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§ 316 Does undefinable definability save matheology?
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WM
Aug 19, 2013, 9:31:17 AM8/19/13
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§ 316 Does undefinable definability save matheology?
On 13 June 2013 user albino (meanwhile deleted) asked in MathOverflow about set theory without the axiom of power set.
http://mathoverflow.net/questions/133597/what-would-remain-of-current-mathematics-without-axiom-of-power-set
I will report also the interesting subsequent discussion about definability.
The power set of every infinite set is uncountable. An infinite set (as an element of the power set) cannot be defined by writing the infinite sequence of its elements but only by a finite formula. By lexical ordering of finite formulas we see that the set of finite formulas is countable. So it is impossible to define all elements of the uncountable power set. [albino]
You say that the powerset of an infinite set is questionable because it must have some undefinable elements. You are presuming that the subsets of a set must all be distinguishable by you, or some entity whose only access to powersets is through formal language. But why is such an assumption warranted? What makes you think that a thing does not exist unless you can define it? Is existence a personal belief?
[Andrej Bauer]
I think that elements of a set must be distinguishable, as Cantor has put it. [albino]
{{I would say that the existence of personal beliefs is a personal belief. And I think that undefinable elements cannot be applied. How would you apply an element that you cannot define? What are elements good for that cannot be applied in mathematics? Matheology. But unfortunately I came to late to take part in this discussion}}
Also, you are confusing "distinguishable" with "distinguishable by a formula". – [Andrej Bauer]
{{He said so but refused to explain the difference. By the way, every distinction in mathematics occurs by a finite formula.}}
I am afraid I have not got the meaning of your sentence you are confusing "distinguishable" with "distinguishable by a formula". If you cannot get hold of a notion other than by a finite formula, how would you distinguish two of them without a finite formula? [albino]
You have focused on this deifnability issue, but that's just not crucial. I heartily recommend that you read Joel Hamkin's post that he linked to in the comments to the question. He explained very well why definability is a deceptive notion. – [Andrej Bauer]
{{I think that a person who distinguishes between "distinguishable" and "distinguishable by a formula" but refuses to explain this, is a deceptive person.}}
… Lastly, concerning your remarks about definability, I refer you as I mentioned in the comments to an answer I wrote to a similar proposal, which I believe show that naive treatment of the concept of definability is ultimately flawed. [Joel David Hamkins]
Logicians, I have learned, take some premises and obtain some conclusions. But they do not judge about truth or practical things. In my opinion "definability" is a practical notion. If I define something and others are able to understand what I have defined, than that something is definable (otherwise it may be undefinable or I am not good enough in defining). But with respect to numbers things are easy. If I say pi or e or 1/4, then these numbers are defined. And it is true, in my opinion, that not more than countably many numbers can be defined by finite strings of bits. [albino]
Albino, I'm not sure to which logicians you are referring with your first comment. Meanwhile, yes, your remarks on definability are the usual naive position on definability. If you ever find yourself inclined to mount a serious analysis of definability, however, then I would suggest that you talk more with logicians. {{Really? Why are modern logicians despised like parias by all kinds of scientists?}} In particular, I would point you toward the initial part of my paper on pointwise definable models of set theory (de.arxiv.org/abs/1105.4597), where we deal with the "Math Tea argument", which is essentially the argument you are advancing. [Joel David Hamkins]
I am sorry, but since Zermelo's “Beweis, daß jede Menge wohlgeordnet werden kann”, we know that we have to distinguish between logical proofs and real proofs. If you had proven that all numbers can be expressed with three digits, I would not believe you. And your proof is rather similar. So I know it is not a real proof. Nevertheless I read your paper up to Theorem 4. It reminds me of Zermelo's “If AC, then well-ordering is possible”. But I am not interested in that kind of logical conclusions but only whether I can do it. [albino]
Oh, I'm very sorry to hear that you aren't interested in logical proof or logical conclusions. I'll leave you alone, then, to undertake your own kind of proof activity. – [Joel David Hamkins]
I do accept logical proof! I accept the logical proof that it is impossible to define more than countably many objects including all numbers. This stands as solid as the proof that with three digits you cannot define more than 999 numbers. I even accept non-constructive proofs like Zermelo's, but not as deciding whether something can be constructed - as was Zermelo original intention. (Compare Fraenkel who said that hitherto nobody could well-order the reals.) To be short: If your proof is correct, then you have found a contradiction. [albino]
{{I don't think it is a good idea to ask logicians what can be done in reality. Zermelo was the first to make a fool out of himself, when insisting and "proving" by insisting that every set can be well-ordered, Hamkins will not be the last one. To put an axiom may be a good idea in order to find out what can be thought - but not what can be done.}}
Regards, WM
On 13 June 2013 user albino (meanwhile deleted) asked in MathOverflow about set theory without the axiom of power set.
http://mathoverflow.net/questions/133597/what-would-remain-of-current-mathematics-without-axiom-of-power-set
I will report also the interesting subsequent discussion about definability.
The power set of every infinite set is uncountable. An infinite set (as an element of the power set) cannot be defined by writing the infinite sequence of its elements but only by a finite formula. By lexical ordering of finite formulas we see that the set of finite formulas is countable. So it is impossible to define all elements of the uncountable power set. [albino]
You say that the powerset of an infinite set is questionable because it must have some undefinable elements. You are presuming that the subsets of a set must all be distinguishable by you, or some entity whose only access to powersets is through formal language. But why is such an assumption warranted? What makes you think that a thing does not exist unless you can define it? Is existence a personal belief?
[Andrej Bauer]
I think that elements of a set must be distinguishable, as Cantor has put it. [albino]
{{I would say that the existence of personal beliefs is a personal belief. And I think that undefinable elements cannot be applied. How would you apply an element that you cannot define? What are elements good for that cannot be applied in mathematics? Matheology. But unfortunately I came to late to take part in this discussion}}
Also, you are confusing "distinguishable" with "distinguishable by a formula". – [Andrej Bauer]
{{He said so but refused to explain the difference. By the way, every distinction in mathematics occurs by a finite formula.}}
I am afraid I have not got the meaning of your sentence you are confusing "distinguishable" with "distinguishable by a formula". If you cannot get hold of a notion other than by a finite formula, how would you distinguish two of them without a finite formula? [albino]
You have focused on this deifnability issue, but that's just not crucial. I heartily recommend that you read Joel Hamkin's post that he linked to in the comments to the question. He explained very well why definability is a deceptive notion. – [Andrej Bauer]
{{I think that a person who distinguishes between "distinguishable" and "distinguishable by a formula" but refuses to explain this, is a deceptive person.}}
… Lastly, concerning your remarks about definability, I refer you as I mentioned in the comments to an answer I wrote to a similar proposal, which I believe show that naive treatment of the concept of definability is ultimately flawed. [Joel David Hamkins]
Logicians, I have learned, take some premises and obtain some conclusions. But they do not judge about truth or practical things. In my opinion "definability" is a practical notion. If I define something and others are able to understand what I have defined, than that something is definable (otherwise it may be undefinable or I am not good enough in defining). But with respect to numbers things are easy. If I say pi or e or 1/4, then these numbers are defined. And it is true, in my opinion, that not more than countably many numbers can be defined by finite strings of bits. [albino]
Albino, I'm not sure to which logicians you are referring with your first comment. Meanwhile, yes, your remarks on definability are the usual naive position on definability. If you ever find yourself inclined to mount a serious analysis of definability, however, then I would suggest that you talk more with logicians. {{Really? Why are modern logicians despised like parias by all kinds of scientists?}} In particular, I would point you toward the initial part of my paper on pointwise definable models of set theory (de.arxiv.org/abs/1105.4597), where we deal with the "Math Tea argument", which is essentially the argument you are advancing. [Joel David Hamkins]
I am sorry, but since Zermelo's “Beweis, daß jede Menge wohlgeordnet werden kann”, we know that we have to distinguish between logical proofs and real proofs. If you had proven that all numbers can be expressed with three digits, I would not believe you. And your proof is rather similar. So I know it is not a real proof. Nevertheless I read your paper up to Theorem 4. It reminds me of Zermelo's “If AC, then well-ordering is possible”. But I am not interested in that kind of logical conclusions but only whether I can do it. [albino]
Oh, I'm very sorry to hear that you aren't interested in logical proof or logical conclusions. I'll leave you alone, then, to undertake your own kind of proof activity. – [Joel David Hamkins]
I do accept logical proof! I accept the logical proof that it is impossible to define more than countably many objects including all numbers. This stands as solid as the proof that with three digits you cannot define more than 999 numbers. I even accept non-constructive proofs like Zermelo's, but not as deciding whether something can be constructed - as was Zermelo original intention. (Compare Fraenkel who said that hitherto nobody could well-order the reals.) To be short: If your proof is correct, then you have found a contradiction. [albino]
{{I don't think it is a good idea to ask logicians what can be done in reality. Zermelo was the first to make a fool out of himself, when insisting and "proving" by insisting that every set can be well-ordered, Hamkins will not be the last one. To put an axiom may be a good idea in order to find out what can be thought - but not what can be done.}}
Regards, WM
S�hne des Lichts Br�der des Schattens
Aug 19, 2013, 4:29:41 PM8/19/13
to
"WM" <muec...@rz.fh-augsburg.de> wrote in message
news:773e5bb2-8f9d-4e89...@googlegroups.com...
� 316 Does undefinable definability save matheology?
<snip crap>
.......boring.......
Virgil
Aug 19, 2013, 7:15:51 PM8/19/13
to
In article <773e5bb2-8f9d-4e89...@googlegroups.com>,
Zermelo only argued that it followed from the axioms of ZFC.
Regards, WM
--
WM <muec...@rz.fh-augsburg.de> wrote:
> I don't think it is a good idea to ask logicians what can be done in
> reality. Zermelo was the first to make a fool out of himself, when insisting
> and "proving" by insisting that every set can be well-ordered
Actually, that is a gross misrepresentation.
> I don't think it is a good idea to ask logicians what can be done in
> reality. Zermelo was the first to make a fool out of himself, when insisting
> and "proving" by insisting that every set can be well-ordered
Zermelo only argued that it followed from the axioms of ZFC.
Regards, WM
--
WM
Aug 20, 2013, 7:57:22 AM8/20/13
to
The latter is correct, since the axiom of choice in fact is naturally true. The only unnatural thing is its application to uncountable sets, since there it would fail - if they existed.
Regards, WM
Virgil
Aug 20, 2013, 4:09:25 PM8/20/13
to
In article <557b8626-7b39-4f5d...@googlegroups.com>,
If the choice is to between th axiom of choice in its full splendor or
infinite sets, the vast majority of mathematicians will opt for infinite
sets.
But even today, WM opted for infinite sets in demanding existence of a
diagonal "matrix":
1
2 1
3 2 1
4 3 2 1
. . . . .
which clearly requires the existence of an infinite set of natuals,
and thus disrtoyes WM's wild weird world of WMytheology.
--
infinite sets, the vast majority of mathematicians will opt for infinite
sets.
But even today, WM opted for infinite sets in demanding existence of a
diagonal "matrix":
1
2 1
3 2 1
4 3 2 1
. . . . .
which clearly requires the existence of an infinite set of natuals,
and thus disrtoyes WM's wild weird world of WMytheology.
--
WM
Aug 21, 2013, 10:47:57 AM8/21/13
to
On Tuesday, 20 August 2013 22:09:25 UTC+2, Virgil wrote:
> In article <557b8626-7b39-4f5d...@googlegroups.com>,
>
> WM <muec...@rz.fh-augsburg.de> wrote:
>
>
>
> > On Tuesday, 20 August 2013 01:15:51 UTC+2, Virgil wrote:
>
> > > In article <773e5bb2-8f9d-4e89...@googlegroups.com>,
>
> > >
>
> > > WM <muec...@rz.fh-augsburg.de> wrote:
>
> > >
>
> > >
>
> > >
>
> > > > I don't think it is a good idea to ask logicians what can be done in
>
> > >
>
> > > > reality. Zermelo was the first to make a fool out of himself, when
>
> > > > insisting
>
> > >
>
> > > > and "proving" by insisting that every set can be well-ordered
>
> > >
>
> > >
>
> > >
>
> > > Actually, that is a gross misrepresentation.
>
> > >
>
> > >
>
> > >
>
> > > Zermelo only argued that it followed from the axioms of ZFC.
>
> >
>
> > Zermelo mainly argued in a very long section of his 1908-paper that the axiom
>
> > of choice is justified as a natural idea and that it has been used without
>
> > stating it explicitly by many mathematicians before he formally invented it.
>
> >
>
> > The latter is correct, since the axiom of choice in fact is naturally true.
>
> > The only unnatural thing is its application to uncountable sets, since there
>
> > it would fail - if they existed.
>
>
>
>
>
> If the choice is to between
admitting that you are very uninformed and not admitting that but insulting others, you will always choose the second alternative.
> In article <557b8626-7b39-4f5d...@googlegroups.com>,
>
> WM <muec...@rz.fh-augsburg.de> wrote:
>
>
>
> > On Tuesday, 20 August 2013 01:15:51 UTC+2, Virgil wrote:
>
> > > In article <773e5bb2-8f9d-4e89...@googlegroups.com>,
>
> > >
>
> > > WM <muec...@rz.fh-augsburg.de> wrote:
>
> > >
>
> > >
>
> > >
>
> > > > I don't think it is a good idea to ask logicians what can be done in
>
> > >
>
> > > > reality. Zermelo was the first to make a fool out of himself, when
>
> > > > insisting
>
> > >
>
> > > > and "proving" by insisting that every set can be well-ordered
>
> > >
>
> > >
>
> > >
>
> > > Actually, that is a gross misrepresentation.
>
> > >
>
> > >
>
> > >
>
> > > Zermelo only argued that it followed from the axioms of ZFC.
>
> >
>
> > Zermelo mainly argued in a very long section of his 1908-paper that the axiom
>
> > of choice is justified as a natural idea and that it has been used without
>
> > stating it explicitly by many mathematicians before he formally invented it.
>
> >
>
> > The latter is correct, since the axiom of choice in fact is naturally true.
>
> > The only unnatural thing is its application to uncountable sets, since there
>
> > it would fail - if they existed.
>
>
>
>
>
> If the choice is to between
>
>
> But even today, WM opted for infinite sets in demanding existence of a
>
> diagonal "matrix":
>
>
>
> 1
>
> 2 1
>
> 3 2 1
>
> 4 3 2 1
>
> . . . . .
>
>
>
I assumed the existence of an actually completed infinite set |N in order to inform every not actually completely stupid mathematician that such a set cannot exist without contradiction, namely violating inclusion monotony.
But some will remain uninformed because they are unable to grasp even this simplest variant of my proof that matheology is nonsense, that every Student in the first Semester can understand.
Regards, WM
Virgil
Aug 21, 2013, 4:38:17 PM8/21/13
to
In article <3dabd740-3fad-461b...@googlegroups.com>,
those teachers have the power of grading those students, but will think
for themselves when such power ceases.
--
WM <muec...@rz.fh-augsburg.de> wrote:
> I assumed the existence of an actually completed infinite set |N in order to
> inform every not actually completely stupid mathematician that such a set
> cannot exist without contradiction, namely violating inclusion monotony.
There is no such thing as "inclusion monotony" outside of
> I assumed the existence of an actually completed infinite set |N in order to
> inform every not actually completely stupid mathematician that such a set
> cannot exist without contradiction, namely violating inclusion monotony.
WM's wild weird world of WMytheology.
>
>
> But some will remain uninformed because they are unable to grasp even this
> simplest variant of my proof that matheology is nonsense, that every Student
> in the first Semester can understand.
Students will always claim to understand their teachers for as long as
> simplest variant of my proof that matheology is nonsense, that every Student
> in the first Semester can understand.
those teachers have the power of grading those students, but will think
for themselves when such power ceases.
--
WM
Aug 22, 2013, 10:34:06 AM8/22/13
to
On Wednesday, 21 August 2013 22:38:17 UTC+2, Virgil wrote:
> In article <3dabd740-3fad-461b...@googlegroups.com>,
>
> WM <muec...@rz.fh-augsburg.de> wrote:
>
>
>
> > I assumed the existence of an actually completed infinite set |N in order to
>
> > inform every not actually completely stupid mathematician that such a set
>
> > cannot exist without contradiction, namely violating inclusion monotony.
>
>
>
> There is no such thing as "inclusion monotony" outside of
Thank you for that answer. So in matheology my argument is invalid. I already suspected something like that. I argue only in a theory where inclusion monotony holds. (By the way, Google shows 510 results, not all from me.)
> In article <3dabd740-3fad-461b...@googlegroups.com>,
>
> WM <muec...@rz.fh-augsburg.de> wrote:
>
>
>
> > I assumed the existence of an actually completed infinite set |N in order to
>
> > inform every not actually completely stupid mathematician that such a set
>
> > cannot exist without contradiction, namely violating inclusion monotony.
>
>
>
> There is no such thing as "inclusion monotony" outside of
Regards, WM
Virgil
Aug 22, 2013, 3:43:51 PM8/22/13
to
In article <71d38911-2355-486d...@googlegroups.com>,
Also none of the first couple of pages of them were about anything
mathematical at all.
Most of them seemed to be about certain notions in electrical circuit.
design.
It there were any mathematical principle outside of Wolkenmuekenheim
known as "inclusion monotony" there would be at least 100,000 Google
hits.
And also WM should be able to define and explain his principle of
"inclusion monotony", which so far he he has been careful NOT to do.
--
mathematical at all.
Most of them seemed to be about certain notions in electrical circuit.
design.
It there were any mathematical principle outside of Wolkenmuekenheim
known as "inclusion monotony" there would be at least 100,000 Google
hits.
And also WM should be able to define and explain his principle of
"inclusion monotony", which so far he he has been careful NOT to do.
--
FredJeffries
Aug 22, 2013, 3:52:13 PM8/22/13
to
On Thursday, August 22, 2013 12:43:51 PM UTC-7, Virgil wrote:
>
> > > There is no such thing as "inclusion monotony" outside of
>
> > > There is no such thing as "inclusion monotony" outside of
> Also none of the first couple of pages of them were about anything
> mathematical at all.
http://books.google.com/books?id=8URaBg1palkC&pg=PA306&lpg=PA306
> mathematical at all.
Virgil
Aug 22, 2013, 6:04:35 PM8/22/13
to
In article <c1a36c8f-b3f3-4186...@googlegroups.com>,
It certainly does not seem relevant to WM's misuse of the term.
--
--
Message has been deleted
WM
Aug 23, 2013, 8:10:02 AM8/23/13
to
On Thursday, 22 August 2013 21:43:51 UTC+2, Virgil wrote:
>
> And also WM should be able to define and explain his principle of
>
> "inclusion monotony", which so far he he has been careful NOT to do.
>
> --
A sequence of sets is called inclusion monotonic, if and only if every set contains all elements that its predecessor contains.
>
> And also WM should be able to define and explain his principle of
>
> "inclusion monotony", which so far he he has been careful NOT to do.
>
> --
A sequence of sets is called strictly inclusion monotonic, if and only if every set conatins all elements that ist predecessor contains and at least one further element.
The sequence of FIS of |N is strictly inclusion monotonic.
Regards, WM
Virgil
Aug 23, 2013, 8:55:58 PM8/23/13
to
In article <5b57b3a6-75b5-49e9...@googlegroups.com>,
Those definitions of "inclusion monotonic" and "strictly inclusion
monotonic"do not define any PRINCIPLE of inclusion monotony, and
certainly do not prohibit a set from being the union of a strictly
monotonic sequence of its proper subsets, such as |N being the union of
all its FISONs.
Thus even in Wolkenmuekenheim the union of all FISONs is |N.
--
monotonic"do not define any PRINCIPLE of inclusion monotony, and
certainly do not prohibit a set from being the union of a strictly
monotonic sequence of its proper subsets, such as |N being the union of
all its FISONs.
Thus even in Wolkenmuekenheim the union of all FISONs is |N.
--
Virgil
Aug 23, 2013, 9:00:08 PM8/23/13
to
In article <ad054335-a319-4344...@googlegroups.com>,
Is "strongly" the same as "strictly", and if not, what is the differnce?
And while WM has defined "inclusion monotonic" he has not defined or
otherwise explained any PRINCIPLE of inclusion monotony.
So we still do not have any idea what he means by it.
--
WM <muec...@rz.fh-augsburg.de> wrote:
> On Thursday, 22 August 2013 21:43:51 UTC+2, Virgil wrote:
>
>
> >
> On Thursday, 22 August 2013 21:43:51 UTC+2, Virgil wrote:
>
>
> >
> > And also WM should be able to define and explain his principle of
> >
> > "inclusion monotony", which so far he he has been careful NOT to do.
> >
> > --
>
> >
> > "inclusion monotony", which so far he he has been careful NOT to do.
> >
> > --
>
> A sequence of sets is called inclusion monotonic, if and only if every set
> contains all elements that its predecessor contains.
>
> A sequence of sets is called strictly inclusion monotonic, if and only if
> every set conatins all elements that ist predecessor contains and at least
> one further element.
>
> The sequence of FIS of |N is strongly inclusion monotonic.
> contains all elements that its predecessor contains.
>
> A sequence of sets is called strictly inclusion monotonic, if and only if
> every set conatins all elements that ist predecessor contains and at least
> one further element.
>
Is "strongly" the same as "strictly", and if not, what is the differnce?
And while WM has defined "inclusion monotonic" he has not defined or
otherwise explained any PRINCIPLE of inclusion monotony.
So we still do not have any idea what he means by it.
--
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