A question had been put in MathStackExchange: Why is ZFC called free of contradictions?

This question got some upvotes but 4 more downvotes whereas the answer by Noah Schweber that I will comment on got at least 13 upvotes.

In 3d euclidean geometry three points exist which do not lie on one and the same straight line. Supplementing Hilbert's axioms by the axiom "Three points always lie on a common straight line" makes this system contradictory.
        It is provably impossible to assign meaning to all elements of an uncountable set. So there exist at least two elements of every uncountable set which cannot be identified and distinguished and cannot reasonably be put in an order. But in consequence of the axiom of choice every set can be well-ordered. That means that two elements which cannot be identified, distinguished, and put in an order can be identified, distinguished, and put in an order.
        Why is ZFC nevertheless called free of contradictons?

NS: First of all, we can distinguish objects from each other, even if they're undefinable individually! Think about the real numbers. These are uncountable, so lots of real numbers are undefinable; however, any two real numbers can be distinguished from each other by saying which one is bigger.

WM: Every real number that you can compare with anything is defined (by your choice) and is thus definable. Undefinable real numbers cannot be put into any processing unit. How would you do so? Try to use an infinite digit sequence? Good luck! Every digit you may use belongs to a finite set whereas infinitely many will follow.

NS: The more fundamental problem, however, essentially boils down to the difference between definability without parameters and definability with parameters. A parameter is just some object, which is incorporated into a definition despite not being part of the standard language of set theory. E.g. if a is some set, then the formula
        "Ay (y in x <==> Ez (y in z & z in a))"
defines the union of a, Ua, using a itself. This isn't an outright definition of the union of aa in the usual sense, but it is a definition with parameters (namely, the parameter here is a).

WM: It is not a definition at all.

NS: Now we get to your question. A well-ordering of an uncountable set (say, |R) is itself not necessarily definable in any good sense

WM: Until here the answer is correct.

NS: and in fact we can prove that any well-ordering of RR must be really hard to define

WM: In fact it has been proven to be impossible.

NS: most elements of an uncountable collection are undefinable - so most ordinals are undefinable! So there's no reason to believe that we can get a genuine definition of a from this definition-with-parameters.

WM: Even until this point everything is true.

NS: How many definitions-with-parameters are there?

WM: Why is this admittedly insufficient topic elaborated further?

NS: Well, there's as many of these as there are parameters - that is, there are exactly as many definitions-with-parameters as there are objects in our universe, so we're not going to "run out." And indeed every object a is definable by the formula-with-parameters "x=a." This is of course a very odd notion of definability, but it's the one corresponding to the kind of definability you get from a well-ordering of a set.

WM: Very odd is an euphemism. It is not a definition.

NS: So the apparent contradiction is only coming from the conflation of the notion of definability without parameters and definability with parameters.

WM: Apparent contradiction? Now he gets truly wrong.

NS: There's a natural way you might try to get around this issue: argue that every element of a well-ordered set must be definable, since otherwise "the least undefinable element" defines an undefinable element of that set! (This is essentially the least interesting number paradox, and is similar to the Berry and Richard paradoxes.)

WM: Of course this is a contradiction, nit a "natural way to get around this issue".

NS: Where this breaks down is the interesting fact that "definable" is not definable: there is no way to write the formula "x is definable" in the language of set theory.

WM: Therefore we use mathematics and find that all words of all languages are a countable set. What words are real definitions and what words are nonsense is irrelevant since every subset of a countable set is a countable set.

NS: This is essentially due to Tarski's undefinability theorem.

WM: That is an attempt to veil the contradiction. It is irrelevant here because we cover all possible words in all possible languages.

WM: There are countably many words in every language, and there are countable many languages (in fact finitely many, because every language has to be stored in a finite domain of the real universe).

NS: And in fact - and very surprisingly - even though intuitively there must be lots of undefinable elements in any model of ZFC, this turns out to not necessarily be the case!

WM: Here a paper by Hamkins is quoted which entails the sentence: "if ZFC is consistent, then there are continuum many pointwise definable models of ZFC". One cannot honestly argue that this proves the definability of all elements.

WM: Of course you cannot take an undefinable elements and use it as parameter. First you would have to define it.

NS: That's simply false. You really should read a text on basic model theory before you claim to have found a contradiction in ZFC.

WM: Every language is countable. Otherwise you could not use it, i.e. have a list of words, and more important, you could not define the words as you cannot define all elements of any uncountable set. You have only finite strings of 0 and 1 in the internet any everywhere else. The relevant language contains only countably many words. Try to produce more. Fail. Model theory is irrelevant if it cannot sow this. You cannot compare undefinable real numbers because all decimal representations that you can use have a finite string of decimals. You cannot even think of them as individuals.

NS: You're conflating a language with a usable language - or, more precisely, you're making an ontological assumption that every mathematical object is "knowable" in some sense.

WM: Using usable language is the basic principle of mathematics and also of set theory. Every axiom and theorem of ZFC is expressed in usable language. Try to express some mathematics in a not usable language. Try to send something via internet in a not usable language. Fail.

NS There's no justification for this, though. If you take this as your starting point, then of course there are only countably many mathematical objects; but that's not a background assumption of mathematical practice. And in particular, it can't be turned into an actual contradiction from the ZFC axioms - all you can do is show that the ZFC axioms contradict your view of what the mathematical universe is.

WM: In mathematics we prove assertions. Therefore, take an undefinable element as a parameter. Show how you do it. Fail

NS: If you truly think you can produce a contradiction from the ZFC axioms, I encourage you to try to produce a computer-verified proof of 0=1 directly from the ZFC axioms.

WM: A proof will never convince hard boiled set theorists believing in undefinable entities of anything. Consider my first paragraph. Of course there can be not finitely definable straight lines in geometry so that always three points lie on one of them. No problem, since they are undefinable. But you can believe in their existence since no computer can falsify this existence if you accept it.

NS: [...] the OP advocates a philosophical position which might be called physicalism (I've not heard a specific name for this, although it's not too rare):

WM: Hopefully this position is upheld by all mathematicians.

NS: namely, that there is a connection between mathematical existence and physical reality in that mathematical objects can only be said to exist if they are "representable" in the physical universe.

WM:  It is not "physicalism" but simply rational to require that everything used in mathematics is definable. Everything else is theology. Computer-verified proofs do never establish undefinable results. Computers don't believe in theology. You can see your error when you try to take two undefinable real numbers. How would you "take" them and put them into a computer? Zermelo proved that "every set can be well-ordered". Try it. Fail

NS: The existence of uncountable sets is incompatible with the physicalist philosophy (at least barring wildly implausible new discoveries about the universe we live in). But this is completely uncontroversial.

WM: No it isn't at all. Even in the first part of this answer it was incidentally assumed that every real numbers cnan be taken, i.e., can be defined.

NS: So the OP has only argued for "ZFC is false" insofar as they have argued for "physicalism is true" - and they haven't done that at all.

WM: Remember what has been said about definitions with parameters. Now it seems that these non-things are the bricks where mathematics is based upon.

Regards, WM

WM lied as usual, he left out part of one answer, Here is this text and the complete answer of NS ( I fear he thinks
that the abbreviation
NS is very funny in the same wy as he thinks that "KZ" is funny abbrevation for "Kardinalzahl" in German.

>
> WM: In mathematics we prove assertions. Therefore, take an undefinable element as a parameter. Show how you do it. Fail
>

> NS: It's pretty clear at this point that you're more interested in pushing that point of view than in understanding exactly what's going on here at the level of actually rigorous mathematics, so I'm going to bow out now. I encourage you to read an introductory text in model theory, again. If you truly think you can produce a contradiction from the ZFC axioms, I encourage you to try to produce a computer-verified proof

  of $0=1$ directly from the ZFC axioms. Your failure in that regard will be instructive. – Noah Schweber If you truly

Am Donnerstag, 13. April 2017 16:06:15 UTC+2 schrieb pirx42:
> WM  left out part of one answer

Of course, otherwise the text wolud become boring.

Do you like very odd notions of definability and support unusable languages? If so why don't you post in one of them?

Regards, WM

On Thursday, 13 April 2017 08:31:39 UTC-5, WM  wrote:

> NS: First of all, we can distinguish objects from each other, even if they're undefinable individually! Think about the real numbers. These are uncountable, so lots of real numbers are undefinable; however, any two real numbers can be distinguished from each other by saying which one is bigger.

That right there has so many errors. I stopped reading after this.

It's impossible to distinguish objects from each other unless they are ***well defined***, not only defined in some whimsical fashion like the BIG STUPID does.

Of course the set of real numbers is uncountable because you can't count objects that don't exist.

Am Donnerstag, 13. April 2017 17:20:48 UTC+2 schrieb John Gabriel:
> On Thursday, 13 April 2017 08:31:39 UTC-5, WM  wrote:
>
> > NS: First of all, we can distinguish objects from each other, even if they're undefinable individually! Think about the real numbers. These are uncountable, so lots of real numbers are undefinable; however, any two real numbers can be distinguished from each other by saying which one is bigger.
>
> That right there has so many errors. I stopped reading after this.
>
> It's impossible to distinguish objects from each other unless they are ***well defined***, not only defined in some whimsical fashion like the BIG STUPID does.

I understand that nobody will take the trouble to read through this long discussion. Perhaps I should have picked only this point:

NS: This isn't an outright definition [...] in the usual sense, but it is a definition with parameters. [...] A well-ordering of an uncountable set (say, |R) is itself not necessarily definable in any good sense [...] most elements of an uncountable collection are undefinable - so most ordinals are undefinable! So there's no reason to believe that we can get a genuine definition [...] from this definition-with-parameters.

But then this "not genuine definition, not in any good sense, not outright definition" is taken as the cornerstones of mathematics.

Regards, WM

I dont think maori macrons are as blond as WM:
http://maori.typeit.org/

WM wrote:
> A question had been put in MathStackExchange: Why is ZFC called free of contradictions?

...

> Am Donnerstag, 13. April 2017 16:06:15 UTC+2 schrieb pirx42:
>> WM  left out part of one answer
>
> Of course, otherwise the text wolud become boring.

Of course, Herr Mueckenheim, we all know here your exemplary
honesty in quoting your interlocutors and opponents fairly.

This strict honesty of yours is the likely reason you didn't post
any link to the discussion page on Stack Exchange, or is it
plain shyness?

Anyway, here it is:

https://math.stackexchange.com/questions/2230794/why-is-zfc-called-free-of-contradictions

On Thursday, April 13, 2017 at 10:22:17 PM UTC+2, Python wrote:

> WM wrote:
>
> Of course, Herr Mueckenheim, we all know here your exemplary
> honesty in quoting your interlocutors and opponents fairly.
>
> This strict honesty of yours is the likely reason you didn't post
> any link to the discussion page on Stack Exchange, or is it
> plain shyness?
>
> Anyway, here it is:
>
> https://math.stackexchange.com/questions/2230794/why-is-zfc-called-free-of-contradictions

And, of course, we all know _who_ asked this idiotic question...

Herr Mueckenheim is definitely a very shy person.

I wanted to pick on this already. Basically the problem is that
the one who posted the question and gave the 3D geometry example
seems not to understand what contradiction free means.

For a system to have a contradiction, by Gödel completness,
it means to have no model at all. Because contradication means
that some statements in the system are not simultaneously satisfiable,
and hence we will never have a model.

On the other hand when a system has one model M1, and for an other
model M2, this model is not satisfied in the system, then it only
means, that the system selects some models, what we actually
want by a system.

Otherwise why would we use a system, than for anything else than
to encircle an intended model we have in mind? Its like blowing
smoke rings and look where they go. Of course this can also happen
when we use a background theory ZFC or some such, and

if we modify it by additional axioms. Its the same with a maori
macron, it changes the spelling of the underlying letter.
https://en.wikipedia.org/wiki/M%C4%81ori_language#Name

I doubt that an expert answer, like the one from Noah Schweber
makes any sense. The OP who posted the MSE question, obviously
is trolling basic notions, of introductory courses.

And then to camouflage his incompetence, he posts some academic
looking reference citation out of the blue, using creative notions
such as "identified" with introducing them sufficiently.

Corr.:
such as "identified" without introducing them sufficiently.
And without explaining how they should live in the system
itself or outside in the model theory?

WM <wolfgang....@hs-augsburg.de> writes:

> A question had been put in MathStackExchange: Why is ZFC called free
> of contradictions?
>
> This question got some upvotes but 4 more downvotes whereas the answer
> by Noah Schweber that I will comment on got at least 13 upvotes.
>
> In 3d euclidean geometry three points exist which do not lie on one
> and the same straight line. Supplementing Hilbert's axioms by the
> axiom "Three points always lie on a common straight line" makes this
> system contradictory.  

And a formal proof of the contradiction is easily exhibited using the
axioms of Euclidean geometry, just as Russell did with Frege's
version of set theory.

Yet when it is suggested that WM that he should follow this excellent
plan with his simple proofs of contradiction in ZF, by actually using
the axioms of ZF, he tells us that he will not dirty his hands.

Enough said.


>
> Regards, WM

--
Alan Smaill

There are a few things that work differently in 3D than in 2D.

For example in 2D:
    Not every graph can be drawn without vertex crossing
    each other, this is only possible for the subclass of
    so called planar graphs.

While in 3D:
    I guess one might try to send each edge i, to the
    coordinates (i, i^2, i^3), and there is no crossing,
    for any graph. Correct? This is folkore right? Or is
    there a reference somewhere. (*)

On the other hand what WM probably wanted to cite was:
"Of any three points on a straight line there exists no
more than one that lies between the other two"
Which I guess is true in 2D and 3D, right?

(*)
Which would simplify the routing problem tremedously,
although causing a 3D too much space usage problem:
https://en.wikipedia.org/wiki/Routing_%28electronic_design_automation%29

On Thursday, 13 April 2017 12:58:03 UTC-5, WM  wrote:
> Am Donnerstag, 13. April 2017 17:20:48 UTC+2 schrieb John Gabriel:
> > On Thursday, 13 April 2017 08:31:39 UTC-5, WM  wrote:
> >
> > > NS: First of all, we can distinguish objects from each other, even if they're undefinable individually! Think about the real numbers. These are uncountable, so lots of real numbers are undefinable; however, any two real numbers can be distinguished from each other by saying which one is bigger.
> >
> > That right there has so many errors. I stopped reading after this.
> >
> > It's impossible to distinguish objects from each other unless they are ***well defined***, not only defined in some whimsical fashion like the BIG STUPID does.
>
> I understand that nobody will take the trouble to read through this long discussion.

You are wasting time with the morons who frequent this trash heap. It's best just to keep posting and not responding to the idiots. You cannot convince religious fundamentalists. I also doubt that many new visitors come here.

But I keep posting to prevent these morons from perverting the truth and continuing to spread their syphilitic ideas. If I succeed in discouraging others from visiting this trashy site, then at least I have saved a them from the infection of terminal stupidity.

We would need to look at the lines induced by two
edges i1 and i2, in vector coordinates:

    Line from edge i1 to edge i2 is a line from
    point (i1,i1^2,i1^3) to point (i2,i2^2,i2^3)

https://math.stackexchange.com/questions/404440/what-is-the-equation-for-a-3d-line

And then show that an intersection of two such lines
is impossible for different edge pairs. Which is
related to matrices and ranks.

He He

Happy easter, go figure this out...who wants a cookie?

Plus its not about lines, but line segments, you
have also constraints. So in paramterized representation:

   x = a*t + b

There is also a constraint such as 0 <= t <= 1.

On Friday, April 14, 2017 at 10:29:21 AM UTC+2, John Gabriel wrote:

> [...] If I succeed in discouraging others from visiting this trashy site,

> then at least I have saved a them from the infection of terminal stupidity.

Yeah, you have proven to be a complete asshole full of shit. Obviously an antisocial psychopath.

Am Donnerstag, 13. April 2017 23:39:05 UTC+2 schrieb burs...@gmail.com:

> For a system to have a contradiction, by Gödel completness,
> it means to have no model at all.

ZFC has no model at all. But it is sufficient already to have no *countable* model. Skolem proved his theorem: "Every proposition in normal form either is a contradiction or is already satisfiable in a finite or denumerably infinite  domain."

A lot of irrelevant text deleted. Please explain why a model containing omega can be countable "from outside". Of course I know that you can't.

Regards, WM

Am Donnerstag, 13. April 2017 23:47:51 UTC+2 schrieb burs...@gmail.com:

> I doubt that an expert answer, like the one from Noah Schweber
> makes any sense.

It does not make any sense to first claim that definition with parameters isn't an outright definition [...] in the usual sense and that a well-ordering of an uncountable set is itself not necessarily definable in any good sense so that most elements of an uncountable collection are undefinable and most ordinals are undefinable and there's no reason to believe that we can get a genuine definition [...] from this definition-with-parameters.

And then this mess is taken as the cornerstones of mathematics.

Experts of that kind and their adorers are abusers mathematics.

Regards, WM

Am Freitag, 14. April 2017 08:30:06 UTC+2 schrieb Alan Smaill:
> WM <wolfgang....@hs-augsburg.de> writes:

> > In 3d euclidean geometry three points exist which do not lie on one
> > and the same straight line. Supplementing Hilbert's axioms by the
> > axiom "Three points always lie on a common straight line" makes this
> > system contradictory.  
>
> And a formal proof of the contradiction is easily exhibited using the
> axioms of Euclidean geometry,

No. You forget that there could be undefinable straight lines L or such only definable with parameters L = a, just like the ordinals in set theory. Then there is no contradiction provable.

Regards, WM

Sounds pretty much complete nonsense.

WM schrieb:

Because you used skolem paradox/löwenheim skolem downward
with a countable language?

If for example you don't use a countable language assumption,
you get other results.

Skolem paradix/löwenheim skolem downward is derived from
some henkin completness theorem,

which uses language based term models.

Garbage in, garbage out.

WM schrieb:

Wrong.

The axioms prove that that there are three points not on a straight
line.  Definability is irrelevant.
From the added statement of the negation of the formalised claim,
there is an immediate contradiction.

The axioms do not care about the models or cardinality.

And, by the way, an ordinal like omega + omega has no need
of undefined entities, and what is more has a fine characterisation
in terms of potential infinity.

He is also ignorant of the fact that there are no axioms in Euclidean geometry:

https://www.youtube.com/watch?v=kzcVmXVpk-g

https://www.youtube.com/watch?v=ARzchjd8eZg

>
> Regards, WM

On Friday, April 14, 2017 at 3:36:50 PM UTC+2, John Gabriel wrote:

> He is also ignorant of the [brain fart] that there are no axioms in Euclidean
> geometry ...

That's fine. See https://en.wikipedia.org/wiki/Euclidean_geometry#Axioms

You ignorant fucking moron. Stupid as always. How many times must I tell you that Wikipedia is written by those who are intellectually inferior to me? Moron.

Well, just this once. I'll respond.

Wikipedia Moronica: "Let the following be postulated":

That line, O you infinitely stupid baboon, does not appear in the Greek text.  The only thing that precedes those requirements (not axioms or postulates) is the word 'oroi', which means "definitions".

Say I anal wart, did you even bother to read my article or watch the videos?

The answers to all your NULL objections are contained therein. You will get no further response because you are too stupid.

"To draw a straight line from any point to any point."
"To produce [extend] a finite straight line continuously in a straight line."
"To describe a circle with any centre and distance [radius]."
"That all right angles are equal to one another."
The parallel postulate: "That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles."

On Friday, April 14, 2017 at 5:49:11 PM UTC+2, John Gabriel wrote:

> Wikipedia  [...]: "Let the following be postulated":

I guess this refers to the word "Ηιτήσθω".

> The only thing that precedes those requirements [...] is

> the word 'oroi', which means "definitions".

Not so, it's the word "aitemata"

See: http://farside.ph.utexas.edu/Books/Euclid/Elements.pdf

"The aitemata are the things demanded, the postulates."
http://mysite.du.edu/~etuttle/classics/nugreek/lesson17.htm

"Each postulate is an axiom—which means a statement which is accepted without proof— specific to the subject matter, in this case, plane geometry."
http://aleph0.clarku.edu/~djoyce/elements/bookI/bookI.html#guide

Am Freitag, 14. April 2017 14:35:05 UTC+2 schrieb Alan Smaill:
> WM <wolfgang....@hs-augsburg.de> writes:
>
> > Am Freitag, 14. April 2017 08:30:06 UTC+2 schrieb Alan Smaill:
> >> WM <wolfgang....@hs-augsburg.de> writes:
> >
> >> > In 3d euclidean geometry three points exist which do not lie on one
> >> > and the same straight line. Supplementing Hilbert's axioms by the
> >> > axiom "Three points always lie on a common straight line" makes this
> >> > system contradictory.  
> >>
> >> And a formal proof of the contradiction is easily exhibited using the
> >> axioms of Euclidean geometry,
> >
> > No. You forget that there could be undefinable straight lines L or
> > such only definable with parameters L = a, just like the ordinals in
> > set theory. Then there is no contradiction provable.
>
> Wrong.
>
> The axioms prove that that there are three points not on a straight
> line.

That is impossible if most straight lines have no definition other than "with parameters".

>  Definability is irrelevant.
Obviously wrong.

> From the added statement of the negation of the formalised claim,
> there is an immediate contradiction.

Same holds for real numbers: In mathematics they are finitel definable or not existing.
>
> The axioms do not care about the models or cardinality.

Correct. Therefore every model of ZFC is as uncountable as the model we are sitting in.
>
> And, by the way, an ordinal like omega + omega has no need
> of undefined entities,

Of course not. Why should it? Undefined entities are alomost all real numbers if our universe i9s uncountable.

> and what is more has a fine characterisation
> in terms of potential infinity.

No. Remember Noah Schweber: The existence of uncountable sets is incompatible with the physicalist philosophy ... But this is completely uncontroversial.

Regards, WM

Am Freitag, 14. April 2017 13:44:13 UTC+2 schrieb j4n bur53:
> Because you used skolem paradox/löwenheim skolem downward
> with a countable language?
>
> If for example you don't use a countable language assumption,
> you get other results.

Of course, complete nonsense. Languages are, by definition, based on lists of words. Lists are countable.

Try to use an uncountable language. Fail.

Regards, WM

Am Freitag, 14. April 2017 13:40:22 UTC+2 schrieb j4n bur53:

> WM schrieb:

> > No. You forget that there could be undefinable straight lines L or such only definable with parameters L = a, just like the ordinals in set theory. Then there is no contradiction provable.

> Sounds pretty much complete nonsense.

It is complete nonsense. Just a "definition with parameters".

Regards, WM

Am Freitag, 14. April 2017 20:02:54 UTC+2 schrieb Ralf Bader:

> WM wrote:

> >> The axioms prove that that there are three points not on a straight
> >> line.
> >
> > That is impossible if most straight lines have no definition other than
> > "with parameters".
>
> This is very well possible. In Hilbert's "Grundlagen der Geometrie", 11th
> ed., Axiom I.3 says: "Auf einer Geraden gibt es stets wenigstens zwei
> Punkte. Es gibt wenigstens drei Punkte, die nicht auf einer Geraden liegen."
> That is, "On a line, there are always at least two points. There are at
> least three points which do not lie on a single line."

That concerns finitely defined straight lines.

>
> OK, so the assertion "The axioms prove that that there are three points not
> on a straight line." might be considered wrong, as there is no proof
> required, because the assertion is just one of Hilbert's axioms.
>

If you consider only finitely defined real numbers, then their set is countable and all are in trichotomy with each other. Of course this is provable.
The assumption of uncountably many reals "defined by parameters" is in contradiction with the axiom of trichotomy. This contradiction is as clear as that with the three points on a straight line and Hilbert's axioms.

Try to get a clue why I devised that example.

Regards, WM

On Friday, 14 April 2017 11:59:57 UTC-5, Me  wrote:
> On Friday, April 14, 2017 at 5:49:11 PM UTC+2, John Gabriel wrote:
>
> > Wikipedia  [...]: "Let the following be postulated":
>
> I guess this refers to the word "Ηιτήσθω".

You guess wrong. There is no modern equivalent for the word Ηιτήσθω.

>
> > The only thing that precedes those requirements [...] is
> > the word 'oroi', which means "definitions".
>
> Not so, it's the word "aitemata"

It's aitnmata for the 5 requirements but oroi for the definitions.

>
> See: http://farside.ph.utexas.edu/Books/Euclid/Elements.pdf
>

No one understands the Elements as I do.

> "The aitemata are the things demanded, the postulates."
> http://mysite.du.edu/~etuttle/classics/nugreek/lesson17.htm
>
> "Each postulate is an axiom—which means a statement which is accepted without proof— specific to the subject matter, in this case, plane geometry."
> http://aleph0.clarku.edu/~djoyce/elements/bookI/bookI.html#guide

Wrong. Each aithma is a requirement or claim which does not solicit belief or axiom.

"Ηιτήσθω" was probably used the same way as "λεγω" which means "I say". "Ηιτήσθω" can mean "I claim" or "I suppose" or "I presume". The fact of the matter is that there is not a single Greek alive today who could tell you exactly what that word meant to the Ancient Greeks. It is never used in modern Greek. A modern Greek word is "απαιτώ" and it means "I claim".

While the word "Ηιτήσθω" is used only once in the text, this does not mean that the other language translations got it correct.

If anything, I got it correct because I am the first to prove that there are no axioms or postulates in Greek mathematics.

On Friday, April 14, 2017 at 10:23:25 PM UTC+2, John Gabriel wrote:

> "Ηιτήσθω" was probably used the same way as "λεγω" which means "I say". "Ηιτήσθω" can mean "I claim" or "I suppose" or "I presume". The fact of the matter is that there is not a single Greek alive today who could tell you exactly what that word meant to the Ancient Greeks. It is never used in modern Greek. A modern Greek word is "απαιτώ" and it means "I claim".
>
> While the word "Ηιτήσθω" is used only once in the text, this does not mean that the other language translations got it correct.

Completely agree with you. I'd say that in THIS CASE you really know what you are talking about. Thanx for sharing your thoughts.

How do you like this one:

"The Greek present perfect tense indicates a past action with present significance. Hence, the 3rd-person present perfect imperative 'Ηιτήσθω
could be translated as “let it be postulated”, in the sense “let it stand as postulated”, but not “let the postulate be now brought forward”. The
literal translation “let it have been postulated” sounds awkward in English, but more accurately captures the meaning of the Greek."

Source:
http://farside.ph.utexas.edu/Books/Euclid/Elements.pdf

From another source:

"The aithemata are the things demanded, the postulates. The third-person imperative has no expressed subject; it means 'let it be conceded that'"

http://mysite.du.edu/~etuttle/classics/nugreek/contents.htm

Ηιτήσθω is a first person verb, NOT a 3rd person present perfect imperative, as claimed by Heath and others.

There aren't many rules in Greek, but one rule that is consistent, is that verbs in the first person end in omega(ω). So there is no way, this can mean "Let it be postulated". If it meant "postulate", then it would mean "I postulate ...".  There is no "Let it ..." possible.

The link (nugreek) you provided states "it is requested or required from any point to draw a line to another point".

It might also mean: "I assume that a straight line can be drawn from any point to another point". But this assumption is not a passive assumption that rests on belief or axiom, any more than assuming there are road signs requires axioms. One does not need axioms or postulates to "assume" anything. While one can assume things without proof, this does not mean that proof does not exist.

It is preposterous to think even for one moment that Euclid would assume requirements that are not based on well-formed concepts. Everything that comes after these definitions is based on the same. The clarity and meaning were well understood. It is not possible for the subsequent text to have been correct if the Greek understanding was vague or somewhat based on belief or chance.

The Greeks were atheists by character and questioned everything. There is not a chance they would have left anything to belief.

So, whatever Ηιτήσθω meant exactly, it would have been declined as follows:

Ηιτήσθω      (I followed by verb)
Ηιτήσθεις    (You followed by verb)
Ηιτήσθει     (He, she or it followed by verb)
Ηιτήσθουμε   (We followed by verb)
Ηιτήσθετε    (You plural followed by verb)
Ηιτήσθουν    (They followed by verb)

The fact that I have defined it as demonstrated in my articles and videos is proof that the Ancient Greeks knew it. To persist in calling these postulates or axioms is asinine.

On Friday, 14 April 2017 15:51:37 UTC-5, Me  wrote:

> On Friday, April 14, 2017 at 10:23:25 PM UTC+2, John Gabriel wrote:

> > While the word "Ηιτήσθω" is used only once in the text, this does not mean that the other language translations got it correct.
>
> Completely agree with you. I'd say that in THIS CASE you really know what you are talking about. Thanx for sharing your thoughts.

In every case I really know what I am talking about and everyone else does not. Chuckle.

WM wrote:

> Am Freitag, 14. April 2017 20:02:54 UTC+2 schrieb Ralf Bader:
>> WM wrote:
>
>> >> The axioms prove that that there are three points not on a straight
>> >> line.
>> >
>> > That is impossible if most straight lines have no definition other than
>> > "with parameters".
>>
>> This is very well possible. In Hilbert's "Grundlagen der Geometrie", 11th
>> ed., Axiom I.3 says: "Auf einer Geraden gibt es stets wenigstens zwei
>> Punkte. Es gibt wenigstens drei Punkte, die nicht auf einer Geraden
>> liegen." That is, "On a line, there are always at least two points. There
>> are at least three points which do not lie on a single line."
>
> That concerns finitely defined straight lines.

Those axioms are about lines, not "finitely defined straight lines". That
you are probably too stupid to grasp this does not change facts.
In fact none of Hilbert's axioms allows to "define" any line. There is no
such thing as an empty set or the number one, as initial constants, from
which other constants can be defined by various procedures like taking the
power set or the successor number. There can not be, because the space of
euclidean geometry is homogeneous.

>> OK, so the assertion "The axioms prove that that there are three points
>> not on a straight line." might be considered wrong, as there is no proof
>> required, because the assertion is just one of Hilbert's axioms.
>>
> If you consider only finitely defined real numbers, then their set is
> countable and all are in trichotomy with each other. Of course this is
> provable.

Of course this is not provable because it is wrong.

> The assumption of uncountably many reals "defined by parameters"
> is in contradiction with the axiom of trichotomy. This contradiction is as
> clear as that with the three points on a straight line and Hilbert's
> axioms.
>
> Try to get a clue why I devised that example.

You devised that example because you are too stupid to grasp abstract
notions beyond "finitely defined" instances. And therefore, by principal
reeasons, you are too stupid for mathematics in general. This is a fact,
proven beyond any reasonable doubt. By the way, also physical space has a
lot of homogeneity. Insofar as you are provably too stupid to grasp this
concept you are also too stupid for physics.

Am Samstag, 15. April 2017 16:58:36 UTC+2 schrieb Ralf Bader:
> WM wrote:
>
> > Am Freitag, 14. April 2017 20:02:54 UTC+2 schrieb Ralf Bader:
> >> WM wrote:
> >
> >> >> The axioms prove that that there are three points not on a straight
> >> >> line.
> >> >
> >> > That is impossible if most straight lines have no definition other than
> >> > "with parameters".
> >>
> >> This is very well possible. In Hilbert's "Grundlagen der Geometrie", 11th
> >> ed., Axiom I.3 says: "Auf einer Geraden gibt es stets wenigstens zwei
> >> Punkte. Es gibt wenigstens drei Punkte, die nicht auf einer Geraden
> >> liegen." That is, "On a line, there are always at least two points. There
> >> are at least three points which do not lie on a single line."
> >
> > That concerns finitely defined straight lines.
>
> Those axioms are about lines, not "finitely defined straight lines".

That is the same as the evolution of real numbers. Before mathematics got perverted by matheologians, there were simply "real numbers".

Regards, WM

On Thursday, 13 April 2017 12:58:03 UTC-5, WM  wrote:
 
> NS: This isn't an outright definition [...] in the usual sense, but it is a definition with parameters. [...] A well-ordering of an uncountable set (say, |R) is itself not necessarily definable in any good sense [...] most elements of an uncountable collection are undefinable - so most ordinals are undefinable! So there's no reason to believe that we can get a genuine definition [...] from this definition-with-parameters.
>
> But then this "not genuine definition, not in any good sense, not outright definition" is taken as the cornerstones of mathematics.

Indeed. As the BIG STUPID has been brainwashed to believe:

All ""real numbers"" can be represented as strings of infinite digits.

Obvious delusions:

1. Unrefiable points on a mythical number line corresponding to "real numbers".

2. Definitions that rely on ***intuition*** rather than facts and evidence.

3. Definitions that defy the very simplest arithmetic:

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

and somehow give rise to "ultimate ratios".  

"Faith is the substance of things hoped for; the evidence of things unseen" - from the Bible. Chuckle.

Things unseen like unusable languages, undefinable elements, and more of those "non-things" (as Cantor called them) only in order to avoid withdrawal symptoms with the addicts of this nonsense.

Regards, WM

Lets make a very simple example for the Augsburg Crank
Institute, why we don't need existing things or even
identifiable things. An example from physics:

Ulam and others made models of a new rocket engine,
the idea is to detonate nuclear bombs, this was called
the Orion project:

https://en.wikipedia.org/wiki/Project_Orion_%28nuclear_propulsion%29

https://www.youtube.com/watch?v=-5IviadEChM

Do we need to ask Euclid before making such models?
Or what? Was this thing already in language? If yes,
as a success story or a failure, who will ever know?

WM put the buttle down. Your idea of a non-imaginative
ape like mathematician, to find a contradiction in ZFC,
will never work, thats not how its works.

On Sunday, April 16, 2017 at 9:51:26 PM UTC+2, burs...@gmail.com wrote:
>
> Lets make a very simple example for the Augsburg Crank

> Institute, why we don't need [...]

> identifiable things. An example from physics:

Moreover, what if uncountable many things actually existed (in reality/space-time)? Is there a *logical* reason which would prevent that? I don't think so.

Actually, that's the reason why we allow for such theories in math: there's no *logical* reason to exclude them from our considerations, it seems.

Am Sonntag, 16. April 2017 22:51:49 UTC+2 schrieb Me:
> On Sunday, April 16, 2017 at 9:51:26 PM UTC+2, burs...@gmail.com wrote:
> >
> > Lets make a very simple example for the Augsburg Crank
> > Institute, why we don't need [...]
> > identifiable things. An example from physics:
>
> Moreover, what if uncountable many things actually existed (in reality/space-time)? Is there a *logical* reason which would prevent that? I don't think so.

Nevertheless there is a reason. The smallest realizable distance is given by the wavelength of a photon containing all the energy of the accessible universe.

>
> Actually, that's the reason why we allow for such theories in math: there's no *logical* reason to exclude them from our considerations, it seems.

There is the reason that a language needs to be usable. But that is in contradiction with defining all elements of uncountable sets. A very instructive aspect underlining this has been added to the newest issue of
https://www.hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf
p. 303ff

Regards, WM

On Tuesday, April 18, 2017 at 2:46:32 AM UTC+2, David Bernier wrote:

> This is most unusual for JG.

It's a psychological problem. He's certainly no idiot, he just behaves like an idiot. :-(

Lookup αξιωματα .
https://el.wikipedia.org/wiki/Αξιώματα_της_Ευκλείδειας_γεωμετρίας

Für Anna
http://www.math.uoc.gr/~pamfilos/Summer2012/Pamfilos_Geometry.pdf

BTW: Pamfilos did a Strang LinAlg translation to Greek. Ho Ho

Bablefish http://www.babelfish.de/ translates this for me:

Αξιώματα της Ευκλείδειας γεωμετρίας = Axioms of Euclidean geometry

Am Dienstag, 18. April 2017 03:26:05 UTC+2 schrieb burs...@gmail.com:

:-|

There have been many editions of The Elements over the ages.

The Dane Heiberg used primarily a Greek manuscript discovered
by François peyrard in the Vatican Library in the early 1800's .

I googled:

J. L. heiberg euclid  manuscripts François peyrard

David Bernier

Um, no. You can't look up Ηιτήσθω  anywhere. There is no dictionary that has that word any longer. It's never used in modern Greek and not even found in Ancient Greek dictionaries.

The modern Greeks do not understand mathematics as well as I do. I am not really a pure Greek even though Greek is my first language.

There are no axioms or postulates in Greek mathematics.

https://drive.google.com/open?id=0B-mOEooW03iLRjVxZERCa2R5Tlk

On Wednesday, 19 April 2017 12:49:00 UTC-5, David Bernier  wrote:
> On 04/17/2017 09:05 PM, Me wrote:
> > On Tuesday, April 18, 2017 at 2:46:32 AM UTC+2, David Bernier wrote:
> >
> >> This is most unusual for JG.
> >
> > It's a psychological problem. He's certainly no idiot, he just behaves like an idiot. :-(
> >
>
> :-|
>
> There have been many editions of The Elements over the ages.

And no one has understood the Elements as well as I.

The Greek (Pamfilos) who wrote this:

http://users.math.uoc.gr/~pamfilos/Summer2012/Pamfilos_Geometry.pdf

is an idiot.

All one needs to do is draw a line (from between Pamfilo's eyes) separating the two sides of his skull and it becomes pretty clear this PhD is a πέος:

http://users.math.uoc.gr/~pamfilos/PagePics/mypc.jpg

The moron claims that the foundations were cleared up 2200 after Euclid. But what can one expect from someone who is a product of the BIG STUPID?

This is what you will notice if you look up the word in any dictionary:

Το λεξικό δεν βρήκε καμία λέξη.

Direct translation: The dictionary did not find any word.

Common language: No match found.

Chuckle. You unbelievably stupid morons.

Everybody was waiting for this reaction of you. It doesn't
astonish me that you piss on Greek people as well.

So bird brain John Gabriel birdbrains, go back into
the jungle where you came from, to your ape friends...