A clarification for JG and WM to once and for all explain them how cardinals work

[shio...@googlemail.com]

This is a thread just for you two in which we will, if you dare to reply at least, go over this topic step by step.

If you find some issue in a step, please ONLY reply to that issue so we save space and can go over it thouroughly.

0)

As a preword: Every possible approach to mathematics (and there is more than one) needs axioms, so statements which we simply see as true.
If we do not at least assume something exists, like the empty set, or the natural numbers, or anything like that, we cannot do math simply because we will never be able to prove that anything of the things we talk about exists.

The only thing that is important about the axioms we have to choose thus is that they a) do not contradict each other and b) are useful.
For example, the axiom 'the empty set exists' alone would be completely useless, because that would be the only statement ever provable in that math.

In ZFC, we have the axiom of infinity, stating that there is a set that contains the empty set and for every element x of it, also x U {x} is in that set (thus instantly making that set infinite, since that means there are infinitely many unique elements in it).
For the axiom of infinity, see here: https://en.wikipedia.org/wiki/Axiom_of_infinity

Now, unless you can show me how the axiom of infinity contradicts any of the other set axioms, you'd have to accept that fact simply because it is an axiom.
You can say you don't like this axiom, but you cannot say it is logically wrong or whatever, because as an axiom, it is one of the things you simply have to choose as true to do math.
Even if we chose silly axioms like the axiomset that only states the empty set exist i mentioned before, as long as it is logically coherent, everything is fine.
So unless you can show me how that axiom directly contradicts anything else, let's go on.

1)

The first step is to show you that in fact the natural numbers are an infinite set.
That should usually go without saying, but i feel like you would debate that fact by flat out stating that no infinite set exists.
The natural numbers can be defined in the following way:
https://en.wikipedia.org/wiki/Set-theoretic_definition_of_natural_numbers

As you can see, the natural numbers have to exist due to the axiom of infinity (because they are the smallest set that satisfies the definition of the set constructed in the axiom of infinity, and thus as a subset of that set has to exist. This follows from the axiom of specification https://en.wikipedia.org/wiki/Axiom_schema_of_specification . This axiom states that any subset of a set definable by a logical property (in FO) is a set. And the natural numbers are definable by such an FO property as being the smallest of the sets proclaimed by the axiom of infinity).

Now, all we did until now is using definitions and axioms.
So we only used things that are definitely right in ZFC (and once again, unless you can show me a SETTHEORETICAL contradiction onf ZFC until this point, you cannot deny their truth) and gave names to some things we know exist.

By that, we know that the natural numbers exist and we know that they are an infinite set (since we defined it as the smallest set satisfying the definition of the axiom of infinty and that set has to be infinite since it has to contain infinitely many unique elements per definition).

2)

Now, we define cardinals. We have the natural numbers at our disposal now and define the cardinality of a finite set as the number of its elements.
That is very unproblematic.
However, we know that the naturals are not finite.
There cannot be any finite cardinality we can give them since it would contain the corresponding number and the sequence from 1 to the successor of that number would be a set with a bigger cardinality than the number we wanted to assign the natural numbers.

By now, we simply know one thing: The cardinality of the naturals can NOT be finite.
However, we wish to extend our cardinality definition to infinite sets.
And that is not hard. We can simply call the cardinality of the naturals aleph_0.
Whether this makes sense or not is a secondary question, because we may.
There is nothing stopping us. Naming a thing is NEVER wrong in math.
You could name the set that contains all others sets biggle boggle, and it would not be a problem.
Such a set does not exist in ZFC, but that simply means that biggle boggle does not exist and nothing is wrong or lost by giving that nonexistant set a silly name.

So, we call our cardinality of the naturals aleph_0.
Aleph_0 can not be any natural number because we already verified that the naturals are infinitely big and thus cannot have a finite cardinality.

However, we wish to compare different infinite cardinalities with each other.
Thus we give cardinality a braoder definition: Two sets have the same cardinality if and only if there is a bijection between them.
That holds true for finite sets in any case, and here we can, once again, see, that a finite set cannot have the same cardinality as the naturals do because there can never be a surjective map from any finite map into the naturals.

3) With our definition, we just have to check whether two sets are in bijection with each other.
For the rationals and naturals, that is true.
We can create a list of all rational numbers with cantor's diagonal scheme (we go 1, 1/2, 1/3 ,2/3, 1/4, 2/4 and so on going through all proper fractions) and thus we know that the cardinality of the rationals must be aleph_0, thus the same as the one of the naturals, BECAUSE THEY ARE IN BIJECTION.

Now, there are two good ways to show that not all infinite sets are in bijection with each other: The power set of the naturals and the reals.

I know that you take issue with real numbers and them being infinitely long, so i decided to give you the proof for power sets:
https://en.wikipedia.org/wiki/Cantor%27s_theorem

I can explain it with some more explanation if you wish to, but this proof is logically sound and does not have anything to do with real numbers.
However, with it we can show that in fact the cardinality of the power set of the natural numbers is strictly bigger then the one of the naturals.

Thus we see that in fact it makes sense to define cardinality with bijections.
We can clearly show that this seperates finite sets from infinite ones and it does seperate even infinite cardinalities from each other in a way that is useful for quite some logical applications.



_______________

So, that is it. You can ask for further explanations or try to find contradiction in that if you want to, but there is really nothing done here except using axioms and definitions.
The only proof is that a power set is bigger than the corresponding set, and you would also have to find a mistake in that proof before you can claim anything here is false.

[Julio Di Egidio]

On Friday, December 30, 2016 at 5:46:52 AM UTC+1, shio...@googlemail.com wrote:

> the natural numbers have to exist due to the axiom of infinity (because they
> are the smallest set that satisfies the definition of the set constructed in
> the axiom of infinity, and thus as a subset of that set has to exist.

That's all fine, but now prove (in ZFC) that such set, the smallest set that
satisfies the axiom, does not contain any infinite number. Hint: you
cannot, by the Loewenheim–Skolem theorem.

> The only proof is that a power set is bigger than the corresponding set, and
> you would also have to find a mistake in that proof before you can claim
> anything here is false.

In light of the above, in bijection with what exactly??

I'll give you a counter thesis: the standard set of natural numbers (the one
with finite elements only) can only exists, i.e. be valid, in the context of
strictly finitary mathematics (where, strictly speaking, it is not even a set).

A suggestion for you: learn the difference between potential and actual
infinity in mathematics, IOW the difference between "infinity" in finitary vs.
in infinitary mathematics. Then look up Ross-Littlewood paradox and understand
why standard mathematics just does not work in the infinite case. Then go back
to all Cantor theorems and realise that a real number is not just a sequence
of digits, it is the limit of a sequence, hence so long Cantorian theory.

Not necessarily in that order: of course I have made a long story short, take
it mainly as an attempt to begin a conversation...

Julio

[shio...@googlemail.com]

""
> > the natural numbers have to exist due to the axiom of infinity (because they
> > are the smallest set that satisfies the definition of the set constructed in
> > the axiom of infinity, and thus as a subset of that set has to exist.
>
> That's all fine, but now prove (in ZFC) that such set, the smallest set that
> satisfies the axiom, does not contain any infinite number. Hint: you
> cannot, by the Loewenheim–Skolem theorem.
> ""

How? I know by construction that the natural numbers do not contain any infinite number, because the only sets included in it are sets of the form
s_{i+1}={s_i,{s_i}} for s_0= empty.
I constructed a set that definitely only contains finite sets and that satisfies the axiom.
Löwenheim skolem makes no statement about that, too.

"
> I'll give you a counter thesis: the standard set of natural numbers (the one
> with finite elements only) can only exists, i.e. be valid, in the context of
> strictly finitary mathematics (where, strictly speaking, it is not even a set).
> ""

That is, strictly, not true. At least the statement you made above is not correct, we do in fact know for sure that the naturals as defined by me are only containing finite sets.

"
> A suggestion for you: learn the difference between potential and actual
> infinity in mathematics, IOW the difference between "infinity" in finitary vs.
> in infinitary mathematics. Then look up Ross-Littlewood paradox and understand
> why standard mathematics just does not work in the infinite case. Then go back
> to all Cantor theorems and realise that a real number is not just a sequence
> of digits, it is the limit of a sequence, hence so long Cantorian theory.
>
> Not necessarily in that order: of course I have made a long story short, take
> it mainly as an attempt to begin a conversation...
""

As i said: Before you can tell me such a thing, you'd have to rigourusly show me the truth of your statements above.
You didnt give me a counter thesis, you simply made a statement that you did not back up so far.

[Julio Di Egidio]

On Friday, December 30, 2016 at 7:30:20 AM UTC+1, shio...@googlemail.com wrote:

> On Friday, December 30, 2016 at 6:44:36 AM UTC+1, Julio Di Egidio wrote:
> > On Friday, December 30, 2016 at 5:46:52 AM UTC+1, shio...@googlemail.com wrote:
> >

> > > the natural numbers have to exist due to the axiom of infinity (because they
> > > are the smallest set that satisfies the definition of the set constructed in
> > > the axiom of infinity, and thus as a subset of that set has to exist.
> >
> > That's all fine, but now prove (in ZFC) that such set, the smallest set that
> > satisfies the axiom, does not contain any infinite number. Hint: you
> > cannot, by the Loewenheim–Skolem theorem.
>
> How? I know by construction that the natural numbers do not contain any
> infinite number, because the only sets included in it are sets of the form
> s_{i+1}={s_i,{s_i}} for s_0= empty.
> I constructed a set that definitely only contains finite sets and that
> satisfies the axiom.
> Löwenheim skolem makes no statement about that, too.

Courtesy Wikipedia, << The result implies that first-order theories are unable
to control the cardinality of their infinite models >>. I am not an expert,
so you might even be right that it is not Loewenheim–Skolem that I should be
invoking, OTOH that there exist perfectly good non-standard models of first-
order Peano arithmetic is pretty well know.

Then you have snipped the direct question/objection to you, here it is again:

> > The only proof is that a power set is bigger than the corresponding set, and
> > you would also have to find a mistake in that proof before you can claim
> > anything here is false.
>
> In light of the above, in bijection with what exactly??

Next my counter-thesis:

> > I'll give you a counter thesis: the standard set of natural numbers (the one
> > with finite elements only) can only exists, i.e. be valid, in the context of
> > strictly finitary mathematics (where, strictly speaking, it is not even a set).
> >

> > A suggestion for you: learn the difference between potential and actual
> > infinity in mathematics, IOW the difference between "infinity" in finitary vs.
> > in infinitary mathematics. Then look up Ross-Littlewood paradox and understand
> > why standard mathematics just does not work in the infinite case. Then go back
> > to all Cantor theorems and realise that a real number is not just a sequence
> > of digits, it is the limit of a sequence, hence so long Cantorian theory.
> >
> > Not necessarily in that order: of course I have made a long story short, take
> > it mainly as an attempt to begin a conversation...
>
> As i said: Before you can tell me such a thing, you'd have to rigourusly show me the truth of your statements above.
> You didnt give me a counter thesis, you simply made a statement that you did not back up so far.

You are confused there: thesis is to demonstration as theorem is to proof. And
I did not just give you a thesis, in fact I gave you some rationale and plenty
of hints and hooks. Moreover, you pretend you cover centuries of logic and
mathematics and sort out all debates with a bunch of links to Wikipedia, and I
am supposed to address all that with a strictly formal, totally rigurous, and
absolutely comprehensive treatment, aren't I? And you are also the one
reminding others about logic? Then use it...

You know (you even mention usefulness at some point), for one thing "my" theory
does solve most paradoxes, i.e. not just e.g. Ross-littlewood but even Burali-
Forti: and while I should certainly prove that to you at some point, for now
my point just is how can *you* defend "your" theory at all...

Julio

[WM]

Am Freitag, 30. Dezember 2016 05:46:52 UTC+1 schrieb shio...@googlemail.com:


> However, we know that the naturals are not finite.

Yes. Their magnitudes grow infinitely. But there is no limit.

> There is nothing stopping us.

Perhaps, but there is something sometimes stopping sensible people, namely reason.

> Naming a thing is NEVER wrong in math.

Wrong. Naming the cardinality of the empty set 4711 is wrong - as wrong as naming the cardinality of all FISONs or their union greater than the cardinalities of all FISONs.

For all n in |N: A(n) = A(1) U A(2) U ... U A(n-1) U {n} is finite.

> You could name the set that contains all others sets biggle boggle, and it would not be a problem.
> Such a set does not exist in ZFC, but that simply means that biggle boggle does not exist and nothing is wrong or lost by giving that nonexistant set a silly name.
>
> So, we call our cardinality of the naturals aleph_0.

A set with crdinality aleph_0 contains infinitely many more elements than every finite set. This is contradicted by the fact that shifting all rows of the arithmogeometrical figure of FISONs
1
1, 2
1, 2, 3
...
into the first row. There cannot more elements be gathered than are in all finite rows, i.e., less than aleph_0.

>

> Now, there are two good ways to show that not all infinite sets are in bijection with each other: The power set of the naturals and the reals.
>
> I know that you take issue with real numbers and them being infinitely long,

Infinite sequences simply do not define anything, let alone numbers.

> so i decided to give you the proof for power sets:

The power set axiom is not absolute but relative. In many universes the power set is externally countable. This means that some subsets of |N are missing. In fact this holds for all universes. Therefore the Hessenberg-set does not exist and cannot prove anything. See "The relativeness of the power set" in
https://www.hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf

> https://en.wikipedia.org/wiki/Cantor%27s_theorem
>
> I can explain it with some more explanation if you wish to, but this proof is logically sound and does not have anything to do with real numbers.

Here I explain why this "proof" is invalid:
https://www.hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf
On Hessenberg's proof (I) - (III)

> The only proof is that a power set is bigger than the corresponding set

That is true for finite sets only. Infinities cannot be different.

>, and you would also have to find a mistake in that proof before you can claim anything here is false.

See above, I found three mistakes.

Regards, WM

[John Gabriel]

On Thursday, 29 December 2016 20:46:52 UTC-8, shio...@googlemail.com wrote:
> This is a thread just for you two in which we will, if you dare to reply at least, go over this topic step by step.
>
> If you find some issue in a step, please ONLY reply to that issue so we save space and can go over it thouroughly.
>
> 0)
>
> As a preword: Every possible approach to mathematics (and there is more than one) needs axioms, so statements which we simply see as true.

Fail you idiot. There is no need for axioms in mathematics. You should avoid trying to talk about mathematics because you know NOTHING about it.

PLONK.

<excrement erased>

[burs...@gmail.com]

Am Freitag, 30. Dezember 2016 11:36:46 UTC+1 schrieb John Gabriel:

By no need you mean your nonsense videos
bird brain John Gabriel bridbrain?

[burs...@gmail.com]

Poor city of augsburg, not a single correct proof
by WM in the below PDF. Just a huge pile of
incomprehensible brable.

Is your PDF a childrens book? About the big brown
wolf set theory? This is usually what comes
from augsburg:

Peter und der Wolf | Augsburger Puppenkiste 1967
https://www.youtube.com/watch?v=GHf1msJ-I0c

So they thought they get a math clown as well?

[Ross A. Finlayson]

A clarification: in ZF set theory, the other axioms
"expand" comprehension (defining new sets as exist from
other sets) and the axiom of regularity (and thus the
axiom of infinity) "restrict" comprehension, removing
from the domain of discourse, or universe of sets, what
would be ir-regular/non-well-founded sets, or extra-
ordinary sets as Russell calls them.

So, some have that ZF's "Axiom of Infinity" isn't
necessarily "true", not because of rejecting its
expansion of comprehension, instead rejecting its
restriction of comprehension, of a larger
universe of sets.

This has various implications then about ordering
before counting or vice-versa, about successor and
powerset.

Goedel establishes that ZF isn't both complete and
consistent. Then, some "larger" (or, lesser) true
theory is, then that some would have that axioms
only expand comprehension, for pure logic, while
"axioms" that restrict comprehension are actually
conditions on objects, not fundamentally so.

[WM]

Am Freitag, 30. Dezember 2016 15:34:32 UTC+1 schrieb burs...@gmail.com:
> Poor city of augsburg, not a single correct proof
> by WM in the below PDF. Just a huge pile of
> incomprehensible brable.

Ii is incomprehensible to fools, yes, but comprehensible to intelligent students.

For instance:
- limit(1/n) = 0
- Representing the numbers 1/n as points on the real line yields the same limit.
- Connecting ever point with the point 1 does not change the limit.

Therefore the limit of the sequence of intervals [1/n, 1] is [0, 1].

Only matheologians are unable to follow this argument and to confuse the infinite union (0, 1] of the terms and the limit [0, 1] of the sequence.

Regards, WM

[Ross A. Finlayson]

No, it's that there is a definition they choose
that agrees with others they choose, that then
it is clear that other reasonable _extensions of
the definition of the pair-wise operation to
any/each/every/all_ variously sees the transfer
principle hold.

There _are_ various considerations of where
the transfer principle (or specifically anti-
transfer principle) holds, others where it does
not. Here the ZF adherents have anti-transfer,
because an otherwise reasonable interpretation
breaks other things of the ZF adherent. That
said, your scatter-shot rejection is ineffective
because you can't just say "ZF is wrong", but
you have to demonstrate this using all the same
definitions.

Then, your only hope of a correct rejection of
ZF adherency is a "Theory of Everything" like
the Null Axiom Theory. And, because Goedel shows
that ZF is inconsistent or incomplete, a true
foundation of mathematics and all includes some
correct rejection of ZF adherency, because there's
more than ZF and a true foundation includes all.


Your compilation of like-minded partial fragments
of incorrect rejections of ZF adherency doesn't
include the sweep principle and axiomless deduction
as expanding comprehension, so it fails. (I've read
through it.)

[burs...@gmail.com]

Am Freitag, 30. Dezember 2016 22:46:18 UTC+1 schrieb WM:
> Am Freitag, 30. Dezember 2016 15:34:32 UTC+1 schrieb burs...@gmail.com:
> > Poor city of augsburg, not a single correct proof
> > by WM in the below PDF. Just a huge pile of
> > incomprehensible brable.

I am talking about the proofs in your PDF. Take for
example On Hessenberg's proof (III), highlighting
of the word IS by me:

"Each permutation IS a well-ordering. One of them would
be the well-ordering of Q that is simultaneously the well-
ordering by size. This is a contradiction. Like Hessenberg's
assumption."

I have never seen that permutations ARE wellordered
automatically. You can check this in the finite
already. Take the following set and order it along
the natural order:

1 2 3

Now do this permutstion:

2 1 3

The natural order isn't preserved. You need something
that doesnt say IS wellordered, but something that
says HAS a wellorder.

Simply put a permutation isnt a Homomorphism for
an ordering. In the above the permutation f is:

f(1) = 2
f(2) = 1
f(3) = 3

For such a permutation f being a Homomorphism we
need an source and a traget ordering. Since you use
the verb IS, I assume that you assume that the

source and target ordering are the same, <. So
to be a Homomorphism then we would need to have:

a < b iff f(a) < f(b)

Which is obviously not the case for example for a=1
and b=1. Since f(a)=2 and f(b)=1, and not 2<1. You
could define a wellordering by:

f(a) <_2 f(b) :<=> a <_1 b

But since you admit a bijection of N and Q, in
the same paragraph you even say THE bijection of
N and Q, but I dont know which one you mean.

Its also pretty irrelevant which one it is, since
if you have a bijection g : N -> Q, this automaticall
defines a well ordering:

p/q <_2 r/s :<==> g^-1(p/q) <_1 g^-1(r/s)

Which means your conclusion is wrong. But you are
probably drawing the conclusion from the density of
Q. But this density would be based on <_2, it is

usually based on <_3 defined as follows
(normalized s,q > 0):

p/q <_3 r/s :<==> p*s < r*q

So you are confusing a property bound to <_3, density,
with a property <_2 you habe just constructed. So WMs
thinking has a severe lack of abstraction.

So I guess I better read Charles Boukovski at least
he was funny when drunk.

(*)
https://en.wikipedia.org/wiki/Homomorphism

[burs...@gmail.com]

> Q. But this density would be based on <_2, it is

Corr:
Q. But this density wouldn't be based on <_2, it is

[burs...@gmail.com]

Maybe its not density you have in mind. Who knows
the proof doesnt say exactly. Maybe that Q isn't
bounded from below.

Lets call it property X, which prevents that
we can call <_3 a well ordering.

Basically because of property X we can I guess show
that for any permutation f and bijection g, <_2 will
never coincidence with <_3.

[shio...@googlemail.com]

""
> Courtesy Wikipedia, << The result implies that first-order theories are unable
> to control the cardinality of their infinite models >>. I am not an expert,
> so you might even be right that it is not Loewenheim–Skolem that I should be
> invoking, OTOH that there exist perfectly good non-standard models of first-
> order Peano arithmetic is pretty well know.
> ""

But that is not what you said nor does it say one cannot prove that the naturals are infinite.
Löwenheim Skolem states that every theory in FO that has some infinite model that is countable also has an uncountable infinite model.
However, that is not a problem in our case. In fact, you cannot define the natural numbers by property alone (and thus we didn't do that, since there are definitely uncountable sets for which the property of infinity holds).
But when you construct the set as having only finite sets elements, then löwenheim skolem is of no concern to us, since we are not exclusively using a property to define the set.
We use the axiom of infinity not to construct it, but to show that it exists (since it has to be a subset of the set that we know exists by the axiom of infinity).
Like that, we are not running into a problem.

""
> Then you have snipped the direct question/objection to you, here it is again:
> ""

I snipped it because i didnt understand what you mean here.
I think i explained the bijection or lack of bijection thouroughly, so what do you mean here?

""
> You are confused there: thesis is to demonstration as theorem is to proof. ""

So what's it worth then? Your words were just a nonbacked up statement.

"
> I did not just give you a thesis, in fact I gave you some rationale and plenty
> of hints and hooks. Moreover, you pretend you cover centuries of logic and
> mathematics and sort out all debates with a bunch of links to Wikipedia, and I
> am supposed to address all that with a strictly formal, totally rigurous, and
> absolutely comprehensive treatment, aren't I? And you are also the one
> reminding others about logic? Then use it...""

No, you said a thing that is very probably false without explaining why you think it is correct.
And yes, i am going on as formal as i can by providing links to what exactly i mean simply to save space, links that simply tell other people here the definitions of the things we talk about (since they often understand them wrong) while you do not even do that.

All you did by now was using löwenheim skolem wrong and saying things you do not back up with anything, calling it 'hints'.

[shio...@googlemail.com]

""
> Yes. Their magnitudes grow infinitely. But there is no limit.""

Being infinitely big is exactly defined by there being no finite limit of cardinality.
If there is no natural number that is strictly bigger than the number of elements in a set, then the set is infinite.

""
> Wrong. Naming the cardinality of the empty set 4711 is wrong - as wrong as naming the cardinality of all FISONs or their union greater than the cardinalities of all FISONs.""

You have to understand the difference of definition and theorem.
Naming the cardinality of the empty set 4711 when we already defined its cardinality in another way would be wrong.

Defining the cardinality of the empty set as 4711 when we didnt define it before is completely logically sound and we could do it.
We could even, if we want to, define the cardinality of the empty set with that number, then start counting down and say a set with 1 element has cardinality 4710 and start counting negatively once we hit 0.

A definition is just a naming of things that may or may not be a particularly useful definition.
It is however never wrong unless you defined it in another way before.

Since this is the first definition of the concept of cardinality i gave, it cannot be wrong.

Let us start there, because i am not going to answer your other stuff before you understood that one issue.

You have to accept that definitions are inherently correct because they are nothing more than names for a certain thing that may or may not exist.
That can be a function, a relation or another kind of object.

In our case, we defined cardinality by giving sets the cardinality of the number of their elements if they are finite, calling the cardinality of the infinite numbers aleph_0 (since it cannot be finite) and then saying every set has cardinality aleph_0 if it is in bijection with the natural numbers.

We say sets are of equal cardinality if they are in bijection.

Those are just definitions which do not contradict any prior definition, because this is the one and only definition we ever had for cardinality.

That is something you have to accept before you can understand the rest.

I will say it again: You have to see the difference between theorem and definition.
Saying the empty set has cardinality 4711 after you defined cardinality of finite sets by 'number of elements' is a wrong theorem, since you are not defining the cardinality here but rather making the statement that regarding the second definition, the cardinality is 4711.
But this is wrong because the second definition states otherwise.

Without the second definition and with only the third, saying it has cardinality 4711 is a inherently correct statement because that is how you defined the cardinality.

Once again: Naming a thing is never wrong. Only once you chose definitions and then make statements about the objects of which you have defined properties, only then you can make wrong statements.

[shio...@googlemail.com]

Okay, so then prove me that the number 1 exists without using any theorems please.

[John Gabriel]

Read:

https://www.linkedin.com/pulse/how-we-got-numbers-john-gabriel-1?trk=prof-post

[burs...@gmail.com]

Am Samstag, 31. Dezember 2016 00:46:04 UTC+1 schrieb John Gabriel:

By "fail you idiot" you are talking about your
new "calculus", you even don't master yourself

bird brain John Gabriel birdbrain. BTW the
chees cake factory is calling you.

[shio...@googlemail.com]

I have read that, and i already explained you the mistakes you do in there, namely not defining what a magnitude is.
You say "A magnitude is the idea of size, dimension or extent".
That is not a mathematical definition, that is ambigous at best.

And if a number is a 'measure of size' then obviously sqrt(2) would be a number

Furthermore: You do not get to decide what a number is. You could name a mathematical object number if you chose to, that does not change the fact that sqrt(2) is a welldefined ringelement in actual math (in which we do not have a definition for 'number' since we are much more general in our approach by now).

But unless you can gave me a definition of the word number that either uses FO or other definitions definable in FO, you have not defined that word.

[John Gabriel]

On Friday, 30 December 2016 16:03:34 UTC-8, shio...@googlemail.com wrote:
> On Saturday, 31 December 2016 00:46:04 UTC+1, John Gabriel wrote:
> > On Friday, 30 December 2016 15:30:40 UTC-8, shio...@googlemail.com wrote:
> > > On Friday, 30 December 2016 11:36:46 UTC+1, John Gabriel wrote:
> > > > On Thursday, 29 December 2016 20:46:52 UTC-8, shio...@googlemail.com wrote:
> > > > > This is a thread just for you two in which we will, if you dare to reply at least, go over this topic step by step.
> > > > >
> > > > > If you find some issue in a step, please ONLY reply to that issue so we save space and can go over it thouroughly.
> > > > >
> > > > > 0)
> > > > >
> > > > > As a preword: Every possible approach to mathematics (and there is more than one) needs axioms, so statements which we simply see as true.
> > > >
> > > > Fail you idiot. There is no need for axioms in mathematics. You should avoid trying to talk about mathematics because you know NOTHING about it.
> > > >
> > > > PLONK.
> > > >
> > > > <excrement erased>
> > >
> > > Okay, so then prove me that the number 1 exists without using any theorems please.
> >
> > Read:
> >
> > https://www.linkedin.com/pulse/how-we-got-numbers-john-gabriel-1?trk=prof-post
>
> I have read that, and i already explained you the mistakes you do in there, namely not defining what a magnitude is.

You are wrong.

"A magnitude is the idea of size, dimension or extent".

That is a valid definition.

So I suggest you try again.

[shio...@googlemail.com]

""
> "A magnitude is the idea of size, dimension or extent".
>
> That is a valid definition.
>
> So I suggest you try again.
>""

No it is not a valid definition.
If it is, formulate it in FO here and now.

[shio...@googlemail.com]

Or any other formal system, really, but i do not know you know more than at best FO and AL.

[John Gabriel]

Yes, it is a formal definition you idiot!

You can't rubbish any part of it.

[John Gabriel]

On Friday, 30 December 2016 16:13:23 UTC-8, shio...@googlemail.com wrote:
> Or any other formal system, really, but i do not know you know more than at best FO and AL.

You arrogant piece of shit. One doesn't have to use YOUR systems to define anything properly. You do not know any mathematics.

[John Gabriel]

Ironically this idiot isn't just disagreeing with me, but with the unbelievably brilliant Ancient Greeks. Of course he wouldn't have a clue. He has never read the Elements. Chuckle.

[shio...@googlemail.com]

""Yes, it is a formal definition you idiot!

You can't rubbish any part of it.

> You arrogant piece of shit. One doesn't have to use YOUR systems to define anything properly. You do not know any mathematics.

Ironically this idiot isn't just disagreeing with me, but with the unbelievably brilliant Ancient Greeks. Of course he wouldn't have a clue. He has never read the Elements. Chuckle. ""

I gave you the option of using any formal system you like.
But you cannot even do that.
In math we have a minimum of rigour applied to the phrases we use.
That minimum is that it can be defined (and then is defined) in a formal system like FO.
Anything else leaves room to ambiguity, much like the definition you choose.

The fact that you cannot formulate what a number is with common definitions or FO just means that you did not actually define it.

Let me guess: YOu do not know what FO even is, right?

[John Gabriel]

On Friday, 30 December 2016 16:47:03 UTC-8, shio...@googlemail.com wrote:
> ""Yes, it is a formal definition you idiot!
>
> You can't rubbish any part of it.
>
>
> > You arrogant piece of shit. One doesn't have to use YOUR systems to define anything properly. You do not know any mathematics.
>
>
> Ironically this idiot isn't just disagreeing with me, but with the unbelievably brilliant Ancient Greeks. Of course he wouldn't have a clue. He has never read the Elements. Chuckle. ""
>
> I gave you the option of using any formal system you like.

One does not need a formal system to define anything.

> But you cannot even do that.
> In math we have a minimum of rigour applied to the phrases we use.

You don't know any MATH. A "minimum of rigour" - bwaaaa haaaaa haaaa.

How do you measure this O dimwit? Chuckle. Are their rigour units???

> That minimum is that it can be defined (and then is defined) in a formal system like FO.

First order logic is not required to define anything in mathematics. Perhaps in mythmatics, but not mathematics.

> Anything else leaves room to ambiguity, much like the definition you choose.

You don't understand. So I can see why you say there is ambiguity. In fact, there is no ambiguity whatsoever. A magnitude is an idea and one that its very primitive. That being so, it is not possible to define it in terms of anything else. It is merely an idea or a concept. Not all ideas and concepts are fully defined. Those which are primitive are used to derive other ideas. If you knew anything about object oriented programming, then the closest analogy would be a virtual class.

>
> The fact that you cannot formulate what a number is with common definitions or FO just means that you did not actually define it.

Again, your assertion and still wrong. Did it even occur to you O stupid one, that first order logic didn't exist when the foundations of mathematics were laid?

>
> Let me guess: YOu do not know what FO even is, right?

Let me see, presumptuous again. You are starting to get on my nerves with your intolerable stupidity.

[shio...@googlemail.com]

""
> One does not need a formal system to define anything.
> ""

Yes you do, elseways it is not a mathematical definition because it bears ambigouity.

""
> First order logic is not required to define anything in mathematics. Perhaps in mythmatics, but not mathematics.""

I said 'formal system', and that in fact is a requirement.
FO is just the easiest one of those with some expressive power.
And in fact mathematics is completely built on set theory and formal systems (most commonly FO).
One could use other axioms and another formal system, but i left you that option.
You didnt take it because you probably do not know what a formal system even is.

""> You don't understand. So I can see why you say there is ambiguity. In fact, there is no ambiguity whatsoever. A magnitude is an idea and one that its very primitive. That being so, it is not possible to define it in terms of anything else. It is merely an idea or a concept. Not all ideas and concepts are fully defined. Those which are primitive are used to derive other ideas. If you knew anything about object oriented programming, then the closest analogy would be a virtual class.""

There is. Because you cannot modelcheck it.
For a property to be a definition of something, that property has to be verifiable.
For example, the property 'red number' is completely ambigous, because we didnt state what that really means.
'2 is a red number' cannot be checked.
However, if i define what i mean in FO, it is always verifiable whether something holds (for example with sequent calculus).
That is why you have to define such a property in a formal system in which it can be checked whether some structure is a model for the defined property.

With your last sentence you have admitted that your 'idea' is not fully defined.
And that is not acceptable in math, it lacks rigour.

Why would we take a definition that is not completely defined, cannot be defined in FO (and thus not model checked) while we can easily define sqrt(2) completely rigourusly as ringelement?

[genm...@gmail.com]

On Thursday, 29 December 2016 20:46:52 UTC-8, shio...@googlemail.com wrote:

<excrement deleted>

The following link:

http://mathworld.wolfram.com/First-OrderLogic.html

states:

-----------------------------------------------
The set of terms of first-order logic (also known as first-order predicate calculus) is defined by the following rules:

1. A variable is a term.
-----------------------------------------------

There are more rules but I don't need any more than the very first one for my demonstration.

Now I am going to demonstrate what a baboon is this idiot shio...@googlemail.com.

So the imbecile asks me to define a "magnitude" using first order systems.

A magnitude is a very primitive idea. However, the very first rule requires that a "variable" and a "term" are both well defined.

In order for a variable to be well defined, those objects known as "numbers" must have been established. What our moron doesn't realise, is that in my article https://www.linkedin.com/pulse/how-we-got-numbers-john-gabriel-1?trk=prof-post I was deriving the number concept. It would be circular and idiotic to assume it exists by trying to define it using FO logic.

As for the word "term", it is very ambiguous. Also it is nowhere defined in FO logic what it means to be a term.

The BIG STUPID - always stupid

[genm...@gmail.com]

On Friday, 30 December 2016 17:05:21 UTC-8, shio...@googlemail.com wrote:

<Too much crap I couldn't be bothered to respond to ...>

> Why would we take a definition that is not completely defined, cannot be defined in FO (and thus not model checked) while we can easily define sqrt(2) completely rigourusly as ringelement?

It is *completely* defined for the purpose for which it is defined.

You can't define sqrt(2) as a ring element without knowing its properties. But if you know its properties, that does not mean you have defined it. Properties alone do not define an object. Any object also has attributes in addition to properties.

But you are extremely short sighted because you have no clue what it means to be a number. The only way to claim that sqrt(2) is a *number* is to provide a measure of the magnitude that is sqrt(2). Stating its properties as a ring element measures NOTHING. Get it?

[genm...@gmail.com]

I don't know how I allowed myself to be dragged into a discussion with such a moron. It's like the time I proved that Peano's axioms are crap and christensen couldn't understand a thing. A waste of time....

[jenga...@gmail.com]

Oh wow johnny, you really didn't know what FO is XD

You...do know that a variable in logic has nothing to do with numbers...right?
Oh who am i kidding, you just showed you do not.

Did you really just hear the word FO for the first time and then look it up just to spout nonsense about it with your double account?
Yes. Yes you did.

Because in FO, the word term is defined very easily and you WROTE THE DEFINITION IN HERE.
A Variable in FO is an element from the class VAR containing all abstract variables (so for example unqiue letter x,y,z and so on).
For easy of representation, let var me VAR={x_i | i \in IN}.

A term in FO is defined in the following way:

All variables are terms and all functions having the variables as arguments are terms, that means x_1 is a term and f(x_1,x_2) are terms, they are not formulas in FO because they do not have a truth value.
This is completely well defined by these two rules.
A term is any combination of variables and functions without using any other operators (like and, or, negation or relations).

See? I can explain you very thouroughly what i talk about.
You howev

[jenga...@gmail.com]

""
> It is *completely* defined for the purpose for which it is defined.
>
> You can't define sqrt(2) as a ring element without knowing its properties. But if you know its properties, that does not mean you have defined it. Properties alone do not define an object. Any object also has attributes in addition to properties.
>
> But you are extremely short sighted because you have no clue what it means to be a number. The only way to claim that sqrt(2) is a *number* is to provide a measure of the magnitude that is sqrt(2). Stating its properties as a ring element measures NOTHING. Get it?""

True, but i know its properties and i can easily define them.
You however can't.
An element fullfilling the relation x^2=2 and being positive is welldefined.
No problem with that, it can be easily verified whether this holds true.

YOur ambigious definition can not be verified.

I am not shortsighted. The fact remains that you cannot properly define what a number is, nor what a measure of magnitude is, nor what a magnitude is.

There is no formality in your statements, they are things that might make sense in your head, but you cannot give me a proper way to check whether something is a number in your definition.

In any case, your definition of what a number is is completely irrelevant.
Even if you could define the word 'number' in a way that holds for rationals but not for irrationals (which is not hard), it would have no use.

BEcause simply, that definition is something noone in math works with because it bears no merit and does not even make logical sense unless you can define it better.

[jenga...@gmail.com]

Which probably was because you didnt prove it but your lack of skill made you think you did.

[burs...@gmail.com]

Am Samstag, 31. Dezember 2016 02:09:06 UTC+1 schrieb genm...@gmail.com:
> In order for a variable to be well defined

Nope, variables are always well defined, in the
sense that no logic requires some special semantic
considerstion for the use of a variable.

In most logics, its always syntax first then
semantic. Its almost never that the semantic
influences the syntax. Syntactically a variable

is just a symbol v from the set V of variables.
There are other symbols involved in forming
logical formulae, constants c from C, functions
f from F, and relations r from R.

A term in FOL (not in HOL or other logics), is
then inductively defined:
- if v in V then v in T
- if c in C then c in T
- if f in F and f arity n and
t1,..,tn in T then f(t1,..,tn) in T

A formula in FOL (not in HOL or other logics), is
then inductively defined:
- if r in R and r arity n and
t1,..,tn in T then r(t1,..,tn) in F
- if A,B in F then A /\ B in F
- if A,B in F then A \/ B in F
- if A,B in F then A -> B in F
- if A in F then ~A in F
- if A in F and v i V then exists v A in F
- if A in F and v i V then forall v A in F

You are ultra sick JG, bird brain John Gabriel
birdbrain, please see a doctor.

[burs...@gmail.com]

Am Samstag, 31. Dezember 2016 03:27:18 UTC+1 schrieb burs...@gmail.com:
> - if A in F and v i V then exists v A in F
> - if A in F and v i V then forall v A in F

Corr.:
- if A in F and v in V then exists v A in F
- if A in F and v in V then forall v A in F

[burs...@gmail.com]

But FOL is an extremely simplified logic.

Its white washed and a lot of things such
as the following are not built in:
- Its first order and not higher order.
- Its single sorted and not multi sorted.
- Its non-modal and not modal.
- What else...?

So when taking a math text book, most often,
it wouldn't be economical to try to translate

it to FOL.

[John Gabriel]

On Friday, 30 December 2016 17:27:49 UTC-8, jenga...@gmail.com wrote:
> On Saturday, 31 December 2016 02:09:06 UTC+1, genm...@gmail.com wrote:
> > On Thursday, 29 December 2016 20:46:52 UTC-8, shio...@googlemail.com wrote:
> >
> > <excrement deleted>
> >
> > The following link:
> >
> > http://mathworld.wolfram.com/First-OrderLogic.html
> >
> > states:
> >
> > -----------------------------------------------
> > The set of terms of first-order logic (also known as first-order predicate calculus) is defined by the following rules:
> >
> > 1. A variable is a term.
> > -----------------------------------------------
> >
> > There are more rules but I don't need any more than the very first one for my demonstration.
> >
> > Now I am going to demonstrate what a baboon is this idiot shio...@googlemail.com.
> >
> > So the imbecile asks me to define a "magnitude" using first order systems.
> >
> > A magnitude is a very primitive idea. However, the very first rule requires that a "variable" and a "term" are both well defined.
> >
> > In order for a variable to be well defined, those objects known as "numbers" must have been established. What our moron doesn't realise, is that in my article https://www.linkedin.com/pulse/how-we-got-numbers-john-gabriel-1?trk=prof-post I was deriving the number concept. It would be circular and idiotic to assume it exists by trying to define it using FO logic.
> >
> > As for the word "term", it is very ambiguous. Also it is nowhere defined in FO logic what it means to be a term.
> >
> > The BIG STUPID - always stupid
>
> Oh wow johnny, you really didn't know what FO is XD
>
> You...do know that a variable in logic has nothing to do with numbers...right?

Bullshit. It needn't have anything to do with numbers. But in this case it does.

<too much excrement ignored>

[Julio Di Egidio]

On Saturday, December 31, 2016 at 12:20:28 AM UTC+1, shio...@googlemail.com wrote:
> ""
> > Courtesy Wikipedia, << The result implies that first-order theories are unable
> > to control the cardinality of their infinite models >>. I am not an expert,
> > so you might even be right that it is not Loewenheim–Skolem that I should be
> > invoking, OTOH that there exist perfectly good non-standard models of first-
> > order Peano arithmetic is pretty well know.
> > ""
>
> But that is not what you said

Oh yes, it is.

Then I even doubt you have a degree in mathematics, given the many and
repeated errors in your exposition starting with the very language, not
to mention the fact that you won't answer questions. Either you are a
spammer or you are very confused, maybe both.

Julio

[John Gabriel]

On Friday, 30 December 2016 22:11:23 UTC-8, Julio Di Egidio wrote:
> On Saturday, December 31, 2016 at 12:20:28 AM UTC+1, shio...@googlemail.com wrote:
> > ""
> > > Courtesy Wikipedia, << The result implies that first-order theories are unable
> > > to control the cardinality of their infinite models >>. I am not an expert,
> > > so you might even be right that it is not Loewenheim–Skolem that I should be
> > > invoking, OTOH that there exist perfectly good non-standard models of first-
> > > order Peano arithmetic is pretty well know.
> > > ""
> >
> > But that is not what you said
>
> Oh yes, it is.
>
> Then I even doubt you have a degree in mathematics,

Oh, I am inclined to think he has a degree in mythmatics. It explains why he is as thick as a brick.

[Julio Di Egidio]

On Saturday, December 31, 2016 at 4:16:15 AM UTC+1, burs...@gmail.com wrote:

> But FOL is an extremely simplified logic.

FOL is *the most* elementary mathematical logic,
arguably the most natural, and it is the logic that
is relevant in meta-mathematics and foundations.

Julio

[Ross A. Finlayson]

First-order, why not, zeroeth-order?

Going to higher order is in a sense
adding variables (and quantifying over
them). Then, first-order predicate logic
has all the terms and their collections as
also first-order (instead of as each other).
Then, some zeroeth-order is a primitive
substrate underneath the first-order, about
the constant and variable and otherwise in
terms of foundation (not foundations, plural,
but foundation, sole foundation) of the
logic.

[Julio Di Egidio]

On Friday, December 30, 2016 at 11:02:27 PM UTC+1, Ross A. Finlayson wrote:

> Then, your only hope of a correct rejection of
> ZF adherency is a "Theory of Everything"

Yes, but I'd call it a Theory of All: the door says Mathematics, it's about
Numbers.

> like
> the Null Axiom Theory. And, because Goedel shows
> that ZF is inconsistent or incomplete, a true
> foundation of mathematics and all includes some
> correct rejection of ZF adherency, because there's
> more than ZF and a true foundation includes all.

Adherence which, the claim goes, is the idea that one can do *infinitary*
mathematics with ZF as foundation. It's anti-transfer just as in does-not- transfer.

Julio

[Julio Di Egidio]

On Saturday, December 31, 2016 at 7:27:39 AM UTC+1, Ross A. Finlayson wrote:
> On Friday, December 30, 2016 at 10:13:26 PM UTC-8, Julio Di Egidio wrote:
> > On Saturday, December 31, 2016 at 4:16:15 AM UTC+1, burs...@gmail.com wrote:
> >
> > > But FOL is an extremely simplified logic.
> >
> > FOL is *the most* elementary mathematical logic,
> > arguably the most natural, and it is the logic that
> > is relevant in meta-mathematics and foundations.
>

> First-order, why not, zeroeth-order?

But I did not say "predicate".

Julio

[Ross A. Finlayson]

Yes, and also that: other features (of a like sort) do.

Here the "index variable" or otherwise the prime mover
of the clockworks of a sort sees that other features
of the clock arithmetic revolve at the time and transfer
together, where here this doesn't.

[Julio Di Egidio]

On Saturday, December 31, 2016 at 7:38:30 AM UTC+1, Ross A. Finlayson wrote:
> On Friday, December 30, 2016 at 10:31:02 PM UTC-8, Julio Di Egidio wrote:
> > On Friday, December 30, 2016 at 11:02:27 PM UTC+1, Ross A. Finlayson wrote:
> >
> > > Then, your only hope of a correct rejection of
> > > ZF adherency is a "Theory of Everything"
> >
> > Yes, but I'd call it a Theory of All: the door says Mathematics, it's about
> > Numbers.
> >
> > > like
> > > the Null Axiom Theory. And, because Goedel shows
> > > that ZF is inconsistent or incomplete, a true
> > > foundation of mathematics and all includes some
> > > correct rejection of ZF adherency, because there's
> > > more than ZF and a true foundation includes all.
> >
> > Adherence which, the claim goes, is the idea that one can do *infinitary*
> > mathematics with ZF as foundation. It's anti-transfer just as in does-not- transfer.
>

> Yes, and also that: other features (of a like sort) do.

Either they do, include all, or they don't: there's not a like sort of.

> Here the "index variable" or otherwise the prime mover of the clockworks

Eh no, the index variable is the clock...

Julio

[burs...@gmail.com]

Am Samstag, 31. Dezember 2016 07:27:39 UTC+1 schrieb Ross A. Finlayson:
> First-order, why not, zeroeth-order?

propotional logic can be viewed as a
zero-th order logic. At least one does
need a domain for it.

BTW Gödel was already not using FOL in his
On Formally Undecidable Propositions paper,
according to this transcription.
https://www.research.ibm.com/people/h/hirzel/papers/canon00-goedel.pdf

That he wasn't using FOL can be confirmed here:
http://www.w-k-essler.de/pdfs/goedel.pdf

For example his formulation of the comprehension
axiom, without using set membership relation,
and instead a function application:

∃u.∀v .(u(v) ⇔ a)

Wouldn't be a wellformed as an axiom schema in
FOL. Since u is treated as a variable, but it
occurs in a relation position.

[burs...@gmail.com]

Am Samstag, 31. Dezember 2016 12:32:45 UTC+1 schrieb burs...@gmail.com:
> propotional logic can be viewed as a
> zero-th order logic. At least one does
> need a domain for it.

Corr.:

propotional logic can be viewed as a
zero-th order logic. At least one does

not need a domain for it.

Even QSAT doesn't really need a domain
but it has nevertheless quantifiers.
https://en.wikipedia.org/wiki/True_quantified_Boolean_formula

[burs...@gmail.com]

On a sidenote altough we might consider
propotional logic as zero-th order
father of FOL.

QSAT isn't, since it has a similar feature
like the logic that Gödel was using, we
could for example form:

∃u ~u

And again u is a variable occuring in a
relation position, here a propositional
position.

[WM]

Am Freitag, 30. Dezember 2016 23:22:02 UTC+1 schrieb burs...@gmail.com:


> "Each permutation IS a well-ordering. One of them would
> be the well-ordering of Q that is simultaneously the well-
> ordering by size. This is a contradiction. Like Hessenberg's
> assumption."
>
> I have never seen that permutations ARE wellordered
> automatically.

The natural nunbers are well-ordered in their natural order and in any permutation that preserves this order type automatically. If the bijection from |N to Q is fixed, then the rationals are well-ordered too and change their order as the naturals do, maintainig their order type. The pair q_n becomes the pair q_m.

> Take the following set and order it along
> the natural order:
>
> 1 2 3
>
> Now do this permutation:
>
> 2 1 3
>
> The natural order isn't preserved.

Nevertheless the set is well-ordered as any finite set and any infinite sequence.

> You need something
> that doesnt say IS wellordered, but something that
> says HAS a wellorder.

Wrong. Zermelo has proved that every set can be well-ordered - not that every set has a well-order. [E. Zermelo: "Beweis, daß jede Menge wohlgeordnet werden kann", Math. Ann. 59 (1904) 514-516]

Regards, WM

[Ross A. Finlayson]

It is about features of a like type,
here that are indexed for their limit
with the same "prime mover".

Here if there is transfer or anti-transfer,
maybe the index is just an indicator, about
whether the "inductive index" is ordinary or
extra-ordinary. Then otherwise when there is
one or the other, maybe it is the real concrete
index (as a parameter, invariant, instead of a
variable, as one way to look at it, or as the
variable, among otherwise the template, here
about whether the clock moves or time moves
around it).

[Ross A. Finlayson]

These days at least again "everything can be
written in first order" (not necessarily how
it can, but that it can).

These days Goedel might be re-written in the
Goedel-Bernays-von Neumann set theory that has
the axiom schema (of ZF(C)) for the writing
everything in the first order.

[burs...@gmail.com]

Am Samstag, 31. Dezember 2016 16:36:03 UTC+1 schrieb WM:
> Wrong. Zermelo has proved that every set can be well-ordered
> - not that every set has a well-order. [E. Zermelo: "Beweis,
> daß jede Menge wohlgeordnet werden kann", Math. Ann. 59 (1904) 514-516]

Yes in German "werden kann" and not in German "ist".

Theorem — For every set X, there EXISTS a well-ordering with domain X.

Or you can rephrase it:

Theorem — For every set X, there EXISTS a W (binary relation) subset X x X which is a well-ordering of X.

For well-ordering see, binary relation here:
https://en.wikipedia.org/wiki/Total_order#Strict_total_order

For well-odering see, binary relation involved again:
https://en.wikipedia.org/wiki/Greatest_element

[burs...@gmail.com]

Am Samstag, 31. Dezember 2016 17:06:59 UTC+1 schrieb Ross A. Finlayson:
> > ∃u.∀v .(u(v) ⇔ a)
> >
> > Wouldn't be a wellformed as an axiom schema in
> > FOL. Since u is treated as a variable, but it
> > occurs in a relation position.
>
> These days at least again "everything can be
> written in first order" (not necessarily how
> it can, but that it can).

Its not that easy, some logics are not so
trivial, see for example:

The McKinsey Axiom is not Canonical
http://www.jstor.org/stable/pdf/2274699.pdf?seq=1#page_scan_tab_contents

[burs...@gmail.com]

That there isn't a global wellordering can be easily seen
intuitively. Take the interval [0,1] of rationals, call this
set X_1. Assume it is well ordered by the usual <.

Now remove the zero 0 from X_1, to get X_2. I guess
X_2 is not anymore wellordered by the same <. Since the
zero 0 is missing, no least element anymore.

[Ross A. Finlayson]

With the sweep principle
and well-ordering the reals,
those are not just different
sets, but sets of different
elements. This is where R^bar
and R^dots are different sets,
and while both may be [0,1],
removing any elements eliminates
the properties (of the functions
that so define it) of being R^bar.

[Ross A. Finlayson]

The relevance of the note is that
some people didn't know that there's
a first-order logic for anything in a
similar way (and via similar means) as
where Skolem has a countable model for
anything.

Then in the reference you provide there
is discussion of [Kelley-Morse] as being
the "smallest" normal, modal, logic with
that Scott's diamond and Scott's box exchange
(here, transfer), as about the diamond
and box as quantifiers.

(Or, that's my reading of it.)

[John Gabriel]

I really wish you would stop posting your incoherent babble. You are an unbelievable troll Finlayson. It's exhausting having to mark all your comments as abusive. Please desist.

[John Gabriel]

Okay Shithead, let's play the game. Sorry, I meant Shio. By the way, what is your real name and where do you teach? Or are you too insecure and afraid of what you know to reveal this information?

> If you find some issue in a step, please ONLY reply to that issue so we save space and can go over it thouroughly.

> As a preword: Every possible approach to mathematics (and there is more than one) needs axioms, so statements which we simply see as true.

So I have an issue with your very first statement. It is FALSE that any approach to mathematics requires axioms. In fact, you probably don't know this, but there are no axioms in Euclid's Elements, The Works of Archimedes or any other Ancient Greek texts (which I have studied in the Ancient Greek). The word axiom did not even exist when those documents were written.

You might foolishly argue that certain facts were assumed without proof, but this is false. I have explained this in my videos:

https://www.youtube.com/watch?v=aLTp7Rv65oU

https://www.youtube.com/watch?v=ExMk90-ENs0

https://www.youtube.com/watch?v=Dxc3QUhPDbc

https://www.youtube.com/watch?v=vtq0uyAqG2k

If you don't watch them, that's your problem because I won't be explaining everything here again.

It was Sir Thomas Heath who introduced the words 'axiom' and 'postulate' in his translations. Of course you've never read the Elements (by your own admission) so you wouldn't know. And of course you've never studied the Works of Archimedes which would be way over your head.

I will answer at most one question on this issue before we continue.

[burs...@gmail.com]

My wrong, it rather shows that [0,1] is not well ordered
by the usual <. Hm. The criteria is stronger, every non-empty
subset of X has a least element under the ordering

This would speak for a global well ordering, but I guess
the problem is size, we would need to have W subset VxV.
So the Zermelo theorem just says W subset XxX.

In contrast to Axiom of global choice
https://en.wikipedia.org/wiki/Axiom_of_global_choice

(Note also the reference to other set theories in the
above wiki Article, maybe Ross A. Finlayson wanted to
correct me into this direction)

[Ross A. Finlayson]

The existence of a well-ordering of the reals,
as from a discussion last year on sci.logic,
saw put together that well-ordering the reals
has them countable in their normal ordering.

This is EF or sweep, a well-ordering of (the
unit interval of) the real numbers.

Otherwise there would be a contradiction of
the gaplessness of the complete ordered field
or the denseness of the ordered field, with
usual modern foundations or "modern mathematics".

[burs...@gmail.com]

To not get into a reals discussion and since the
context is anyway WMs Hessenberg (III) where there
is a funny statement about bijection N <-> Q

I avoided reals anyway, I wrote I hour ago:

> That there isn't a global wellordering can be easily seen
> intuitively. Take the interval [0,1] of rationals, call this
> set X_1. Assume it is well ordered by the usual <.

You see in the above I wrote rationals, no reals.

[Ross A. Finlayson]

Yes, I noticed, and changed the context to reflect
instead about the properties of the unit interval
of reals, instead of the ordered fields' rationals
that fall in the unit interval.

This was to make a point that there's more than one
set-theoretical model of the reals.

[WM]

Am Samstag, 31. Dezember 2016 17:16:57 UTC+1 schrieb burs...@gmail.com:

> Am Samstag, 31. Dezember 2016 16:36:03 UTC+1 schrieb WM:
> > Wrong. Zermelo has proved that every set can be well-ordered
> > - not that every set has a well-order. [E. Zermelo: "Beweis,
> > daß jede Menge wohlgeordnet werden kann", Math. Ann. 59 (1904) 514-516]
>
> Yes in German "werden kann" and not in German "ist".

After it has been well-ordered it is well ordered. In particular every sequence is well-ordered and every permutation that is constructed from it by transpositions.

Regards, WM

[WM]

Am Samstag, 31. Dezember 2016 20:00:06 UTC+1 schrieb burs...@gmail.com:
> To not get into a reals discussion and since the
> context is anyway WMs Hessenberg (III) where there
> is a funny statement about bijection N <-> Q
>
> I avoided reals anyway, I wrote I hour ago:
>
> > That there isn't a global wellordering can be easily seen
> > intuitively. Take the interval [0,1] of rationals, call this
> > set X_1. Assume it is well ordered by the usual <.

That there isn't the Hessenberg set can be easily seen. Assume it would exist in a countable model. Like every model of ZF, without exception, also this model contains all natural natural numbers of the big model "we are sitting in". Therefore also the Hessenberg set would exist when the countable model is observed from outside. Then the countable model would appear uncountable even from outside.

>
> You see in the above I wrote rationals, no reals.

Hessenberg claims the set of all natural numbers n such that n is not an element of f(n). This set is as real as the well-ordering of all rational numbers by magnitude.

Regards, WM

[Ross A. Finlayson]

It is another example of the difference between
the pair-wise and the total, and if you would
address it reasonably it's in a framework about
what is called the "transfer principle" as about
that which applies to each applies to all (or not).

This is where a permutation generator over an
infinite sequence via transpositions needs to
start from all these infinitely many distinct
seeds of a sequence to so exhaust or have
written each permutation.

Then, it is a simpler problem with this extra work,
than the pointless flailing you arrive at in your
pair-wise ignorance.

Sisyphus is a myth of a task that is always undone,
here instead you need that he arrives at the top.

[John Gabriel]

Finlayson's two cents... Sheesh, what an absolute idiot.

[burs...@gmail.com]

Am Sonntag, 1. Januar 2017 19:24:32 UTC+1 schrieb John Gabriel:

By absolute idiot you are talking about you and your
new "calculus". Don't know how piecewise functions work?

[shio...@googlemail.com]

On Saturday, 31 December 2016 06:58:55 UTC+1, John Gabriel wrote:
> On Friday, 30 December 2016 17:27:49 UTC-8, jenga...@gmail.com wrote:
> > On Saturday, 31 December 2016 02:09:06 UTC+1, genm...@gmail.com wrote:
> > > On Thursday, 29 December 2016 20:46:52 UTC-8, shio...@googlemail.com wrote:
> > >
> > > <excrement deleted>
> > >
> > > The following link:
> > >
> > > http://mathworld.wolfram.com/First-OrderLogic.html
> > >
> > > states:
> > >
> > > -----------------------------------------------
> > > The set of terms of first-order logic (also known as first-order predicate calculus) is defined by the following rules:
> > >
> > > 1. A variable is a term.
> > > -----------------------------------------------
> > >
> > > There are more rules but I don't need any more than the very first one for my demonstration.
> > >
> > > Now I am going to demonstrate what a baboon is this idiot shio...@googlemail.com.
> > >
> > > So the imbecile asks me to define a "magnitude" using first order systems.
> > >
> > > A magnitude is a very primitive idea. However, the very first rule requires that a "variable" and a "term" are both well defined.
> > >
> > > In order for a variable to be well defined, those objects known as "numbers" must have been established. What our moron doesn't realise, is that in my article https://www.linkedin.com/pulse/how-we-got-numbers-john-gabriel-1?trk=prof-post I was deriving the number concept. It would be circular and idiotic to assume it exists by trying to define it using FO logic.
> > >
> > > As for the word "term", it is very ambiguous. Also it is nowhere defined in FO logic what it means to be a term.
> > >
> > > The BIG STUPID - always stupid
> >
> > Oh wow johnny, you really didn't know what FO is XD
> >
> > You...do know that a variable in logic has nothing to do with numbers...right?
>
> Bullshit. It needn't have anything to do with numbers. But in this case it does.
>
> <too much excrement ignored>

You didnt ignore, you simply cannot follow me anymore because you are no actual mathematician XD

Admit it please: You are unable to do linear algebra, you do not know what logic even is and....really what is left johnnyboy?

You cannot even define the words you use XD

[shio...@googlemail.com]

On Saturday, 31 December 2016 07:11:23 UTC+1, Julio Di Egidio wrote:
> On Saturday, December 31, 2016 at 12:20:28 AM UTC+1, shio...@googlemail.com wrote:
> > ""
> > > Courtesy Wikipedia, << The result implies that first-order theories are unable
> > > to control the cardinality of their infinite models >>. I am not an expert,
> > > so you might even be right that it is not Loewenheim–Skolem that I should be
> > > invoking, OTOH that there exist perfectly good non-standard models of first-
> > > order Peano arithmetic is pretty well know.
> > > ""
> >
> > But that is not what you said
>
> Oh yes, it is.
>
> Then I even doubt you have a degree in mathematics, given the many and
> repeated errors in your exposition starting with the very language, not
> to mention the fact that you won't answer questions. Either you are a
> spammer or you are very confused, maybe both.
>
> Julio

It is not what you have said. I explained to you in detail why your statement was wrong, in any case.

You however cannot tell me what 'errors' i have made.

I answered every question you gave me.

And unlike you, i understand what löwenheim skolem implies.
So, really, what credibility do you have?

So far you just spouted phrases without meaning and ran away like a coward when i asked you to clarify your statements.
Pathetic XD

[Ross A. Finlayson]

Well I passed linear algebra
and have quite a grip on logic
and it looks we have a bit different idea
of what "fundamental" is and
what cardinals are and what they aren't.

Whether "infinite sets are equivalent" or
"ZF is inconsistent", eventually, about that
restrictions of comprehension are not true axioms
but simply conditions on objects within a larger theory,
with objects that are all that they are,
your schoolbook reckoning of the definitions,
and there's nothing wrong with that,
isn't so necessarily compelling
to a fundamental point of view.


We, and here "we" is people with quite a strong
and current mathematical education, aren't so
much interested in what we already know, as,
the truth, whatever it may well be. Now,
reasoning should tell you that there are
certain properties of a theory as of an
inclusive theory of everything for it to
be properly a foundation, with all that
is necessary, for mathematics, that it's
_the_ foundation. So, you should know ZF
is at best incomplete, then that there's a
true foundation under it if it's at best
consistent, of which it is but a fragment,
then for that being mathematical truth and
of interest to us.

We're all pretty much familiar with pretty
much your entire curriculum, so, what's of
interest is the news, as it were, not,
something we already know (and not something
we already know is not).

[shio...@googlemail.com]

""
> So I have an issue with your very first statement. It is FALSE that any approach to mathematics requires axioms. In fact, you probably don't know this, but there are no axioms in Euclid's Elements, The Works of Archimedes or any other Ancient Greek texts (which I have studied in the Ancient Greek). The word axiom did not even exist when those documents were written."""

But they did implicitly use axioms.
Elseways they could not have done anything.
For example, the greeks most certainly thought that 1+1 equals 2.
That however is already an assumption without an axiom and cannot be verified, because you do not even know whether '1' exists.

They would at least have to start out with 'the natural numbers exist'.
They might not have formulated them, but formally, they need it.

You would have to explain me elseways why the number 1 exists and how it is constructed.

The same goes for greek geometry. They wouldnt even know that a thing like the triangle does exist.

So yes, you can do math without knowing anything about set theory, but you have to implicitly assume things in that case and that exactly is the meaning of an axiom system: Things we can assume without proving them.

At least the greeks never proved that 1 exists, because they didnt think they had to.

[John Gabriel]

On Sunday, 1 January 2017 17:52:08 UTC-8, shio...@googlemail.com wrote:
> ""
> > So I have an issue with your very first statement. It is FALSE that any approach to mathematics requires axioms. In fact, you probably don't know this, but there are no axioms in Euclid's Elements, The Works of Archimedes or any other Ancient Greek texts (which I have studied in the Ancient Greek). The word axiom did not even exist when those documents were written."""
>
> But they did implicitly use axioms.

No shithead. They didn't use axioms either implicitly or explicitly.

You are sounding more and more like the idiot Dan Christensen. Is this just another of the douchebag's aliases? I think so.

> Elseways they could not have done anything.
> For example, the greeks most certainly thought that 1+1 equals 2.

Yes. Must be Dan Christensen. You are hereby PLONKED for good you dumb bastard.

[Me]

On Monday, January 2, 2017 at 2:52:08 AM UTC+1, shio...@googlemail.com wrote:

> But they did implicitly use axioms.

Actually, Euclid used them explicitely (though he didn't call them axioms). See http://aleph0.clarku.edu/~djoyce/java/elements/bookI/bookI.html#guide

[John Gabriel]

Euclid couldn't have used them because whatever Euclid used was not an axiom. Everything Euclid used was provable.

David Joyce is an idiot.

[shio...@googlemail.com]

I dont know what plonked means, but as i see it, your post contains no argument.
Must be because you are a coward and cannot admit when you are wrong.
Oh well, not like i didnt know XD

[Me]

On Monday, January 2, 2017 at 3:43:42 AM UTC+1, John Gabriel wrote:
> On Sunday, 1 January 2017 18:40:23 UTC-8, Me wrote:
> > On Monday, January 2, 2017 at 2:52:08 AM UTC+1, shio...@googlemail.com wrote:
> >
> > > But they did implicitly use axioms.
> >
> > Actually, Euclid used them explicitely (though he didn't call them axioms).
> > See http://aleph0.clarku.edu/~djoyce/java/elements/bookI/bookI.html#guide
> >
> Euclid couldn't have used them because whatever Euclid used was not an axiom.

Bullshit.

"Book 1 contains Euclid's 10 axioms (5 named 'postulates'—including the parallel postulate—and 5 named 'common notions') ..."

Source: https://en.wikipedia.org/wiki/Euclid's_Elements#Euclid.27s_method_and_style_of_presentation

"Euclid of Alexandria authored the earliest extant axiomatic presentation of Euclidean geometry and number theory."

Source: https://en.wikipedia.org/wiki/Axiomatic_system#Axiomatic_method

> Everything Euclid used was provable.

Sure, it might be "provable" (from certain other axioms, that is). On the other hand:

"Following the definitions, postulates, and common notions, there are 48 propositions. Each of these propositions includes a statement followed by a proof of the statement. Each statement of the proof is logically justified by a definition, postulate, common notion, or an earlier proposition that has already been proven."

Source: http://aleph0.clarku.edu/~djoyce/java/elements/bookI/bookI.html#guide

> David Joyce is an idiot.

Sure...

[John Gabriel]

On Sunday, 1 January 2017 19:09:12 UTC-8, Me wrote:
> On Monday, January 2, 2017 at 3:43:42 AM UTC+1, John Gabriel wrote:
> > On Sunday, 1 January 2017 18:40:23 UTC-8, Me wrote:
> > > On Monday, January 2, 2017 at 2:52:08 AM UTC+1, shio...@googlemail.com wrote:
> > >
> > > > But they did implicitly use axioms.
> > >
> > > Actually, Euclid used them explicitely (though he didn't call them axioms).
> > > See http://aleph0.clarku.edu/~djoyce/java/elements/bookI/bookI.html#guide
> > >
> > Euclid couldn't have used them because whatever Euclid used was not an axiom.
>
> Bullshit.
>
> "Book 1 contains Euclid's 10 axioms (5 named 'postulates'—including the parallel postulate—and 5 named 'common notions') ..."

I rest my case. Anyone reading this will know you are an idiot.

They are not postulates or axioms and these words were not even in existence then. They were first introduced by Thomas Heath.

You are too stupid and insignificant for me to waste my time trying to convince you. Die in your ignorance fool! Chuckle.

[shio...@googlemail.com]

No actually euclids formulations are widely regarded as the first prototype of axioms.
MOst people who read it know that.

[burs...@gmail.com]

Maybe JG is mentioning Thomas Heath as he is an english
math historian. So giving his, JGs, aim of constructing
a BIG STUPID theory, it is logical to attack Thomas Heath.

But it might be that there are older math historians, maybe
in latin language, french language or german language, that
documented the use of poatulates, so that he won't be able

to ascribe axioms as a translation artefact of Thomas Heath.
Well as a start, to refute JG here, we might sift trough:

1) William Kneale, Martha Kneale, The Development of Logic
2) Bochen'ski, Formale Logic
3) What else...?

Maybe we find something in the bibliography. Anyway recommended
reading for JG, greatest mathematician of all time, since they
connect the points to modern logic.

[burs...@gmail.com]

Bingo:

Friedrich Ueberweg, 1826 - 1871
https://de.wikipedia.org/wiki/Friedrich_Ueberweg

I guess he is before

Thomsd Heath, 1861 - 1940
https://de.wikipedia.org/wiki/Thomas_Heath

I have even a copy of his "System der Logik",
grabbed it from a local Antiquariat (Klio Zurich or
another one) some years ago. Not yet sure what he
is saying about Greek Axioms, so far I used it
to collect dust.

[John Gabriel]

Idiot. What do you know? By your own admission you've never read it and you know more than Euclid? Bwaaa haaaa haaa.

> MOst people who read it know that.

Most people are unbelievably stupid. I am not "most people" you fucking idiot!
I know what I am talking about, unlike the BIG STUPID of which you are a faithful lackey.

Listen junior, you haven't been on here long enough otherwise you could have read what I have written in this regard on this forum. Heck, you can search for it if you are not lazy. An axiom is basically a definition that has stood the test of time.

[burs...@gmail.com]

> An axiom is basically a definition that has stood the test of time.

An axiom need not have the form of a definition.
What are you talking about bird brain John Gabriel birdbrain?

[WM]

Am Montag, 2. Januar 2017 02:52:08 UTC+1 schrieb shio...@googlemail.com:
> ""
> > So I have an issue with your very first statement. It is FALSE that any approach to mathematics requires axioms. In fact, you probably don't know this, but there are no axioms in Euclid's Elements, The Works of Archimedes or any other Ancient Greek texts (which I have studied in the Ancient Greek). The word axiom did not even exist when those documents were written."""
>
> But they did implicitly use axioms.
> Elseways they could not have done anything.
> For example, the greeks most certainly thought that 1+1 equals 2.

2 is defined so: If you take a rigid body and another rigid body, then you have two rigid bodies.

> That however is already an assumption without an axiom and cannot be verified, because you do not even know whether '1' exists.

Do you know at least whether you exist? If a thing exist, then 1 exists. How depraved must present mathematics be if it spoils innocent newbies by such a mess of nonsense. Do you know how ridiculous you behave? Here is more of your ilk:

To the question "what is 2 + 3" a French primary school pupil replied: "3 + 2, since addition is commutative". He did not know what the sum was equal to and could not even understand what he was asked about!

Another French pupil (quite rational, in my opinion) defined mathematics as follows: "there is a square, but that still has to be proved".

Judging by my teaching experience in France, the university students' idea of mathematics (even of those taught mathematics at the École Normale Supérieure - I feel sorry most of all for these obviously intelligent but deformed kids) is as poor as that of this pupil.

[V.I. Arnold: "On teaching mathematics" (1997), Mathematics in Palais de Découverte in Paris on 7 March 1997, Translated by A.V. Goryunov]
http://pauli.uni-muenster.de/~munsteg/arnold.html

> They would at least have to start out with 'the natural numbers exist'.
> They might not have formulated them, but formally, they need it.
>
> You would have to explain me elseways why the number 1 exists and how it is constructed.

Sorry noone will be able to explain anything to such a perverted brain.

>
> At least the greeks never proved that 1 exists, because they didnt think they had to.

No. They knew that only fools can think they have to prove 1 exists.

Regards, WM

[burs...@gmail.com]

Am Montag, 2. Januar 2017 17:46:38 UTC+1 schrieb WM:
> Do you know at least whether you exist?

You must use a special logic, if you need
to show the existence of a constant,

maybe this is also why you think there
are multiple {1}. In ZFC + FOL, you can

just use a constant symbol, every constant
symbol has a value. Of course you cannot

sit on your lazy ass, you need to do something
so that the symbols get their correct meaning.

One way, use Neumann Ordinals, and convention
0 = {}, 1 = {0}, 2 = {0,1}, etc..

These conventions will be part of the premisses,
so basically you work:

ZFC + Neumann |- 1 < 2

Lesson for any student: Always define your axioms
and conventions, before you go on drawing conclusions.

Later on we might see the advantage here:

R |/- exists x (x^2 = -1)

C |- exists x (x^2 = -1)

But since WM and JG have never teached a single
minute to some students, good grace, no harm was made.

[burs...@gmail.com]

Or with respected to the limited brain capacity
of WM and JG, lets make a simpler example:

N |/- exists x (x+1 = 0)

Z |- exists x(x+1 = 0)

[WM]

Am Montag, 2. Januar 2017 18:04:22 UTC+1 schrieb burs...@gmail.com:

> Am Montag, 2. Januar 2017 17:46:38 UTC+1 schrieb WM:
> > Do you know at least whether you exist?
>
> You must use a special logic, if you need
> to show the existence of a constant,

Only fools can believe that they have to prove the existence of 1.

>
> maybe this is also why you think there
> are multiple {1}. In ZFC + FOL, you can

Drop this silly topic. Have you found a counter-argument meanwhile to my proof that everything in the universe belongs to one and the same countable set because there is a surjection from the countable set of all rational spatio-temporal co-ordinates on the set of all events including all elements of mathematics ever mentioned, defined, or proved?

Nobody has yet found a contradiction. Try it.

Regards, WM

[WM]

Am Montag, 2. Januar 2017 18:04:22 UTC+1 schrieb burs...@gmail.com:

> Lesson for any student: Always define your axioms
> and conventions, before you go on drawing conclusions.

Does mathematics require axioms? Occasionally logicians inquire as to whether the current "Axioms" need to be changed further, or augmented. The more fundamental question – whether mathematics requires any Axioms – is not up for discussion. That would be like trying to get the high priests on the island of Okineyab to consider not whether the Divine Ompah's Holy Phoenix has twelve or thirteen colours in her tail (a fascinating question on which entire tomes have been written), but rather whether the Divine Ompah exists at all. Ask that question, and icy stares are what you have to expect, then it's off to the dungeons, mate, for a bit of retraining. Mathematics does not require "Axioms". The job of a pure mathematician is not to build some elaborate castle in the sky, and to proclaim that it stands up on the strength of some arbitrarily chosen assumptions. The job is to investigate the mathematical reality of the world in which we live. For this, no assumptions are necessary. Careful observation is necessary, clear definitions are necessary, and correct use of language and logic are necessary. But at no point does one need to start invoking the existence of objects or procedures that we cannot see, specify, or implement. The difficulty with the current reliance on "Axioms" arises from a grammatical confusion [...] People use the term "Axiom" when often they really mean definition. Thus the "axioms" of group theory are in fact just definitions. We say exactly what we mean by a group, that's all. [N.J. Wildberger: "Set Theory: Should You Believe?"]

> But since WM and JG have never teached a single
> minute to some students,

Here is one of my scripts:
"Die Geschichte des Unendlichen", 7th ed., Maro Augsburg (2011)
In words: seventh edition.

Your opinions in general are as wrong and provably as wrong as your statement here.

Regards, WM

[Harry Stoteles]

BTW, the constant semantics thing has nothing to do
with infinite vs finite, or countable vs uncountable,
the error happens also in the finite,

how many rio grande are there? Can you do it correctly,
what logic is need so that this fails?

ZFC |- {rio_grande} = {rio_grande}

Or even this fails, which doesn't in standard ZFC+FOL,
when rio_grande is a constant symbol?

ZFC |- |{rio_grande}| = 1

What logic or reformulation is needed? You have no clue
WM right? Not the fainthest idea of logic right? BTW I
think there is a typo:

> Maro Verlag, Augsburg (2011)

It should read:

Moron Verlag, Augsburg (2011)

[Ross A. Finlayson]

The universe contains itself.

It's fair to believe
that there's an origin,
but then it's everywhere,
and you are not it.

It's fair to consider the domain principle,
that there is a universe in the theory,
and clearly it would be it's own powerset,
that's called Cantor's paradox and is among
reasons why ZF is a platform, not a foundation,
because it so excludes the universe for its
inner consistency.

With physical objects as mathematical objects,
there are not finitely many or there would only
be a point, so there are infinitely many (and
not just unboundedly many). Then, the functions
between physical objects as mathematical objects,
are also physical objects, and all their combinations
is as the powerset. So, the universe of physical
objects as mathematical objects may offer a
counterexample to ZF as foundation, instead as
platform, but, not for the reasons you say.

You strike people as an embittered retro-finitist,
but your platform is clearly not a foundation.

That's about WM, then, then there are ZF adherents,
and they should know that ZF was a candidate for
foundations, and these days it's known there's
more to it than that, and mathematics still seeks
a proper foundation besides its various proper
platforms.

It's clear these days that a foundation for (all
of) mathematics would need fulfill all of what a
putative "Theory of Everything" would so fulfill.

[Ross A. Finlayson]

To be more fair to Wildberger, it seems that his
opinion is that the platform and its immediate,
local, inner consistency is the relevant neighborly
concern, basically as willing to eschew foundations
for simple tractable means. That may be suitable
for applications and as a platform, but it's not
foundations, and his demurral of needing a foundation
from inside the box is no proper "rejection" of
foundations.

Then as about axioms vis-a-vis definitions, as a matter
of semantics his definitions are either derived (from
other definitions) or axiomatic. Then about whether
a theory can do without axioms, or, "axiomlessly", is
quite another matter for that philosophy can derive
proper objects from nothing more than the very act
of establishing and maintaining reason. This is usually
as referenced from canon and the Hegelian dialectic.

[John Gabriel]

On Monday, 2 January 2017 08:46:38 UTC-8, WM wrote:
> Am Montag, 2. Januar 2017 02:52:08 UTC+1 schrieb shio...@googlemail.com:
> > ""
> > > So I have an issue with your very first statement. It is FALSE that any approach to mathematics requires axioms. In fact, you probably don't know this, but there are no axioms in Euclid's Elements, The Works of Archimedes or any other Ancient Greek texts (which I have studied in the Ancient Greek). The word axiom did not even exist when those documents were written."""
> >
> > But they did implicitly use axioms.
> > Elseways they could not have done anything.
> > For example, the greeks most certainly thought that 1+1 equals 2.
>
> 2 is defined so: If you take a rigid body and another rigid body, then you have two rigid bodies.

That is a set "theoretesque" definition, because you are presuming the knowledge of counting numbers.

S = {(rigid body 1), (marble)}

2 = n(S)

It measures nothing except to count how many objects.

However, the Euclidean derivation defines a number as the measure of a magnitude. The von Neumann derivation already assumes the existence of counting numbers. The perfect Euclidean derivation assumes nothing.

>
> > That however is already an assumption without an axiom and cannot be verified, because you do not even know whether '1' exists.

The definition of 1 by Euclid (Book VII) is vague:

A unit is that by virtue of which all things are one.

However, a unit is simply the ratio of a chosen magnitude for measure compared with itself.

https://www.linkedin.com/pulse/how-we-got-numbers-john-gabriel-1?trk=prof-post

>
> Do you know at least whether you exist? If a thing exist, then 1 exists. How depraved must present mathematics be if it spoils innocent newbies by such a mess of nonsense. Do you know how ridiculous you behave? Here is more of your ilk:
>
> To the question "what is 2 + 3" a French primary school pupil replied: "3 + 2, since addition is commutative". He did not know what the sum was equal to and could not even understand what he was asked about!

There is a joke about that in the US too. High school students graduate and can state the commutative and other properties, but can't do simple arithmetic without a calculator.

>
> Another French pupil (quite rational, in my opinion) defined mathematics as follows: "there is a square, but that still has to be proved".

Absurd. The symptom of a square is 4 points with any two points equally spaced from each other.

>
> Judging by my teaching experience in France, the university students' idea of mathematics (even of those taught mathematics at the École Normale Supérieure - I feel sorry most of all for these obviously intelligent but deformed kids) is as poor as that of this pupil.

We have a fine example of this here: Jean Pierre Messager (aka Python or YBM).

>
> [V.I. Arnold: "On teaching mathematics" (1997), Mathematics in Palais de Découverte in Paris on 7 March 1997, Translated by A.V. Goryunov]
> http://pauli.uni-muenster.de/~munsteg/arnold.html
>
> > They would at least have to start out with 'the natural numbers exist'.
> > They might not have formulated them, but formally, they need it.
> >
> > You would have to explain me elseways why the number 1 exists and how it is constructed.

I do this in my article.

>
> Sorry noone will be able to explain anything to such a perverted brain.
> >
> > At least the greeks never proved that 1 exists, because they didnt think they had to.

Actually they did. They started off with ratios of magnitudes. A ratio p:q is just a comparison of the magnitudes p and q. If we have p:p, then the comparison is that the magnitudes are equal. If we then choose p to be the measuring stick or mass or volume, etc, then p becomes the unit.

[John Gabriel]

On Monday, 2 January 2017 10:03:08 UTC-8, WM wrote:
> Am Montag, 2. Januar 2017 18:04:22 UTC+1 schrieb burs...@gmail.com:
>
>
> > Lesson for any student: Always define your axioms
> > and conventions, before you go on drawing conclusions.
>
> Does mathematics require axioms? Occasionally logicians inquire as to whether the current "Axioms" need to be changed further, or augmented. The more fundamental question – whether mathematics requires any Axioms – is not up for discussion. That would be like trying to get the high priests on the island of Okineyab to consider not whether the Divine Ompah's Holy Phoenix has twelve or thirteen colours in her tail (a fascinating question on which entire tomes have been written), but rather whether the Divine Ompah exists at all. Ask that question, and icy stares are what you have to expect, then it's off to the dungeons, mate, for a bit of retraining. Mathematics does not require "Axioms". The job of a pure mathematician is not to build some elaborate castle in the sky, and to proclaim that it stands up on the strength of some arbitrarily chosen assumptions. The job is to investigate the mathematical reality of the world in which we live. For this, no assumptions are necessary. Careful observation is necessary, clear definitions are necessary, and correct use of language and logic are necessary. But at no point does one need to start invoking the existence of objects or procedures that we cannot see, specify, or implement. The difficulty with the current reliance on "Axioms" arises from a grammatical confusion [...] People use the term "Axiom" when often they really mean definition. Thus the "axioms" of group theory are in fact just definitions. We say exactly what we mean by a group, that's all. [N.J. Wildberger: "Set Theory: Should You Believe?"]

Right. A well formed definition is as good as an axiom because it can be checked according to my method described here:

https://www.linkedin.com/pulse/what-does-mean-concept-well-defined-john-gabriel?trk=prof-post

>
> > But since WM and JG have never teached a single
> > minute to some students,
>
> Here is one of my scripts:
> "Die Geschichte des Unendlichen", 7th ed., Maro Augsburg (2011)
> In words: seventh edition.
>
> Your opinions in general are as wrong and provably as wrong as your statement here.

This troll will soon surpass Dan Christensen in terms of annoyance.

>
> Regards, WM

[John Gabriel]

On Monday, 2 January 2017 10:03:08 UTC-8, WM wrote:

Don't go too far WM, the most useful mathematics contains no axioms, only well formed definitions.

[Ross A. Finlayson]

Von Neumann builds ordinals, as a structure of sets,
suitable for counting in simple (and basic) structural
algorithms, but numbers first as the inductive set.

The integers as numbers have all the properties of all
the integer parts of all the numbers, in all their spaces.

Algebra's Z(+,*) the Zahlen are an image of the integers
in usual group theory settings, that 1 and 1/1 and 1.0
have the same integer value.

You can ignore JG because
what is called "1" is just congruency,
and you can ignore WM because
what is called "1" is just an other.

[John Gabriel]

Do you enjoy stating the obvious?

> suitable for counting in simple (and basic) structural
> algorithms, but numbers first as the inductive set.


>
> The integers as numbers have all the properties of all
> the integer parts of all the numbers, in all their spaces.
>
> Algebra's Z(+,*) the Zahlen are an image of the integers
> in usual group theory settings, that 1 and 1/1 and 1.0
> have the same integer value.
>
> You can ignore JG because
> what is called "1" is just congruency,
> and you can ignore WM because
> what is called "1" is just an other.

Von Neuman assumes that the counting numbers are already in place. Nothing you say can refute that you moron! It's not a valid derivation by any standards.

[Ross A. Finlayson]

Ordinals don't need counting numbers first,
and you can ignore saying "1" is just a count.
Von Neumann was careful and direct,
and mostly direct, and carefully,
he built a structure for ordinals as sets,
and they're the same anyway.

Platonists believe mathematics is discovered,
there are many ways to go about it.

Ordinals are cardinals are complements,
one will find the other,
"numbering" and "counting" are not the same thing,
it's a simplification where they are.

Making things simple doesn't always keep them simple,
it's OK for toys and methods and tools,
but the foundations need not that "luxury".

It's a lot to ask,
that the numbers are many and all thing at once,
then it's convenient that they are.

[shio...@googlemail.com]

""
> > No actually euclids formulations are widely regarded as the first prototype of axioms.
>
> Idiot. What do you know? By your own admission you've never read it and you know more than Euclid? Bwaaa haaaa haaa.
> ""

Yes, i do. Because you do not need to read the book of euclid to know what his postulations were.
I rather spend time in reading actual mathbooks and not antique texts that are outdated.

""
> Most people are unbelievably stupid. I am not "most people" you fucking idiot!
> I know what I am talking about, unlike the BIG STUPID of which you are a faithful lackey.""

You are not most people, but unbelievably stupid, that you are.

""
> Listen junior, you haven't been on here long enough otherwise you could have read what I have written in this regard on this forum. Heck, you can search for it if you are not lazy. An axiom is basically a definition that has stood the test of time.
""

No, it is not. That is your crackpot definition of that word, but there is no 'test of time' in math.
If you do not use an axiom, you will never be able to prove that the number 1 exist, because there is nothing you can test.
Everything you would like to talk about does not exist then.

Not a single variable, not a single number or ringelement, not even functions.