[0,n], n in |N}, has limit [0,oo) which is not a compact
> interval and does not contain its upper bound.
>
> Thus the limit, if any exists, of a sequence of triangles need not be a
> triangle.

Nope, thus your math is inadequate at best, period.

Julio

[Julio Di Egidio]

On Friday, July 10, 2015 at 6:03:57 PM UTC+1, Virgil wrote:

> What forces the limit of a sequence of segments to be a segment!

For the umpteenth time: plain logic.

Julio

[Virgil]

In article <e9e6186c-1dd4-4e80...@googlegroups.com>,

Julio Di Egidio <ju...@diegidio.name> wrote:

Show me!

The limit of the increasing sequence of all FISONs, (finite initial
segments of naturals) is not a FISON.

[Virgil]

In article <b9bdd491-43ff-42e8...@googlegroups.com>,

Julio Di Egidio <ju...@diegidio.name> wrote:

> > Thus the limit, if any exists, of a sequence of triangles need not be a
> > triangle.
>
> Nope, thus your math is inadequate at best, period.

IF each triangle in a sequence of symmetric triangles has the same
midpoint but double the prior triangle's edge lengths, the limit is the
entire plane, which is not a triangle at all!

At least not outside of outre systems like WM's worthless world of
WMytheology.

[Dan Christensen]

On Thursday, July 9, 2015 at 11:20:01 AM UTC-4, Dan Christensen wrote:
> On Thursday, July 9, 2015 at 6:56:00 AM UTC-4, WM wrote:
> > Am Mittwoch, 8. Juli 2015 15:50:27 UTC+2 schrieb Dan Christensen:
> > > On Wednesday, July 8, 2015 at 6:15:57 AM UTC-4, WM wrote:
> > > > Am Mittwoch, 8. Juli 2015 07:39:19 UTC+2 schrieb Dan Christensen:
> > > > > On Friday, July 3, 2015 at 9:07:46 AM UTC-4, WM wrote:
> > > > > > On Friday, 3 July 2015 11:37:46 UTC+2, George Greene wrote:
> > > > > > > On Thursday, July 2, 2015 at 2:07:33 PM UTC-4, Julio Di Egidio wrote:
> > > > > > > > Yes, N is a *potentially infinite* set, of course.
> > > > > > >
> > > > > > > N is an ACTUALLY infinite set.
> > > > > >
> > > > > > That is your insane belief - nothing else.
> > > > > >
> > > > >
> > > > > Do you claim that the set of natural numbers is only a potential infinity?
> > > >
> > > > Since there is no other infinity (remember I have contradicted it) and since it is not finite, the claim stands.
> > > >
> > >
> > > So, you do not distinguish a potential infinity from any other sort of infinity.
> >
>
> What WM snipped:
>
> Do you agree that a set S is infinite iff there exists an injection f: S --> S that is not a surjection?
>
> > Since there is no other sort.
> >
>
> Now, please answer the question. Or alternatively...
>
> Do you agree that a set S is FINITE iff for all injections f: S --> S, f is also a surjection?
>
> Or maybe you have your own favourite definition of finite and infinite? In that case, please state them here giving the necessary and sufficient conditions for a set being finite (or infinite).
>
> I hope you are not going to say that definitions, like axioms, are also "rubbish."
>

No comment, WM??? Oh, well...

[X.Y. Newberry]

Dan Christensen wrote:
> DEFINING ACTUAL INFINITY
>
> A set S is said to be actually infinite iff there exists an injective function f: S --> S such that f is not surjective. (There are other, equivalent definitions.)
>
>
> DEFINING POTENTIAL INFINITY
>
> Is it possible to formally define the potentially infinite?
>
> A set S is said to be potentially infinite iff .....????

>
>
> Dan
>
> Download my DC Proof 2.0 software at http://www.dcproof.com
> Visit my Math Blog at http://www.dcproof.wordpress.com
>

One distinction is this. Suppose we put a ball in a vase at 1/2s,
1/2+1/4s, 1/2+1/4+1/8 ... etc. In case of actual infinity there will be
aleph_0 balls in the vase at t = 1s. In case of potential infinity the
state of the vase at t = 1s is undefined.

--
X.Y. Newberry

If Jack says ‘What I am saying at this very moment is not true’, we can
successfully and truly assert that he did not utter a truth: ‘What Jack
said is not true’. But it is hardly conceivable that Jack’s utterance is
true by virtue of its success in attributing non-truth to itself.

Haim Gaifman

---
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[Dan Christensen]

On Sunday, August 2, 2015 at 10:50:57 PM UTC-4, Newberry wrote:
> Dan Christensen wrote:
> > DEFINING ACTUAL INFINITY
> >
> > A set S is said to be actually infinite iff there exists an injective function f: S --> S such that f is not surjective. (There are other, equivalent definitions.)
> >
> >
> > DEFINING POTENTIAL INFINITY
> >
> > Is it possible to formally define the potentially infinite?
> >
> > A set S is said to be potentially infinite iff .....????
> >
> >
> > Dan
> >
> > Download my DC Proof 2.0 software at http://www.dcproof.com
> > Visit my Math Blog at http://www.dcproof.wordpress.com
> >
>
> One distinction is this. Suppose we put a ball in a vase at 1/2s,
> 1/2+1/4s, 1/2+1/4+1/8 ... etc. In case of actual infinity there will be
> aleph_0 balls in the vase at t = 1s. In case of potential infinity the
> state of the vase at t = 1s is undefined.
>

Can you use this idea to formally define the necessary and sufficient conditions for a set being "potentially infinite" as required above?

Dan

[Virgil]

In article <mpmkrf$u4p$1...@dont-email.me>,

"X.Y. Newberry" <newbe...@gmail.com> wrote:

> Dan Christensen wrote:
> > DEFINING ACTUAL INFINITY
> >
> > A set S is said to be actually infinite iff there exists an injective
> > function f: S --> S such that f is not surjective. (There are other,
> > equivalent definitions.)
> >
> >
> > DEFINING POTENTIAL INFINITY
> >
> > Is it possible to formally define the potentially infinite?
> >
> > A set S is said to be potentially infinite iff .....????
> >
> >
> > Dan
> >
> > Download my DC Proof 2.0 software at http://www.dcproof.com
> > Visit my Math Blog at http://www.dcproof.wordpress.com
> >
>
> One distinction is this. Suppose we put a ball in a vase at 1/2s,
> 1/2+1/4s, 1/2+1/4+1/8 ... etc. In case of actual infinity there will be
> aleph_0 balls in the vase at t = 1s. In case of potential infinity the
> state of the vase at t = 1s is undefined.

Is the state at t = 1s merely undefined or is it totally undefineable in
a world limited to only potential infiniteness?

[X.Y. Newberry]

It is undefined with respect to the stated conditions. And there is no
state such that anything has aleph_0 or more of anything. In particular
there is no vase with aleph_0 balls at t = 1s or at any time for that
matter.

Not sure if I understand your question. We can define that at t = 1s a
vase has 5 balls.

[X.Y. Newberry]

In ZFC all infinite sets are actually infinite, but ZFC does not have
the capability to define "potential infinity." You can define a
constructive set theory, in which implicitly all infinite sets are
potentially infinite. I do not know if you can define a super-theory
that encompasses both, and can define potential infinity. What would it
be useful for?

I am giving this example because I think it is tangible enough.

> Dan

[Virgil]

In article <mpmrcl$diq$1...@dont-email.me>,

Then there canot be any sequnces which converge to a limit value that is
not one of the members of the sequence.

And an monotone increasing bounded sequence either need not or cannot
have any limit value?

[WM]

Am Montag, 3. August 2015 06:48:20 UTC+2 schrieb Newberry:


> In ZFC all infinite sets are actually infinite,

No, this is simply a wrong interpretation of the axiom of infinity, based upon Cantor's theological ideas.

> but ZFC does not have
> the capability to define "potential infinity."

Zermelo's axiom of infinity defines a potentially infinite set.

Regards, WM

[Virgil]

In article <4175fcaf-3579-44a4...@googlegroups.com>,

WM <wolfgang.m...@hs-augsburg.de> wrote:

> Am Montag, 3. August 2015 06:48:20 UTC+2 schrieb Newberry:
>
>
> > In ZFC all infinite sets are actually infinite,
>
> No, this is simply a wrong interpretation of the axiom of infinity, based
> upon Cantor's theological ideas.

The axiom of infinity, at least all versions of it outside of WM's
witless worthless wacky world of WMytheology, require the existence of a
set satisfying the Peano properties, which makes it actually infinite as
a set which can inject to a proper subset of itself!

>
> > but ZFC does not have
> > the capability to define "potential infinity."
>
> Zermelo's axiom of infinity defines a potentially infinite set.

A set like that which injects to a proper subset of itself is always
actually infinite, at least everywhere outside of WM's witless worthless
wacky world of WMytheology. And what goes on inside of that witless
worthless wacky world of WMytheology is not proper mathemtqics at all!

[Dan Christensen]

I can think of no use for it, but the notion seems to preoccupy a number of cranks and trolls here. I would be nice to finally shut them down though. They go on an on about it, but can offer no formal definitions -- only examples of roughly what they think it might entail.

[Dan Christensen]

You call it a potentially infinite set, but you still cannot define what that means. Put up or shut up, WM!

[FredJeffries]

On Monday, August 3, 2015 at 12:56:34 AM UTC-7, WM wrote:
>
> Zermelo's axiom of infinity defines a potentially infinite set.

Zermelo's axiom of infinity does not define ANYTHING