Some axioms like the axiom of choice

[WM]

- Axiom of well-ordering: Every set can be well-ordered. (This axiom is not constructive. In most cases provably no set theoretic definition of a well-order can be found.)

- Axiom of three points on a line: Every triple of points belongs to a straight line. (But in most cases provably no geometrical construction can be given.)

- Axiom of ten even primes: There are 10 even prime numbers. (But provably no arithmetical method to find them is available.)

- Axiom of prime number triples: There is a second triple of prime numbers besides (3, 5, 7). (But provably this second triple is not arithmetically definable.)

- Axiom of meagre sum (AMS): There is a set of n different positive natural numbers with sum n*n/2. (This axiom is not constructive. Provably no such set can be found.)

- Axiom of ultimate mathematics simplification: All mathematical problems are solved by whatever I declare as the solution. (This axiom is guaranteed not less useful than the axiom of choice.)

Regards, WM

[mitch]

chuckle

Since you like to be such a critic, why not spend a few
paragraphs giving your affirmative views on the nature
of mathematics, the nature of logic, and what relation
might exist between them?

It is too easy to be a critic (and even easier to be
a troll).

mitch

[shio...@googlemail.com]

You didn't understand what an axiom is.
The point about choice is that choice cannot be disproven, thus we can assume it as true without coming to a contradiction.

However it is proveable that there are no even prime numbers except 2, so having this as an axiom would lead to a contradictory axiom system, which choice does NOT.

[WM]

Am Mittwoch, 11. Oktober 2017 18:04:43 UTC+2 schrieb shio...@googlemail.com:


> The point about choice is that choice cannot be disproven,

In fact it cannot be disproven since there is nothing uncountable. But if there was an uncountable set, then its elements could not be counted. Is that clear so far?

> thus we can assume it as true without coming to a contradiction.

A contradiction would exist for uncopuntable sets because to one cannot choose what cannot be counted.

>
> However it is proveable that there are no even prime numbers except 2, so having this as an axiom would lead to a contradictory axiom system, which choice does NOT.

It is provable that objects that have no material existence cannot be dealt with other than by labelling them. Alas there are only countably many labels.

Of course that is a contradiction as already Cantor recognized: "Infinite definitions" (that do not happen in finite time) are non-things (Undinge). If König's theorem was true, according to which all "finitely definable" real numbers form a set of cardinality 0, this would imply that the whole continuum was countable [G. Cantor, letter to D. Hilbert (8 Aug 1906)]

But set theory has blocked set theorists, even intelligent one, so strongly that they cannot dare to think of a contradiction and remain blind in this respect.

Reggards, WM

[WM]

Am Mittwoch, 11. Oktober 2017 17:49:52 UTC+2 schrieb mitch:


> Since you like to be such a critic, why not spend a few
> paragraphs giving your affirmative views on the nature
> of mathematics, the nature of logic, and what relation
> might exist between them?
>

Logic rules mathematics. Here is an important statement:
For every n in |N: Every n belongs to a finite initial segment and does neither require nor allow to consider the set |N as actually closed.

The quantification "For all n in |N" (although applicable to qualitative statements like all natural numbers are integers) should be avoided completely as it leads to acontradictions like:
For all n in |N: there is no "all" n in |N.

Correct is this:
For every n in |N: n belongs to a finite initial sequence and is followed by a (potentially) infinite sequence.

It is not necessary to understand more, but these facts should be grasped. Then there is no contradictory mathematics possible.

Regards, WM

[shio...@googlemail.com]

> > The point about choice is that choice cannot be disproven,
>
> In fact it cannot be disproven since there is nothing uncountable. But if there was an uncountable set, then its elements could not be counted. Is that clear so far?

There are uncountable sets, we have been over that. i doN#t think you will accept that, but it is not important for this matter.
What is proven is that choice does not contradict the other axioms.

>
> > thus we can assume it as true without coming to a contradiction.
>
> A contradiction would exist for uncopuntable sets because to one cannot choose what cannot be counted.

No. Simply no.
There are uncountable sets and they do not contradict with choice in any way.

> > However it is proveable that there are no even prime numbers except 2, so having this as an axiom would lead to a contradictory axiom system, which choice does NOT.
>
> It is provable that objects that have no material existence cannot be dealt with other than by labelling them. Alas there are only countably many labels.

Your statement is not a mathematical one, so you miss the point.
Fact is: Your other 'axioms' contradict the axiom system ZF, while choice does not.

> Of course that is a contradiction as already Cantor recognized: "Infinite definitions" (that do not happen in finite time) are non-things (Undinge). If König's theorem was true, according to which all "finitely definable" real numbers form a set of cardinality 0, this would imply that the whole continuum was countable [G. Cantor, letter to D. Hilbert (8 Aug 1906)]
>

Once again, quoting old works of mathematicians who usually use other definitions entirely is a moot point and does not get you anywhere.
Cantor does not agree with what you are saying.

[mitch]

On 10/11/2017 11:49 AM, WM wrote:
> Am Mittwoch, 11. Oktober 2017 17:49:52 UTC+2 schrieb mitch:
>
>
>> Since you like to be such a critic, why not spend a few
>> paragraphs giving your affirmative views on the nature
>> of mathematics, the nature of logic, and what relation
>> might exist between them?
>>
> Logic rules mathematics.

I see this differently. Either the two are
on equal footing, or, mathematics rules logic.

In the latter case, my view is based on concluding
that the arithmetization of mathematics arising from
the "crisis in geometry" is nonsensical. So the
statement could be restated as geometry rules logic.

> Here is an important statement:
> For every n in |N: Every n belongs to a finite initial segment and does neither require nor allow to consider the set |N as actually closed.

I agree to the import of this.

You are generally dismissive of the standard literature
and formalisms. But, there is a source in which you will
find what you are criticizing made explicit.

In Chapter 2, Section 3 of "Set Theory" by Drake, there
is a definition for well-founded relation in which a
second subformula explicitly assuming the existence of
the closure is included.

If one demands a witness for this existential statement,
as does first-order logic, then one is committed to
a completed infinity. But, there are other interpretations
for quantifiers. It is easy enough to think of an
epsilon-delta proof in terms of a game-theoretic
interpretation, although making this precise might not
be easy. This will never suffice for notions of truth
that resolve to the Aristotelian concept of individuals
as primary substance.

Aristotle, himself, held that truth ultimately resides
with perceptions. By characterizing mathematical logic
solely in terms of a syntax/semantics distinction,
philosophers and their logicians have made it difficult
for many mathematicians to accept their analyses.

In my opinion, the trust people place in mathematical
certainty arises from a common relationship to a
spatiotemporal experience and not a faith in "abstract
objects" floating outside of space and time with gods
and demons. If one's conception of truth eliminates
people from the account, one will have silliness.

> The quantification "For all n in |N" (although applicable to qualitative statements like all natural numbers are integers) should be avoided completely as it leads to acontradictions like:
> For all n in |N: there is no "all" n in |N.
>

I will leave this alone as it leads to what your
detractors refer to as "quantifier dyslexia".

I do see a play on words that I had not noticed
before.

> Correct is this:
> For every n in |N: n belongs to a finite initial sequence and is followed by a (potentially) infinite sequence.
>
> It is not necessary to understand more, but these facts should be grasped. Then there is no contradictory mathematics possible.
>

The problem you will always face here with regard
to your detractors is a characterization of potential
infinity.

Because I have no problem "trusting" topology, I am
more inclined to view natural numbers with Gauss as
a system of aliquot and aliquant parts. I can write
a theory on that basis. By contrast, both Skolem and
Brouwer would have me "believe" in natural numbers
outside of a theory and counted one at a time.

Thank you for a straightforward reply to my question.

mitch

[khongdo...@gmail.com]

On Wednesday, 11 October 2017 13:08:31 UTC-6, mitch wrote:
> On 10/11/2017 11:49 AM, WM wrote:
> > Am Mittwoch, 11. Oktober 2017 17:49:52 UTC+2 schrieb mitch:
> >
> >
> >> Since you like to be such a critic, why not spend a few
> >> paragraphs giving your affirmative views on the nature
> >> of mathematics, the nature of logic, and what relation
> >> might exist between them?
> >>
> > Logic rules mathematics.
>
> I see this differently. Either the two are
> on equal footing, or, mathematics rules logic.

I might differ with WM a thing or two but you are definitely
wrong here; the two aren't on equal footing and every mathematics
we're familiar with would need some logic and not the other way around.

[mitch]

Sadly, Nam, the mathematics with which you are familiar
is a more profound vacuum than intergalactic space. It
is even more profound than the vacuum which will remain
after all of the protons decay.

Presumably, your "propositional logic" is associated with
sixteen truth tables. This association is so close that there
are authors who have abandoned axiomatics because of the
decidability that truth tables provide.

In the link,

https://en.wikipedia.org/wiki/File:Free-boolean-algebra-hasse-diagram.svg

you will see an order-theoretic conception of your propositional
connectives.

To me, this is just a representation of a tetrahedron.



A tetrahedron has an open exterior. Let it be denoted
by an empty set.

Let

"FALSE" be denoted by {}.



A tetrahedron has 4 vertices. Let them be denoted by
singleton sets.

Let

"p /\ q" be denoted by {A}

"p /\ ~q" be denoted by {B}

"~p /\ q" be denoted by {C}

"~p /\ ~q" be denoted by {D}



A tetrahedron has 6 edges without endpoints. Let
each edge be denoted by a set formed from the union
of sets denoting its endpoints.

Let

"p" be denoted by {A, B}

"q" be denoted by {A, C}

"p <-> q" be denoted by {A, D}

"p <-> ~q" be denoted by {B, C}

"~q" be denoted by {B, D}

"~p" be denoted by {C, D}



A tetrahedron has 4 faces without boundaries. Let
each face be denoted by a set formed from the union
of sets denoting its edges.

Let

"p \/ q" be denoted by {A, B, C}

"q -> p" be denoted by {A, B, D}

"p -> q" be denoted by {A, C, D}

"~p \/ ~q" be denoted by {B, C, D}



A tetrahedron has an open interior. Let it be denoted
by a set formed from the union of sets denoting its
faces.

Let

"TRUE" be denoted by {A, B, C, D}



Mathematicians respect logic. That is why they communicate
with proofs. But, mathematics is not characterized by proofs
since proofs abound outside of mathematical discourse.

For what this is worth, Kant suggested that mathematical
forms were ideal. The nineteenth century philosophers and
logicians argued against this on the basis that no such
ideal objects could be demonstrated in nature. According
to such critics, the proper analysis lay with linguistic
analysis.

What they gave us is a tetrahedron and "abstract objects"
lying outside of space and time.

Good job. Con-artist trickery and a new religion.

mitch

[khongdo...@gmail.com]

On Wednesday, 11 October 2017 14:50:33 UTC-6, mitch wrote:
> On 10/11/2017 02:23 PM, khongdo...@gmail.com wrote:
> > On Wednesday, 11 October 2017 13:08:31 UTC-6, mitch wrote:
> >> On 10/11/2017 11:49 AM, WM wrote:
> >>> Am Mittwoch, 11. Oktober 2017 17:49:52 UTC+2 schrieb mitch:
> >>>
> >>>
> >>>> Since you like to be such a critic, why not spend a few
> >>>> paragraphs giving your affirmative views on the nature
> >>>> of mathematics, the nature of logic, and what relation
> >>>> might exist between them?
> >>>>
> >>> Logic rules mathematics.
> >>
> >> I see this differently. Either the two are
> >> on equal footing, or, mathematics rules logic.
> >
> > I might differ with WM a thing or two but you are definitely
> > wrong here; the two aren't on equal footing and every mathematics
> > we're familiar with would need some logic and not the other way around.
> >
>
> Sadly, Nam, the mathematics with which you are familiar
> is a more profound vacuum than intergalactic space. It
> is even more profound than the vacuum which will remain
> after all of the protons decay.

Non sequitur babbling. Mathematics doesn't rule (mathematical) logic.

[mitch]

A little fact about your level of knowledge has everything to
do with everything instead of nothing to do with anything.

As for what you snipped, I have demonstrated a reason for
asserting parallelism. Yours is a claim of priority.

Prove it.

mitch

[khongdo...@gmail.com]

Idiotic babbling. Mathematics involves inferences which would require
definitions of rules of inference, among other things. Defining rules of
inference pertains to the domain of logic, not mathematics.

>
> As for what you snipped, I have demonstrated a reason for
> asserting parallelism. Yours is a claim of priority.
>
> Prove it.

Exactly in what _LOGIC framework_ is your word "Prove" defined?

[mitch]

I never denied that. Mathematicians communicate with
proofs. I denied logical priority and you -- with your
inestimable knowledge base -- chose to dispute it.

>>
>> As for what you snipped, I have demonstrated a reason for
>> asserting parallelism. Yours is a claim of priority.
>>
>> Prove it.
>
> Exactly in what _LOGIC framework_ is your word "Prove" defined?
>

That is a good question.

The notion of a principled proof was abandoned at least
as early as Bolzano.

When you allow skeptics and their skepticism to dictate
that mathematics is "formal" what is left except arguments
about what one believes words to mean?

There can be no notion of a proof that is meaningful.

Just ask Mr. Di Egidio or Mr. Greene.

mitch

[khongdo...@gmail.com]

On Wednesday, 11 October 2017 21:48:33 UTC-6, mitch wrote:
> On 10/11/2017 10:03 PM, khongdo...@gmail.com wrote:
> > On Wednesday, 11 October 2017 20:46:32 UTC-6, mitch wrote:

> >> Prove it.
> >
> > Exactly in what _LOGIC framework_ is your word "Prove" defined?
>
> That is a good question.

> There can be no notion of a proof that is meaningful.

So _your_ "Prove it" isn't meaningful at all.

[WM]

Am Mittwoch, 11. Oktober 2017 19:00:33 UTC+2 schrieb shio...@googlemail.com:
> > > The point about choice is that choice cannot be disproven,
> >
> > In fact it cannot be disproven since there is nothing uncountable. But if there was an uncountable set, then its elements could not be counted. Is that clear so far?
>
> There are uncountable sets, we have been over that.

There are not uncountably many paths in the Binary Tree that can be constructed by nodes:

"If all nodes of a path have been deleted, that does not mean you ever deleted the path!!! You could delete all and only the finite paths from the tree, and that would delete all the nodes, yet you would never have deleted any infinite path, and they would all still exist in any case, regardless of what had been deleted! [George Greene, sci logic (22 Jul 2016)]

"You seem to be ignoring the fact that, after you have colored a countable family of pathes, say P0, P1, …, Pn,…, there may be other paths Q that are not on this countable list but have, nevertheless, had all their nodes and edges colored. Perhaps the first node and edge of Q were also in P1, the second node and edge of Q were in P2, etc. [...] by choosing the sequence of Pn's intelligently, you can, in fact, ensure that this sort of thing happens for every path Q." [Andreas Blass, MathOverflow (3 Jul 2013)]

It simply is impossible to give all sets in a constructive way ... and any theory, founded on this assumption, would by no means be a theory of sets. [E. Zermelo, letter to E. Artin (?) (25 May 1930)]

> i doN#t think you will accept that,

Nobody who can think should accept that.

> but it is not important for this matter.

Therefore this matter rapes, besnirches and finally tries to kill mathematics,


> > A contradiction would exist for uncountable sets because to one cannot choose what cannot be counted.
>
> No. Simply no.

Yes, simply yes, because all your possible choices belong to a countable set.

> There are uncountable sets and they do not contradict with choice in any way.

That's the claim of some matheologians, but no sober mind will accept this. I explain it to my students in this simple way:

You choose a thought object by thinking of a name, picture, whatever. But all names, pictures, whatevers belong to a countable set. Everybody not saying so is stupid or cheating.

>

> Fact is: Your other 'axioms' contradict the axiom system ZF, while choice does not.

Why should an unfindable even prime number be a harder contradiction than an undefinable well-order?

>
>
> > Of course that is a contradiction as already Cantor recognized: "Infinite definitions" (that do not happen in finite time) are non-things (Undinge). If König's theorem was true, according to which all "finitely definable" real numbers form a set of cardinality 0, this would imply that the whole continuum was countable [G. Cantor, letter to D. Hilbert (8 Aug 1906)]
> >
>
> Once again, quoting old works of mathematicians who usually use other definitions entirely is a moot point and does not get you anywhere.
> Cantor does not agree with what you are saying.

But we know that his disagreement is resting on a false premise.

Regards, WM

[WM]

Am Mittwoch, 11. Oktober 2017 21:08:31 UTC+2 schrieb mitch:
> On 10/11/2017 11:49 AM, WM wrote:
> > Am Mittwoch, 11. Oktober 2017 17:49:52 UTC+2 schrieb mitch:
> >
> >
> >> Since you like to be such a critic, why not spend a few
> >> paragraphs giving your affirmative views on the nature
> >> of mathematics, the nature of logic, and what relation
> >> might exist between them?
> >>
> > Logic rules mathematics.
>
> I see this differently. Either the two are
> on equal footing, or, mathematics rules logic.

But logic is not only applied in mathematics. It has a broader field.

>
> In the latter case, my view is based on concluding
> that the arithmetization of mathematics arising from
> the "crisis in geometry" is nonsensical. So the
> statement could be restated as geometry rules logic.

I think I can agree and even go farther: Reality rules logic.

>
> > Here is an important statement:
> > For every n in |N: Every n belongs to a finite initial segment and does neither require nor allow to consider the set |N as actually closed.
>
> I agree to the import of this.
>
> You are generally dismissive of the standard literature
> and formalisms. But, there is a source in which you will
> find what you are criticizing made explicit.
>
> In Chapter 2, Section 3 of "Set Theory" by Drake, there
> is a definition for well-founded relation in which a
> second subformula explicitly assuming the existence of
> the closure is included.

That sound good.

>
> If one demands a witness for this existential statement,
> as does first-order logic, then one is committed to
> a completed infinity. But, there are other interpretations
> for quantifiers. It is easy enough to think of an
> epsilon-delta proof in terms of a game-theoretic
> interpretation, although making this precise might not
> be easy. This will never suffice for notions of truth
> that resolve to the Aristotelian concept of individuals
> as primary substance.
>
> Aristotle, himself, held that truth ultimately resides
> with perceptions. By characterizing mathematical logic
> solely in terms of a syntax/semantics distinction,
> philosophers and their logicians have made it difficult
> for many mathematicians to accept their analyses.
>
> In my opinion, the trust people place in mathematical
> certainty arises from a common relationship to a
> spatiotemporal experience and not a faith in "abstract
> objects" floating outside of space and time with gods
> and demons.

All "abstract objects" have to be dealt with by concrete actions of the brain in reality. So they all are mapped to elements of reality.

> > The quantification "For all n in |N" (although applicable to qualitative statements like all natural numbers are integers) should be avoided completely as it leads to acontradictions like:
> > For all n in |N: there is no "all" n in |N.
> >
>
> I will leave this alone as it leads to what your
> detractors refer to as "quantifier dyslexia".
>
> I do see a play on words that I had not noticed
> before.
>
> > Correct is this:
> > For every n in |N: n belongs to a finite initial sequence and is followed by a (potentially) infinite sequence.
> >
> > It is not necessary to understand more, but these facts should be grasped. Then there is no contradictory mathematics possible.
> >
>
> The problem you will always face here with regard
> to your detractors is a characterization of potential
> infinity.

Why? It has been defined in a splendid way by Hilbert:

In analysis we have to deal only with the infinitely small and the infinitely large as a limit-notion, as something becoming, arising, being under construction, i.e., as we put it, with the potential infinite. [D. Hilbert: "Über das Unendliche", Mathematische Annalen 95 (1925) p. 167]

>
> Because I have no problem "trusting" topology, I am
> more inclined to view natural numbers with Gauss as
> a system of aliquot and aliquant parts. I can write
> a theory on that basis.

The confusing property is that in several cases it is possible to talk about all natural numbers. All natural numbers are integers, all have a unique prime factorization, all are rationals with denominator 1 in reduced form.

These true statements are usually quoted when defending the existence of a totality of |N. It is not easy to distinguish here what is possible and what is not.

Regards, WM

[mitch]

Correct.

In the end, proof is effective only with respect
to belief.

Mr. Di Egidio recently made a statement about
not "trusting" topology. But, one of the hallmark
properties of useful modern logics is compactness.

The compactness theorem reflects the fact that
these useful logics are topological in character.

I once explained to Mr. Greene how I formulated
a topology corresponding with the compactness
theorem for propositional logic to which he replied
that one ought not confuse matters simply because
the same word has been used.

But, what one finds in the literature is that the
reason the compactness theorems for propositional
and first-order logic are called compactness theorems
is because of the topologies with which the logics
may be correlated.

I recently found the proof I figured out so many
years ago published in Barwise' essay "First-order
logic" in "The Handbook of Mathematical Logic".
The situation for first-order logic is mentioned
in "Model Theory" by Hodges and explicitly stated
in Keisler's essay "Fundamentals of model theory"
in "The Handbook of Mathematical Logic",

< begin quote >

The compactness theorem is sometimes called the
local theorem or the finiteness theorem. It is
more often called the compactness theorem because
it says that a certain topological space is compact.

< end quote >

I can only conclude that Mr. Di Egidio and Mr. Greene
hold the beliefs that they hold because of their
educational backgrounds with respect to logic. Both
obtain their training in philosophy departments. Both
speak as if facts from the literature and the history
of mathematical logic are irrelevant since the only
form of a proof that they will find convincing is one
that will speak to their personal beliefs.

It is actually the contempt shown by a certain class
of trolls on these newsgroups which has led to my
practical understanding of rhetorical logic versus my
views on what comprises mathematical logic.

I fully concede to the supremacy of skepticism and
the skeptical application of logic through rhetoric.
I have no interest in it and do not think it to be
relevant to the meaning of proof in the foundations
of mathematics. But, it forces one to conclude
that proof is meaningless if one's interlocutors
wish to play such a game.

mitch

[mitch]

On 10/12/2017 01:28 AM, WM wrote:
> Am Mittwoch, 11. Oktober 2017 21:08:31 UTC+2 schrieb mitch:
>> On 10/11/2017 11:49 AM, WM wrote:
>>> Am Mittwoch, 11. Oktober 2017 17:49:52 UTC+2 schrieb mitch:
>>>
>>>
>>>> Since you like to be such a critic, why not spend a few
>>>> paragraphs giving your affirmative views on the nature
>>>> of mathematics, the nature of logic, and what relation
>>>> might exist between them?
>>>>
>>> Logic rules mathematics.
>>
>> I see this differently. Either the two are
>> on equal footing, or, mathematics rules logic.
>
> But logic is not only applied in mathematics. It has a broader field.

Yes and no.

Your statement is one with which I agree. It is one that
Stephen Simpson of reverse mathematics fame has applied to
hold that logic and mathematics are not the same. But within
the foundations of mathematics, the two have been conflated.
The logicists in the continental tradition (Weierstrass,
Dedekind, Cantor) probably did not expect their attempts to
improve mathematics with logic to be undermined by the fact
that logic itself had as many or more problems than mathematics.

If logic is about linguistic analysis through paraphrases
of natural language with formal languages, then I shall
deny it every application of mathematics it uses to declare
logical priority over mathematics. Through such criticism
its uselessness is shown unless the claim of logical
priority is abandoned.

>>
>> In the latter case, my view is based on concluding
>> that the arithmetization of mathematics arising from
>> the "crisis in geometry" is nonsensical. So the
>> statement could be restated as geometry rules logic.
>
> I think I can agree and even go farther: Reality rules logic.

But, one must be careful here about what one means by
reality.

My original science interest had been biology. There are
those who see physics as a truthful account of reality. I
will always ask how an evolved biological organism has any
access to truth. The problem I see is similar to an observation
by Russell. While other sciences attempted to ground themselves
in physics, physics began relying on mathematics.

A respect for modern scientific accounts leads to a fundamental
circularity.

With regard to your statement, Kant had been the earliest critic
of Leibniz' identity of indiscernibles. He declared that identity
in mathematics had been derived from what intuitionists have
called "apartness". That is, two points are different if one
is in a part of space not containing the other.

But, if we wish to understand apartness with respect to modern
scientific accounts, we must view our experience for what
it is, a spatiotemporal continuum divided into space and time
only at the level of subjective experience.

As I read in Kip Thorne's book on black holes, "My space is
your spacetime".

None of the -isms in the foundations of mathematics approach
this problem. They seem to be stuck in the nineteenth and
early twentieth centuries.

I think this is called the identity theory of truth,
although I am uncertain.

For my part, the problem reduces to reverting to the
historical account of mathematical objects as useful
logical fictions. But, one needs a logic interpretable
in the context of fictions. First-order logic is not
that logic.

>
>>> The quantification "For all n in |N" (although applicable to qualitative statements like all natural numbers are integers) should be avoided completely as it leads to acontradictions like:
>>> For all n in |N: there is no "all" n in |N.
>>>
>>
>> I will leave this alone as it leads to what your
>> detractors refer to as "quantifier dyslexia".
>>
>> I do see a play on words that I had not noticed
>> before.
>>
>>> Correct is this:
>>> For every n in |N: n belongs to a finite initial sequence and is followed by a (potentially) infinite sequence.
>>>
>>> It is not necessary to understand more, but these facts should be grasped. Then there is no contradictory mathematics possible.
>>>
>>
>> The problem you will always face here with regard
>> to your detractors is a characterization of potential
>> infinity.
>
> Why? It has been defined in a splendid way by Hilbert:
>
> In analysis we have to deal only with the infinitely small and the infinitely large as a limit-notion, as something becoming, arising, being under construction, i.e., as we put it, with the potential infinite. [D. Hilbert: "Über das Unendliche", Mathematische Annalen 95 (1925) p. 167]
>>
>> Because I have no problem "trusting" topology, I am
>> more inclined to view natural numbers with Gauss as
>> a system of aliquot and aliquant parts. I can write
>> a theory on that basis.
>
> The confusing property is that in several cases it is possible to talk about all natural numbers. All natural numbers are integers, all have a unique prime factorization, all are rationals with denominator 1 in reduced form.
>
> These true statements are usually quoted when defending the existence of a totality of |N. It is not easy to distinguish here what is possible and what is not.

No. It is not.

My formalisms are now at a point where I am
distinguishing between "intension" and "extension".
Universals in a proof are always intensional. So,
one needs to treat "apartness" and "equality" separately
without treating them as logical negations of one
another.

I am out of time now. But, I shall try to explain
a small fragment to you after work.

mitch

[Ben Bacarisse]

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

> Am Mittwoch, 11. Oktober 2017 19:00:33 UTC+2 schrieb shio...@googlemail.com:
>> > > The point about choice is that choice cannot be disproven,
>> >
>> > In fact it cannot be disproven since there is nothing
>> > uncountable. But if there was an uncountable set, then its elements
>> > could not be counted. Is that clear so far?
>>
>> There are uncountable sets, we have been over that.
>
> There are not uncountably many paths in the Binary Tree that can be
> constructed by nodes:

It would be a great coup for you if you could write out the bijection
from N to the paths in the binary tree[1]. Since only things with
finite definitions exist in WMaths it will have a finite definition.
Can you write it down?

[1] Obviously I mean in "your" binary tree -- the one that "can be
constructed by nodes". It looks the same as everyone else's binary tree
but your favourite game is to play with words rather than mathematics
and I want to avoid that so I'm only asking about your binary tree.

<snip>
--
Ben.

[Peter Percival]

khongdo...@gmail.com wrote:

> Idiotic babbling. Mathematics involves inferences which would
> require definitions of rules of inference, among other things.
> Defining rules of inference pertains to the domain of logic, not
> mathematics.

Mathematics was done long before logic, as we now think of it, was invented.

>> As for what you snipped, I have demonstrated a reason for asserting
>> parallelism. Yours is a claim of priority.
>>
>> Prove it.
>
> Exactly in what _LOGIC framework_ is your word "Prove" defined?

Mathematicians proved things long before logic, as we now think of it,
was invented.


--
Do, as a concession to my poor wits, Lord Darlington, just explain
to me what you really mean.
I think I had better not, Duchess. Nowadays to be intelligible is
to be found out. -- Oscar Wilde, Lady Windermere's Fan

[WM]

Am Donnerstag, 12. Oktober 2017 12:08:53 UTC+2 schrieb mitch:

> My formalisms are now at a point where I am
> distinguishing between "intension" and "extension".
> Universals in a proof are always intensional. So,
> one needs to treat "apartness" and "equality" separately
> without treating them as logical negations of one
> another.
>
> I am out of time now. But, I shall try to explain
> a small fragment to you after work.
>

Only a hint: Your broad interest in philosophy could be adequately applied in https://philosophy.stackexchange.com/questions

Regards, WM

[WM]

Am Donnerstag, 12. Oktober 2017 12:24:55 UTC+2 schrieb Ben Bacarisse:
> WM <wolfgang.m...@hs-augsburg.de> writes:
>
> > Am Mittwoch, 11. Oktober 2017 19:00:33 UTC+2 schrieb shio...@googlemail.com:
> >> > > The point about choice is that choice cannot be disproven,
> >> >
> >> > In fact it cannot be disproven since there is nothing
> >> > uncountable. But if there was an uncountable set, then its elements
> >> > could not be counted. Is that clear so far?
> >>
> >> There are uncountable sets, we have been over that.
> >
> > There are not uncountably many paths in the Binary Tree that can be
> > constructed by nodes:
>
> It would be a great coup for you if you could write out the bijection
> from N to the paths in the binary tree[1].

Map every finite path on its last node. Understand that real numbers are simply limits.

> Since only things with
> finite definitions exist in

true mathematics

> [1] Obviously I mean in "your" binary tree -- the one that "can be
> constructed by nodes".

Try to imagine a path that cannot be constructed by nodes. Note that by definition a path is a construct by certain nodes.

In my humble opinion the following statements

"If all nodes of a path have been deleted, that does not mean you ever deleted the path!!! You could delete all and only the finite paths from the tree, and that would delete all the nodes, yet you would never have deleted any infinite path, and they would all still exist in any case, regardless of what had been deleted! [George Greene, sci logic (22 Jul 2016)]

"You seem to be ignoring the fact that, after you have colored a countable family of pathes, say P0, P1, …, Pn,…, there may be other paths Q that are not on this countable list but have, nevertheless, had all their nodes and edges colored. Perhaps the first node and edge of Q were also in P1, the second node and edge of Q were in P2, etc. [...] by choosing the sequence of Pn's intelligently, you can, in fact, ensure that this sort of thing happens for every path Q." [Andreas Blass, MathOverflow (3 Jul 2013)]

It simply is impossible to give all sets in a constructive way ... and any theory, founded on this assumption, would by no means be a theory of sets. [E. Zermelo, letter to E. Artin (?) (25 May 1930)]

are the summit of matheology.

I cannot recommend to share those opinions. They rape, besmirch and finally try to kill mathematics.

Regards, WM

[Shobe, Martin]

The consequences of those beliefs.

Martin Shobe

[Ben Bacarisse]

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

> Am Donnerstag, 12. Oktober 2017 12:24:55 UTC+2 schrieb Ben Bacarisse:
>> WM <wolfgang.m...@hs-augsburg.de> writes:
>>
>> > Am Mittwoch, 11. Oktober 2017 19:00:33 UTC+2 schrieb shio...@googlemail.com:
>> >> > > The point about choice is that choice cannot be disproven,
>> >> >
>> >> > In fact it cannot be disproven since there is nothing
>> >> > uncountable. But if there was an uncountable set, then its elements
>> >> > could not be counted. Is that clear so far?
>> >>
>> >> There are uncountable sets, we have been over that.
>> >
>> > There are not uncountably many paths in the Binary Tree that can be
>> > constructed by nodes:
>>
>> It would be a great coup for you if you could write out the bijection
>> from N to the paths in the binary tree[1].
>
> Map every finite path on its last node. Understand that real numbers
> are simply limits.

What is the last node on the left? Is the left-hand edge not a path? I
have to ask because anything is possible in WMaths.

<snip>

> Try to imagine a path that cannot be constructed by nodes. Note that
> by definition a path is a construct by certain nodes.

The left-hand edge is constructed by nodes. It is not in the image of
your bijection.

<snip>
--
Ben.

[shio...@googlemail.com]

> > There are uncountable sets, we have been over that.
>
> There are not uncountably many paths in the Binary Tree that can be constructed by nodes:
>

They are uncountable and have nothing to do (generally speaking) with binary trees.

> > i doN#t think you will accept that,
>
> Nobody who can think should accept that.
>

Except each and every fields medal winner and 99% of the working mathematicians today and their students.

>

> > There are uncountable sets and they do not contradict with choice in any way.
>
> That's the claim of some matheologians, but no sober mind will accept this. I explain it to my students in this simple way:

No that is a thing that got proven by mathematicians.
Your problem is that you either do not understand proofs or simply do not even want to accept logic itself when it doesn't suit you.

> You choose a thought object by thinking of a name, picture, whatever. But all names, pictures, whatevers belong to a countable set. Everybody not saying so is stupid or cheating.
> >

Not mathematics.

> > Fact is: Your other 'axioms' contradict the axiom system ZF, while choice does not.
>
> Why should an unfindable even prime number be a harder contradiction than an undefinable well-order?

Because there provably is no undefinable prime number. Every prime number is a natural number, every natural number is defineable.
Every prime number except 2 cannot be even because elseways it would provably be divisible through 2.

> > Once again, quoting old works of mathematicians who usually use other definitions entirely is a moot point and does not get you anywhere.
> > Cantor does not agree with what you are saying.
>
> But we know that his disagreement is resting on a false premise.

No. All we know is that outdated works are irrelevant for modern amthematics.
You simply choose to quote (and often misquote) them because you do not like modern mathematics and do not really understand it.

[WM]

Am Freitag, 13. Oktober 2017 11:08:22 UTC+2 schrieb shio...@googlemail.com:
> > > There are uncountable sets, we have been over that.
> >
> > There are not uncountably many paths in the Binary Tree that can be constructed by nodes:
> >
>
> They are uncountable and have nothing to do (generally speaking) with binary trees.

I showed you evidence on the contrary: Greene, Blass, Zermelo. Can't you read?

>
>
> > > i doN#t think you will accept that,
> >
> > Nobody who can think should accept that.
> >
>
> Except each and every fields medal winner

Truth is more important than honours.

> and 99% of the working mathematicians today and their students.

Since you cannot understand proofs, try to trust renowned matheologians, for instance Greene, Blass, Zermelo:

You could construct all and only the nodes from the tree, yet you would never have constructed any infinite path.

Regards, WM

[WM]

Am Donnerstag, 12. Oktober 2017 21:08:59 UTC+2 schrieb Ben Bacarisse:

> > Try to imagine a path that cannot be constructed by nodes. Note that
> > by definition a path is a construct by certain nodes.
>
> The left-hand edge is constructed by nodes. It is not in the image of
> your bijection.

Most paths are not in the bijection because they cannot be defined.

In order to distinguish path A from n paths P1, P2, ..., Pn we need n nodes. It cannot be excluded, that one node a of A is sufficient, to distinguish A from all P1, P2, ..., Pn, but these are not n different paths unless there are nodes that distinguish P1 from P2 to Pn, and P2 from P3 to Pn and so on. In total n - 1 nodes are required to distinguish n paths P1, P2, ..., Pn.

Regards, WM

[shio...@googlemail.com]

""
> > They are uncountable and have nothing to do (generally speaking) with binary trees.
>
> I showed you evidence on the contrary: Greene, Blass, Zermelo. Can't you read?
""

You didn't. You talked about a thing that has nothing to do with the thing you claimed.
Obviously i would disregard that. It is like you trying to prove that there are no prime numbers by talking about vectors.

> > Except each and every fields medal winner
>
> Truth is more important than honours.

Indeed, but you keep claiming everybody except some few agrees with you.
Which ain't true, everyone who honestly looks at this matter with a proper backround education about these things disagrees with you.

> > and 99% of the working mathematicians today and their students.
>
> Since you cannot understand proofs, try to trust renowned matheologians, for instance Greene, Blass, Zermelo:
>

But i do understand proofs and these people do not agree with you.

> You could construct all and only the nodes from the tree, yet you would never have constructed any infinite path.

Which has nothing to do with your statement. I adressed the wrong thing syou like to write about graphs in the other thread.

[WM]

Am Freitag, 13. Oktober 2017 11:37:30 UTC+2 schrieb shio...@googlemail.com:
> ""
> > > They are uncountable and have nothing to do (generally speaking) with binary trees.
> >
> > I showed you evidence on the contrary: Greene, Blass, Zermelo. Can't you read?
> ""
>
> You didn't. You talked about a thing that has nothing to do with the thing you claimed.
> Obviously i would disregard that. It is like you trying to prove that there are no prime numbers by talking about vectors.

I said, and the three quoted references agree, that not all paths can be defined by nodes.

> > You could construct all and only the nodes from the tree, yet you would never have constructed any infinite path.
>
>
> Which has nothing to do with your statement.

It is precisely my text, written and claimed by myself.

Regards, WM

[mitch]

True.

But, mathematics is not a social science. Aristotle
did not write Prior Analytics, Posterior Analytics,
and Topics as a mathematical treatise. The study of
logic and proof is a study of human behavior and not
mathematics.

At best, the only use for such "consequences" is to
categorize how well an individual's beliefs cohere.

As for mathematics, all sorts of individuals with all
sorts of beliefs use mathematical justifications. Hilbert's
formalist program had been directed toward ensuring that
those uses are not compromised by mathematics itself. This
is a far cry from the conventionalism that has assumed the
name of formalism because of the legitimate study of
"formal" logical calculi.

There is much to criticize about the promotion of
first-order logic as mathematical logic.

mitch

[shio...@googlemail.com]

> I said, and the three quoted references agree, that not all paths can be defined by nodes.

Which says nothing about the existance of uncountable sets, which is why i still won't grace it with an answer in this thread because I wont let you change the topic.

> > > You could construct all and only the nodes from the tree, yet you would never have constructed any infinite path.
> >
> >
> > Which has nothing to do with your statement.
>
> It is precisely my text, written and claimed by myself.
>

Your statement was that there are no uncountable sets, which this has nothing to do with, period.

[WM]

Am Freitag, 13. Oktober 2017 12:50:17 UTC+2 schrieb shio...@googlemail.com:
> > I said, and the three quoted references agree, that not all paths can be defined by nodes.
>
> Which says nothing about the existance of uncountable sets,

And Elvis is alive too. But what kind of existence does he enjoy?

> Your statement was that there are no uncountable sets,

My statement here is that not uncountably many infinite paths in the Binary Tree can be distinguished. That has been proven. Further I claim that they do not exist because a thought does not exist if nobody can think it.

Regards, WM

[Ben Bacarisse]

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

> Am Donnerstag, 12. Oktober 2017 21:08:59 UTC+2 schrieb Ben Bacarisse:
>
>> > Try to imagine a path that cannot be constructed by nodes. Note that
>> > by definition a path is a construct by certain nodes.
>>
>> The left-hand edge is constructed by nodes. It is not in the image of
>> your bijection.
>
> Most paths are not in the bijection because they cannot be defined.

I just defined a path not in your supposed bijection: the left-hand
edge. It is defined by nodes -- those that are powers of 2.

<snip>
--
Ben.

[mitch]

Thank you. But my degree is in mathematics. My interests
lie with mathematics. And, my knowledge of philosophy may
be best summarized with the adage:

"Know your enemy"

I have run across apartness in three distinct contexts
in my deliberations over the foundations of mathematics.

The first had been in a formalization of sets based on
the presentation of them as collections. Logicians object
to such things with contemptuous remarks about "second-order".
But, two objects must be distinct if one is in a set in which
the other is not.

However, this first depended upon a defined identity that could
be eliminated from the formal theory. So, I sought a conception
of identity that could not be eliminated. I found it in
Tarski's axiom,

AxAy( x = y <-> Ez( x = z /\ z = y ) )

Obviously, if you substitute the right hand side for any
occurrence of the left hand side in a given formula, the
relation on the left is not eliminated from the resulting
formula.

It turns out that Tarski's axiom is related to apartness
in intuitionism with the formula,

AxAy( Ez( x = z /\ z = y ) -> x = y )

Before I knew this, I had formulated this as a schema to
warrant substitutivity in a logical calculus.

On such an account, Tarski's axiom is a mathematical
axiom relating the identity statements of a mathematical
discourse to the logical axioms through the apartness
schema.

But Tarski's axiom only admits proof of transitivity. So,
I devised an axiom for symmetry based upon a simple idea
commonly used to describe convex domains in real analysis.

As you are aware, one parameterizes a line segment with
a linear function over the interval [0,1]. Hence, there is
a base point and a traversal of the segment.

One could, of course, write a different parameterization
so that the segment is traversed first in one direction and
then in the other.

Based on this, I wrote the sentence,

AxAy( x =/= y <-> ( ~( x =/= x ) /\ ( ( ~( x =/= x ) -> y =/= x ) /\ ( y
=/= x -> x =/= y ) ) ) )

with the intention of relating the two with an exclusive
disjunction,

AxAy ~( x = y <-> x =/= y )

The problem here is that '=' is associated with an existential
claim while '=/=' is not. Or, more precisely, '=' is "extensional"
and '=/=' is "intensional" in relation to the context Frege had
used them when developing his notions of mathematical logic.

Russell's paper "On denoting" gives a proper account of how the
second must be put into relation with the first.

And, when you trace this back through Leibniz and Aristotle, it
becomes clear how this applies to proofs and universal quantification
in proofs.

Your "potential infinity" falls out of the quantitative interpretation
of universal quantifiers.

mitch

[WM]

In potential infinity this path is only the set of its finite initial segments. Each one has its own node.

But those which can be defined can also be mapped. Map the powers of 2 on the finite paths. Map the poers of 3 on the paths defined by finite formulas like that one above.

Regards, WM

[WM]

Am Samstag, 14. Oktober 2017 02:29:20 UTC+2 schrieb mitch:


>
> Your "potential infinity" falls out of the quantitative interpretation
> of universal quantifiers.

Yes. But that's how mathematics works. Therefore the formulation "for all n in |N" is not mathematics. I prefer "for every n in |N". It leaves the completeness question where it belongs.

Regards, WM

[Julio Di Egidio]

On Saturday, October 14, 2017 at 10:27:47 AM UTC+2, WM wrote:
> Am Samstag, 14. Oktober 2017 02:29:20 UTC+2 schrieb mitch:
>
> > Your "potential infinity" falls out of the quantitative interpretation
> > of universal quantifiers.
>
> Yes.

No, it doesn't: it's the domain that is potentially infinite, where it is.

Julio

[mitch]

It is not clear to what you are objecting here. Are you objecting to
WM's statement or mine?

WM is asserting that mathematics is quantitative, whence universal
quantifiers are supposed to be interpreted quantitatively. This is
a natural view. But, like naive set theory, it leads to problems
which are not obvious at first.

I am asserting that quantifiers have alternate interpretations,
and, that, because of the origin and role of universal quantifiers
in the pedagogical structure of proofs, they must be understood
with respect to their intensional form. So, interpreting universal
quantification as a parameter over sets or classes cannot be made
"definite".

This expresses itself in the model theory of set theory where an
assumption of partiality permits one to define an external object
from internal parameters.

Relative to the identity of indiscernibles, one can understand
the universe as that unique object which falls under no concept.
It is distinct from its elements because all of its elements
fall under some concept which the universe does not.

But, philosophers and logicians have largely rejected the identity
of indiscernibles.

Negative free logic satisfies the identity of non-existents.
If one rejects a self-membered universe, the universe is a
non-existent relative to the interpretation of the existential
quantifier as selecting witnesses from the universe. However,
there could be many non-existents because there is no criterion
for uniqueness as with the identity of indiscernibles. So, this
notion of universe cannot be claimed as singular. But, it cannot
be claimed as plural either. It is vague. It is potential.

mitch

[shio...@googlemail.com]

> > Which says nothing about the existance of uncountable sets,
>
> And Elvis is alive too. But what kind of existence does he enjoy?
>

It simply says nothing about the existance of uncountable sets whatever nonsense you write about binary trees, since obviously there are more uncountable sets than the real numbers.

> > Your statement was that there are no uncountable sets,
>
> My statement here is that not uncountably many infinite paths in the Binary Tree can be distinguished. That has been proven. Further I claim that they do not exist because a thought does not exist if nobody can think it.

No, your statement was that there are no uncountable sets.

[Julio Di Egidio]

On Saturday, October 14, 2017 at 3:48:10 PM UTC+2, mitch wrote:
> On 10/14/2017 05:14 AM, Julio Di Egidio wrote:
> > On Saturday, October 14, 2017 at 10:27:47 AM UTC+2, WM wrote:
> >> Am Samstag, 14. Oktober 2017 02:29:20 UTC+2 schrieb mitch:
> >>
> >>> Your "potential infinity" falls out of the quantitative interpretation
> >>> of universal quantifiers.
> >>
> >> Yes.
> >
> > No, it doesn't: it's the domain that is potentially infinite, where it is.
>
> It is not clear to what you are objecting here. Are you objecting to
> WM's statement or mine?

It's perfectly clear that I am objecting to your statement, the only
statement that in fact there is, and only incidentally to WM who is
approving it: I guess because he has to maintain that there is no
totality, not for anything else. And you are wrong: the universal
quantifier quantifies over any totality, but whether a potentially
infinite set is or is not legitimately a totality is a separate
problem.

Julio

[Peter Percival]

khongdo...@gmail.com wrote:
> On Wednesday, 11 October 2017 14:50:33 UTC-6, mitch wrote:
>> On 10/11/2017 02:23 PM, khongdo...@gmail.com wrote:
>>> On Wednesday, 11 October 2017 13:08:31 UTC-6, mitch wrote:
>>>> On 10/11/2017 11:49 AM, WM wrote:
>>>>> Am Mittwoch, 11. Oktober 2017 17:49:52 UTC+2 schrieb mitch:
>>>>>
>>>>>
>>>>>> Since you like to be such a critic, why not spend a few
>>>>>> paragraphs giving your affirmative views on the nature
>>>>>> of mathematics, the nature of logic, and what relation
>>>>>> might exist between them?
>>>>>>
>>>>> Logic rules mathematics.
>>>>
>>>> I see this differently. Either the two are
>>>> on equal footing, or, mathematics rules logic.
>>>
>>> I might differ with WM a thing or two but you are definitely
>>> wrong here; the two aren't on equal footing and every mathematics
>>> we're familiar with would need some logic and not the other way around.
>>>
>>
>> Sadly, Nam, the mathematics with which you are familiar
>> is a more profound vacuum than intergalactic space. It
>> is even more profound than the vacuum which will remain
>> after all of the protons decay.
>
> Non sequitur babbling. Mathematics doesn't rule (mathematical) logic.
>

Look at Shoenfield's Lemma 1 on page 15, and the Formation Theorem and
Lemma 2 over the page. At those points Shoenfield hasn't yet set up a
formal system (he doesn't do that until page 21) but he finds it
necessary (and certainly possible) to prove mathematical theorems. So,
at least in Shoenfield, mathematics rules logic. If you look in other
logic texts you'll see similar things.

[Peter Percival]

Logic (as in sci.logic) is a branch of mathematics and therefore can't
be prior to it. To say that logic is prior to mathematics is like
saying that differential geometry is prior to mathematics.

[WM]

Am Samstag, 14. Oktober 2017 17:02:56 UTC+2 schrieb shio...@googlemail.com:
> > > Which says nothing about the existance of uncountable sets,
> >
> > And Elvis is alive too. But what kind of existence does he enjoy?
> >
>
> It simply says nothing about the existance of uncountable sets whatever nonsense you write about binary trees,

You always defame thinks you cannot understand. Nothing new. Accepted by those who are on the same level of comprehension.

> > My statement here is that not uncountably many infinite paths in the Binary Tree can be distinguished. That has been proven. Further I claim that they do not exist because a thought does not exist if nobody can think it.
>
> No, your statement was that there are no uncountable sets.

That's the obvious conclusion from the fact that otherwise there must be unthinkable thoughts.

Regards, WM

[Julio Di Egidio]

On Saturday, October 14, 2017 at 8:13:14 PM UTC+2, Peter Percival wrote:

> Logic (as in sci.logic) is a branch of mathematics and therefore can't
> be prior to it.

You are fucking stupid...

Julio

[Julio Di Egidio]

On Saturday, October 14, 2017 at 8:17:59 PM UTC+2, WM wrote:

> > No, your statement was that there are no uncountable sets.
>
> That's the obvious conclusion from the fact that otherwise there must be
> unthinkable thoughts.

No, that's your mistake, to confuse the fact that a theory may be
incomplete or even broken with the fact that anything not in that(!)
theory cannot exist, which is simply wrong.

Julio

[shio...@googlemail.com]

> You always defame thinks you cannot understand. Nothing new. Accepted by those who are on the same level of comprehension.
>

No, i simply tell you that you have not proven your original statement.
Like you never do.

> > No, your statement was that there are no uncountable sets.
>
> That's the obvious conclusion from the fact that otherwise there must be unthinkable thoughts.

It is not the 'obvious conclusion', it is in fact a wrong conclusion because most uncountable sets have nothing to do with binary trees.

[mitch]

Why don't you read the chapter from Strawson you recommended
to me?

Strawson attempts to justify how we inductively arrive at universals
in our logic by invoking probabilities. The difficulty here is that
serious mathematicians all have books on probability on their
shelves which use a great deal of calculus ultimately relying
on axiom systems invoking sets.

What you will find on my bookshelf is "Probability" by Shiryaev
for the modern textbook account, "Foundations of Probability" by
Kolmogorov for the account of its first axiomatization, and
"A Philosophical Essay on Probabilities" by LaPlace for some
historical background.

As with Mr. Greene, when you do bother to reveal some source
for your opinions, they fall apart under scrutiny. Now you
suddenly want to talk about universals as having an extensional
import simply because you wish to be disagreeable with me.

Make up your mind.

Unless your non-metaphysical God is feeding you revealed
knowledge, the only source of knowledge is other people's
opinions. The only means of discerning the evolution of
other people's opinions is by tracing through the literature.

You should actually try it some time. You are "stuck" as
a *follower* of Wittgenstein by your own choice. So, when
you attempt to formalize your ideas, you cluelessly stumble
over the mathematics you feel no need to learn.

Should you trace through the relevant literature, you will
arrive at Aristotle's discussion of "abstraction". And,
following it into Leibniz and Frege's criticism of "intensional
logicians", you will arrive at "intension" as an account of
the universal quantifier.

Since you have a devotion to Wittgenstein, you have tossed
out the critical theory of definite description by which
Frege "objectifies" classes as "the extension of concepts".
But, through chosen ignorance, you do not even understand
your choice.

Your personal opinions -- the basis of your personal
"reasoned arguments" -- are objectively irrelevant.

The skepticism of you and your colleagues in philosophy
are responsible for the meaninglessness of your words.
If you manage to circumvent David Hume's argument, we
can talk.

And, good luck with that!

mitch

[Julio Di Egidio]

On Saturday, October 14, 2017 at 9:05:00 PM UTC+2, mitch wrote:
> On 10/14/2017 12:32 PM, Julio Di Egidio wrote:
> > On Saturday, October 14, 2017 at 3:48:10 PM UTC+2, mitch wrote:
> >> On 10/14/2017 05:14 AM, Julio Di Egidio wrote:
> >>> On Saturday, October 14, 2017 at 10:27:47 AM UTC+2, WM wrote:
> >>>> Am Samstag, 14. Oktober 2017 02:29:20 UTC+2 schrieb mitch:
> >>>>
> >>>>> Your "potential infinity" falls out of the quantitative interpretation
> >>>>> of universal quantifiers.
> >>>>
> >>>> Yes.
> >>>
> >>> No, it doesn't: it's the domain that is potentially infinite, where it is.
> >>
> >> It is not clear to what you are objecting here. Are you objecting to
> >> WM's statement or mine?
> >
> > It's perfectly clear that I am objecting to your statement, the only
> > statement that in fact there is, and only incidentally to WM who is
> > approving it: I guess because he has to maintain that there is no
> > totality, not for anything else. And you are wrong: the universal
> > quantifier quantifies over any totality, but whether a potentially
> > infinite set is or is not legitimately a totality is a separate
> > problem.
>
> Why don't you read the chapter from Strawson you recommended
> to me?

You better learn to read before accusing me.

> Strawson attempts to justify how we inductively arrive at universals
> in our logic by invoking probabilities.

Nope, that is utterly irrelevant to the issue of totalities in
*mathematical* systems, which are *strictly deductive* systems,
and it's *the opposite* of what Strawson says about induction,
anyway.

Julio

[Julio Di Egidio]

On Saturday, October 14, 2017 at 9:05:00 PM UTC+2, mitch wrote:

> As with Mr. Greene, when you do bother to reveal some source
> for your opinions, they fall apart under scrutiny.

Under whose scrutiny, you moron? Secondly, indeed I give *reasons* not
references: and the problem with that is all yours and, seriously, of
your lack of a sane methodology before anything else.

Julio

[Julio Di Egidio]

On Saturday, October 14, 2017 at 9:05:00 PM UTC+2, mitch wrote:

> Unless your non-metaphysical God is feeding you revealed
> knowledge, the only source of knowledge is other people's
> opinions. The only means of discerning the evolution of
> other people's opinions is by tracing through the literature.
>
> You should actually try it some time. You are "stuck" as
> a *follower* of Wittgenstein by your own choice. So, when
> you attempt to formalize your ideas, you cluelessly stumble
> over the mathematics you feel no need to learn.
>
> Should you trace through the relevant literature, you will
> arrive at Aristotle's discussion of "abstraction". And,
> following it into Leibniz and Frege's criticism of "intensional
> logicians", you will arrive at "intension" as an account of
> the universal quantifier.
>
> Since you have a devotion to Wittgenstein, you have tossed
> out the critical theory of definite description by which
> Frege "objectifies" classes as "the extension of concepts".
> But, through chosen ignorance, you do not even understand
> your choice.
>
> Your personal opinions -- the basis of your personal
> "reasoned arguments" -- are objectively irrelevant.

What a fucking idiot! I give arguments, and those arguments may be mistaken
but you fucking idiot that call any argument a personal opinion unless
Aristotle has said it first are really a totally clueless slave and moron.

Devotions are *your* problem!
Personal opinions are *your* problem!
Stop projecting *your* bullshit!

Julio

[Julio Di Egidio]

BTW, the fucking idiots who cannot distinguish logic from mathematical
logic better not read Strawson at all.

Julio

[George Greene]

On Wednesday, October 11, 2017 at 8:37:32 AM UTC-4, WM wrote:
> - Axiom of well-ordering: Every set can be well-ordered. (This axiom is not constructive. In most cases provably no set theoretic definition of a well-order can be found.)
THAT'S *WHY*IT*HAS* to be *AN* *A*X*I*O*M*, YOU IDIOT!!!
If a definition were available then you would USE THAT, INSTEAD,
YOU IDIOT! You use the axiom BECAUSE there is no construction!
The construction would constitute THE PROOF of the existence AS A THEOREM,
in which case there COULD BE no AXIOM! Axioms HAVE to be INDEPENCENT OF the
OTHER AXIOMS! They have to NOT be provable from the other axioms!!



> - Axiom of three points on a line: Every triple of points belongs to a straight line. (But in most cases provably no geometrical construction can be given.)

You are completely mis-stating this. In the plane or in space, three points DON'T have to lie on a straight line. WHAT axiom did you EVEN MEAN?

> - Axiom of ten even primes: There are 10 even prime numbers. (But provably no arithmetical method to find them is available.)

That is just false. It is disprovable in PA. Provably there is an arithmetical method to show that NO even number except 2 is prime.

> - Axiom of prime number triples: There
> is a second triple of prime numbers besides (3, 5, 7).
> (But provably this second triple is not arithmetically definable.)

You don't yet know whether this is or isn't "like" the axiom of choice.
The fact that this triple is not arithmetically definable may mean that
YOU WILL NEVER know. The difference between this and the axiom of choice
is that AC has ALREADY been proven independent of ZF.
This conjecture has NOT been proven to be independent of PA (i.e. satisfied
in some nonstandard model of PA, not that I know what nonstandard primes could
even look like).


> - Axiom of meagre sum (AMS): There is a
> set of n different positive natural numbers with
> sum n*n/2. (This axiom is not constructive.
> Provably no such set can be found.)

Damn, you're stupid. It is NOT merely provable that no such set can be FOUND -- IT IS ALSO provable that no such set CAN EXIST. THE SMALLEST possible sum that n different positive natural numbers can have is the sum of 1+2+...+n, which is
n*(n+1)/2, WHICH IS GREATER THAN n*(n/2). SO NO SUCH SET EXITS. Therefore this
IS NOT an axiom -- IT IS THE DENIAL OF A THEOREM. It IS JUST FALSE. The axiom
of choice and the well-ordering axiom (and their other many equivalents) ARE NOT like that -- you canNOT disprove the existence of the thing that they claim exists.


> - Axiom of ultimate mathematics simplification:
> All mathematical problems are solved by whatever
> I declare as the solution.
> (This axiom is guaranteed not less useful than the axiom of choice.)

Well, SURE, this is a very useful axiom and it used to get used all the time --
it was used to create zero, negative numbers, i, imaginary numbers, and complex numbers. It is a defining feature of this field that YOU GET to use this axiom AS LONG AS you don't subsequently derive a contradiction from the existence of the "whatever" that you hereby declared into existence. And guess what: if anybody wants to dispute or refute you, *N*O*T*H*I*N*G* *except* deriving a contradiction COUNTS as a rebtuttal.

If you've got a PROBLEM with this, then it's YOUR problem.

[George Greene]

On Wednesday, October 11, 2017 at 12:49:10 PM UTC-4, WM wrote:
> Logic rules mathematics.

Not in YOUR opinion. Otherwise, you would be willing TO USE it
when talking about mathematics.

> Here is an important statement:
> For every n in |N: Every n belongs to a finite initial segment

Of course

> and does neither require nor allow to consider the set |N as actually closed.

That statement is NOT important. It is also not coherent. It is also not mathematically. YOU CANNOT DEFINED whether any purported "set" is or is not "actually closed". You are NOT SAYING ANYthing.

[Ben Bacarisse]

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

> Am Freitag, 13. Oktober 2017 22:41:55 UTC+2 schrieb Ben Bacarisse:
>> WM <wolfgang.m...@hs-augsburg.de> writes:
>>
>> > Am Donnerstag, 12. Oktober 2017 21:08:59 UTC+2 schrieb Ben Bacarisse:
>> >
>> >> > Try to imagine a path that cannot be constructed by nodes. Note that
>> >> > by definition a path is a construct by certain nodes.
>> >>
>> >> The left-hand edge is constructed by nodes. It is not in the image of
>> >> your bijection.
>> >
>> > Most paths are not in the bijection because they cannot be defined.
>>
>> I just defined a path not in your supposed bijection: the left-hand
>> edge. It is defined by nodes -- those that are powers of 2.
>>
> In potential infinity this path is only the set of its finite initial
> segments. Each one has its own node.

The left-hand edge is one such path.

<snip>
--
Ben.

[mitch]

But, it is entirely relevant to logical systems and the
influence of logicism in the modern literature.

Keep twisting history Mr. Philosopher.

You better learn some mathematics before lecturing
people with mathematics degrees about mathematics.

mitch

[mitch]

Idiots who recommend Strawson should actually understand
his distinction between qualitative identity and quantitative
identity which invokes spatial reasoning as the basis of
numerical difference.

mitch

[Julio Di Egidio]

Which remains totally irrelevant to the present discussion.

> Keep twisting history Mr. Philosopher.
>
> You better learn some mathematics before lecturing
> people with mathematics degrees about mathematics.

Whatever, I have had enough of just bullshit.

Julio

[Julio Di Egidio]

Sure, I'll think about that... moron.

*Plonk*

Julio

[mitch]

On 10/14/2017 02:35 PM, Julio Di Egidio wrote:
> On Saturday, October 14, 2017 at 9:05:00 PM UTC+2, mitch wrote:
>
>> As with Mr. Greene, when you do bother to reveal some source
>> for your opinions, they fall apart under scrutiny.
>
> Under whose scrutiny, you moron?

Mine.

It is the only one that counts when you
present your personal views as truth while
saying that I am wrong.

Elsewhere, you have been asked to clarify the distinction
you make between logic and mathematical logic
in this forum. You had been unable to be clear.

By contrast, I hold that the qualifier "mathematical"
in "mathematical logic" must be grounded in
mathematics.

One aspect of "methodology" you accuse me of not
having is to demarcate one's logical constants, although
the importance of this has its skeptics. You can find
discussion of this in the link,

https://plato.stanford.edu/entries/logical-constants/


Upon recognizing that truth tables relate to one
another as a finite affine plane, I gave names to
truth tables and formulated a finite projective
plane with a "line at infinity" composed of names
associated with quantificational logic,


NTRU : { ABSRD, ALL, OTHER, NO, SOME }

ABSRD : { ABSRD, NTRU, LEQ, XOR, TRU }
LEQ : { ABSRD, NIMP, NIF, IF, IMP }
XOR : { ABSRD, NOR, AND, OR, NAND }
TRU : { ABSRD, DENY, LET, FLIP, FIX }

ALL : { ALL, NTRU, NIMP, NOR, DENY }
NIMP : { ALL, LEQ, IF, AND, FIX }
NOR : { ALL, XOR, NIF, NAND, FLIP }
DENY : { ALL, TRU, IMP, OR, LET }

OTHER : { OTHER, NTRU, NIF, AND, LET }
NIF : { OTHER, LEQ, IMP, NOR, FLIP }
AND : { OTHER, XOR, NIMP, OR, FIX }
LET : { OTHER, TRU, IF, NAND, DENY }

NO : { NO, NTRU, IF, OR, FLIP }
IF : { NO, LEQ, NIMP, NAND, LET }
OR : { NO, XOR, IMP, AND, DENY }
FLIP : { NO, TRU, NIF, NOR, FIX }

SOME : { SOME, NTRU, IMP, NAND, FIX }
IMP : { SOME, LEQ, NIF, OR, DENY }
NAND : { SOME, XOR, IF, NOR, LET }
FIX : { SOME, TRU, NIMP, AND, FLIP }


The use of names here is compatible with the presentation of
projective geometries given through difference sets. The
one above is taken from


| 20 | 19 | 18 | 17 | 16 | 15 | 14 | 13 | 12 | 11 | 10 | 9 | 8 | 7
| 6 | 5 | 4 | 3 | 2 | 1 | 0 |

|----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14
| 15 | 16 | 17 | 18 | 19 | 20 | 0 |
| 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15
| 16 | 17 | 18 | 19 | 20 | 0 | 1 |
| 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20
| 0 | 1 | 2 | 3 | 4 | 5 | 6 |
| 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 0 | 1
| 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| 19 | 20 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11
| 12 | 13 | 14 | 15 | 16 | 17 | 18 |

reorganized as

18 : { 0, 4, 3, 9, 11 }

0 : { 0, 18, 8, 6, 1 }
8 : { 0, 14, 19, 13, 10 }
6 : { 0, 2, 15, 12, 16 }
1 : { 0, 17, 5, 20, 7 }

4 : { 4, 18, 14, 2, 17 }
14 : { 4, 8, 13, 15, 7 }
2 : { 4, 6, 19, 16, 20 }
17 : { 4, 1, 10, 12, 5 }

3 : { 3, 18, 19, 15, 5 }
19 : { 3, 8, 10, 2, 20 }
15 : { 3, 6, 14, 12, 7 }
5 : { 3, 1, 13, 16, 17 }

9 : { 9, 18, 13, 12, 20 }
13 : { 9, 8, 14, 16, 5 }
12 : { 9, 6, 10, 15, 17 }
20 : { 9, 1, 19, 2, 7 }

11 : { 11, 18, 10, 16, 7 }
10 : { 11, 8, 19, 12, 17 }
16 : { 11, 6, 13, 2, 5 }
7 : { 11, 1, 14, 15, 20 }


The finite affine plane in this projective geometry has 20 lines
whose names correspond with names for truth tables except for
"NTRU" (falsity) which exchanges with "ABSRD" (absurdity) related
to quantificational logic by Frege's use of {x : x=/=x } to ground
his definition of natural numbers.

Those 20 names are organized into the ortholattice,


....................................TRU....................................
............................./.../..//\..\.................................
......................../..../.../../....\...\.............................
.................../...../..../..../.........\.....\.......................
............../....../...../....../...............\......\.................
........./......./....../......../.....................\.......\...........
....../......./......./........./...........................\........\.....
.....IF.....NAND......IMP.......OR..........................NO........ALL..
.....)))....|\.\...../../\...(.((...\\....................../\......././...
........)).)|.\.../.\../(.\.(..(.......\.\................/...\..../../....
............|)/\).).(./.\(.\..(..........\..\.........../......\./.../.....
........../.|..(\).)(/...)..\(\............\....\...../......./.\.../......
....../..(..|...(\.)/...)..(.\)...\..........\.....\/....../.....\./.......
./.(........|(....\/...).(..).\...)..\..........\/....\./.........\........
LET.......XOR..FLIP....FIX..LEQ...DENY......../....../...\......./.\.......
.\\....../...\/...\..../.|./..\...././...../...../.\........\.../...\......
..\..\../.../...\.../.\./|.....\./../..../..../...............\/.....\.....
...\.../.\/....../.\/....|.\../.\../.../.../...........\....../..\....\....
....\././....\././....\..|./...\.\/.//.....................\./.......\.\...
.....NIF......AND......NIMP.....NOR........................SOME......OTHER.
......\.......\.......\.........\.........................../......../.....
.........\.......\......\........\...................../......./...........
..............\......\.....\......\.............../....../.................
...................\.....\....\....\........./...../.......................
........................\....\...\..\..../.../.............................
.............................\...\..\\/../.................................
...................................ABSRD...................................


which is an atomic amalgam of a 4-atom Boolean subblock and a 3-atom
Boolean subblock with the complete connective "NOR" as the shared atom.
All of the constants corresponding with truth tables (except "NTRU") are
located at the loci they would have in the free Boolean lattice on two
generators.

I hold quantificational logic to be prior to propositional logic with
propositional logic identifiable as a closed subsystem because of the
structure of the ortholattice.

Truth-functional connectivity is established with a set of axioms
over the 16 truth table names as I have discussed elsewhere. They
describe functions in a manner found in a monograph by Church on
the lambda calculus.

For someone with as little knowledge of mathematics as you have
to accuse me of not having a methodology just because it is not
acceptable to you is laughable.

Go suck on some philosophy somewhere.

mitch

[mitch]

Really?

Recall that I thought it significant that you actually
tried to express yourself at a formal level. With others,
that did not go so well.

What I criticize here is the regurgitation of "facts" from
what is taught in philosophy departments. Those are not
arguments.

And, I personally do not think that I ever should have
had to mention any historical references. It is
because of "facts" asserted in philosophy departments
that I had to embark on a "fact-checking" excursion.

>
> Devotions are *your* problem!
> Personal opinions are *your* problem!
> Stop projecting *your* bullshit!

Bullshit? You could learn from my bullshit.

Let me give you an example of an "argument"
in mathematical logic. I actually write such
things when needed because that is what
mathematicians do.

You claim to "not trust topology". I posted
the proof below concerning the relations between
language terms in a consistent theory. They
satisfy the axioms of a topological construct
called a "proximity space".

Why would the language terms of a consistent
theory satisfy axioms for what would normally
be considered as arising from a "spatial" theory?
Because no amount of philosophy or logic can
remove geometry from mathematics.

Have you ever posted anything in these forums
that is recognizable as a proof by others?

mitch

===========================================


Let *L* be a language of first-order logic

with equality.



Let *M* be a model of *L* with domain M.



Let *T* be a consistent theory in *L*.



Denote the terms of *L* by T(*L*).



Define a nearness relation on the power

set of T(*L*) by:



A near B if and only if there exists

an (a in A) and a (b in B) such that

a=b is true in *M*.





This nearness relation defines a proximity

space



*P* = < T(*L*), near >



The next 5 statements verify the axioms.





1)



Assertion: A near B implies B near A



Proof: Suppose A near B holds. Then, there

is an (a in A) and a (b in B) such that

a=b in is true in *M*. But, '=' is an

equivalence relation satisfying symmetry.

Thus, b=a is true in *M*. Hence, there is

n (b in B) and an (a in A) such that b=a is

true in *M*. So, B near A holds.





2)



Assertion: (A \/ B) near C if and only if

A near C or B near C



Proof: Suppose (A \/ B) near C. Then, for

some (d in A \/ B) and for some (c in C) one

has that d=c is true in *M*. But, since

(d in A \/ B) holds, one must have (d in A)

or (d in B). If (d in A) holds, then it

follows that A near C holds. If (d in B)

holds, it follows that B near C holds. So,

it is the case that A near C or B near C.



Suppose A near C or B near C. If A near C,

then for some (a in A) and for some (c in C),

one has that a=c is true in *M*. Since

(a in A) it follows that (a in A \/ B). So,

it follows that A \/ B near C.



The case for B near C is similar.





3)



Assertion: A near B implies that neither

A nor B are the empty set.



Proof: Suppose A near B holds. Then for some

(a in A) and for some (b in B) one has that

a=b is true in *M*. The expression 'a=b' is

a well-formed atomic formula of *L* for which

(a in T(*L*)) and (b in T(*L*)). Thus, both

a and b are domain elements, whence neither

A nor B are empty.





4)



Assertion: ~(A near B) implies that there

exists a subset E of T(*L*) such that

~(A near E) and ~(T(*L*)\E near B)



Proof: Suppose ~(A near B). Then for

every (a in A) and for every (b in B)

a=b is not true in *M*.



Define Cl(X) = { y in T(*L*) | there

exists some (x in X) for which y=x is

true in *M* }



Let E = Cl(B)\Cl(A). Then ~(A near E).

For if it were otherwise, there would be

an (a in A) and there would be an (e in E)

such that a=e is true in *M*. By symmetry,

it would be the case that e=a is true in *M*.

But, then (e in Cl(A)) would hold. This

is in contradiction with the definition

of E. Hence ~(A near E).



Since E = Cl(B)\Cl(A) one has that

T(*L*)\E may have among its elements only

those elements of Cl(B) which are in Cl(A).

Since ~(A near B) for every (b in B)

there is no (a in A) such that b=a is

true in *M*. So, no (b in B) can be

in Cl(A). If (y in Cl(B)\B) then there

exists a (b in B) such that y=b is

true in *M*. If for such a y, it is

the case that (y in Cl(A)), then there

exists (a in A) such that y=a is true

in *M*. By symmetry, a=y is true in *M*.



By transitivity one obtains a=b is true

in *M* from a=y is true in *M* and y=b

is true in *M*. But, by assumption,

~(A near B). So, this is a contradiction

and no elements of Cl(B) can be in

T(*L*)\E. Thus ~(T(*L*)\E near B) holds,

as needed to complete the proof.





5)



Assertion: If A /\ B is not empty, then

A near B



Proof: Suppose A /\ B is not empty. Let

x be such that (x in A) and (x in B). Then,

by the reflexive axiom for identity, x=x

is true in *M*. Hence, A near B.





This completes the proofs needed to verify

*P* satsifies the axioms.

[George Greene]

On Saturday, October 14, 2017 at 3:05:00 PM UTC-4, mitch wrote:
> As with Mr. Greene, when you do bother to reveal some source
> for your opinions, they fall apart under scrutiny.

*F*U*C*K* you. You have never caused or even observed ANY other person's opinion
"fall apart". Opinions CAN'T "fall apart". Opinions have to be ABOUT MATTERS of opinion, which BY DEFINITION *CAN'T* be refuted. Things that are subject to refutation by facts are NOT fundamentally MATTERS OF opinion.

[George Greene]

On Saturday, October 14, 2017 at 3:05:00 PM UTC-4, mitch wrote:

> Your personal opinions -- the basis of your personal
> "reasoned arguments" -- are objectively irrelevant.

NObody's "reasoned arguments" HAVE ANYthing to do WITH ANYbody's "opinions"!
What MAKES something a "reasoned argument" is its conformance to some sound and valid argument template - provability is a matter OF SYNTAX, NOT "opinion"!!

[Shobe, Martin]

On 10/14/2017 2:50 PM, George Greene wrote:
> On Wednesday, October 11, 2017 at 8:37:32 AM UTC-4, WM wrote:
>> - Axiom of prime number triples: There
>> is a second triple of prime numbers besides (3, 5, 7).
>> (But provably this second triple is not arithmetically definable.)
>
> You don't yet know whether this is or isn't "like" the axiom of choice.
> The fact that this triple is not arithmetically definable may mean that
> YOU WILL NEVER know. The difference between this and the axiom of choice
> is that AC has ALREADY been proven independent of ZF.
> This conjecture has NOT been proven to be independent of PA (i.e. satisfied
> in some nonstandard model of PA, not that I know what nonstandard primes could
> even look like).

You're giving him too much credit. He's not referring to the things that
mathematicians call prime triplets. The giveaways are he's asking for a
second one and he's calling (3,5,7) one. This falls in the category of
things that can easily be proven false.

Martin Shobe

[mitch]

Irrelevant to you simply because of your choice to be
ignorant of the literature.

You were the one who asserted the meaning of a universal
quantifier. Then you tried to be "right" by differentiating
logic and mathematical logic.

But, when discussing Tarski's semantic conception of truth
and its basis of denotations as the received view for
mathematical logic, you claimed that that I could make
no argument on that basis because analytical truth is the
only notion of truth in logic and philosophy.

You must make up your mind about what you are going to
say consistently instead of blathering how wrong I am every
time you wish to be disagreeable with me.

mitch

[George Greene]

On Wednesday, October 11, 2017 at 4:50:33 PM UTC-4, mitch wrote:
> Sadly, Nam, the mathematics with which you are familiar

You give Nam WAY too much credit. Nam is not familiar with mathematics.


> Presumably, your "propositional logic" is associated with
> sixteen truth tables.

Nam does NOT own propositional logic.
THIS IS *EVERY*BODY's propositional logic.
But the 16 truth-tables would exist WITH IT OR WITHOUT IT.
Propositinoal logic can get by WITH TWO truth-tables (one for Or and one for Not), if you want to go that way. Isomorphisms with other ways ARE constructble.

> This association is so close that there
> are authors who have abandoned axiomatics because of the
> decidability that truth tables provide.

God, what utter horse-shit. But if you actually believe this, and
you actually think it is defensible, then how can you attack Nam for being
familiar with propositional logic based on truth-tables?
"Abandon" is a strong word WHEN ISOMORPHISMS ABOUND between different
approaches. If you are doing something without axioms but what you are doing
is ismorphic to a treatment WITH axioms then you have NOT abandoned anything.

> In the link,
>
> https://en.wikipedia.org/wiki/File:Free-boolean-algebra-hasse-diagram.svg
>
> you will see an order-theoretic conception of your propositional
> connectives.

Which doesn't change the fact that there are 16 binary boolean functions.
This IS NOT an alternative to, or a refutation of, ANY other approach.
It is, however, a gross violation of Occam's Razor if ALL you are TRYING to do
is propositional logic.



> To me, this is just a representation of a tetrahedron.

How in THE HELL do you propose to GET ANYthing ORDER-theoretic out of anything AS SYMMETRIC as a tetrahedron??

[George Greene]

On Wednesday, October 11, 2017 at 4:50:33 PM UTC-4, mitch wrote:

> A tetrahedron has an open exterior.

EVERY closed set has an open exterior.
There is nothing tetrahedron-specific going on here.
And you didn't even SAY "closed tetrahedron". AND YOU SHOULD HAVE, because
if the tetrahedron is solid in 3 dimensions and you TAKE ALL ITS BOUNDARY
POINTS AWAY, what you HAVE LEFT *STILL*APPEARS*TO*BE* a tetrahedron.
Ditto for a solid square minus ITS boundary, or a line-segment minus its endpoints.

> Let it be denoted
> by an empty set.

YOU *CAN'T* do that. If the tetrahedron is solid then its exterior IS ALL THE POINTS IN SPACE that are NOT IN it!! IT *IS*NOT* empty!

[mitch]

Of course opinions fall apart. The expression
refers to how a person's credibility sinks when
their statements do not correspond with the
facts when checked.

You are quite correct, however, that an observer
is not the cause. The utterer causes their own
demise.

Some time ago, I spoke of a proof I formulated
as an undergraduate concerning the relation of
the compactness theorems in logic to the compactness
theorems I learned in real analysis and topology.

In your haste to ridicule, you claimed that I was
stupid for thinking that a word used in separate
disciplines had any correlation.

< begin quote >

The compactness theorem is sometimes called
the local theorem or the finiteness theorem.
It is more often called the compactness theorem
because it says that a certain topological space
is compact.

< end quote >

"Fundamentals of Model Theory"
Jerome Keisler

Printed in "The Handbook of Mathematical Logic"


Keisler goes on to describe that topological
space. He had been referring to the compactness
theorem for first-order logic.

The proof I did, however, had been the one associated
with propositional logic. I found it in print in "The
Handbook of Mathematical Logic". It is in Barwise'
essay, "An Introduction to First-Order Logic",

< begin >

We can give a faster proof by quoting the Tychonoff
Theorem. It hides the basic construction, though,
and thus is less suitable for other constructions
in model theory. Let 2 = { t, f } be the two-element
space with discrete topology and let X = 2^P, the space
of all truth assignments of P with the product topology.
By the Tychonoff theorem X is a compact, Hausdorff space.
Hence, if

*F* = { F_i : i in I }

is an indexed family of closed subsets, and if

Cap_( i in I ) F_i = nullset,

then there is a finite I_0 in I such that

Cap_(i in I_0) = nullset.

For each propositional formula A, let

F_A = { nu in X : nu-bar(A) = t }

[ where nu-bar is the extension of nu over
prime formulas (sentence letters) to composite
formulas ]

We claim that each F_A is clopen (both closed
and open) in X. For A = p a prime formula F_p
is open by the very definition of the product
topology. But,

X - F_p = { nu : nu(p) = f }

is also open, by definition, so F_p is clopen.

For more complicated formulas, the claim follows
by induction on length of formulas and the
following equations:

F_(A \/ B) = F_A cup F_B

F_(A /\ B) = F_A cap F_B

F_(A -> B) = F_B - F_A

F_( ~A ) = X - F_A

This establishes the claim. Now let T be a set of
propositional formulas. By hypothesis, for each
finite T_0 subset T, there is a nu-bar making all
phi in T_0 true, i.e.,

Cap_( A in T_0) F_A =/= nullset

By the compactness of X,

Cap_( A in T ) F_A =/= nullset

Thus, there is a truth assignment nu-bar making
all A in T true.

< end >



Pretty much, George, your just a troll.

mitch

[mitch]

Read Aristotle's distinction between an
argument from principles and an argument
from belief.

Philosophers, their metaphysics, and their
complaints about metaphysics have eradicated
argument from principles from modern treatises
on logic.

The syntax/semantics distinction does not hide
the difference in form. The only thing you have
formalized is argument from belief. And, like
Mr. Di Egidio, your own choice of ignorance blinds
you to this fact.

mitch

[mitch]

On 10/14/2017 09:33 PM, George Greene wrote:
> On Wednesday, October 11, 2017 at 4:50:33 PM UTC-4, mitch wrote:
>> Sadly, Nam, the mathematics with which you are familiar
>
> You give Nam WAY too much credit. Nam is not familiar with mathematics.
>
>
>> Presumably, your "propositional logic" is associated with
>> sixteen truth tables.
>
> Nam does NOT own propositional logic.
> THIS IS *EVERY*BODY's propositional logic.
> But the 16 truth-tables would exist WITH IT OR WITHOUT IT.
> Propositinoal logic can get by WITH TWO truth-tables (one for Or and one for Not), if you want to go that way. Isomorphisms with other ways ARE constructble.
>
>> This association is so close that there
>> are authors who have abandoned axiomatics because of the
>> decidability that truth tables provide.
>
> God, what utter horse-shit.

Again, you are misrepresenting what can be
found in the literature.

I believe it had been one of Quine's books in
which this had been stated explicitly.
Unfortunately, I had to sell a large number
of books when I had some financial problems.

Otherwise, I would quote the book in which
it had been stated.

And, for what it is worth, unary negation is
eliminable by virtue of the complete connectives.
But, when you consider what makes a complete
connective significant, you begin looking at
the entire system as a single system. So,
you look at all 16 truth tables and the relations
between them.

Your statement is certainly true for the kind
of logic promoted by Mr. Di Egidio. When we
paraphrase statements made by people to discover
the nefarious "logical form" by which people
truly communicate -- because they certainly cannot
be communicating with the actual language they
speak to one another -- we will discern logic
words like "or" and "not". But, it would seem
that there really is no agreement on a principled
classification of "logical constant" when you
proceed this way,

https://plato.stanford.edu/entries/logical-constants/




So, for my part, if the word "mathematical" is to have
any meaning in the expression "mathematical logic" then
the fact that mathematical structures provide a principled
demarcation of logical constants is convenient. The
analysis of negation and de Morgan conjugations
as involution over the system of truth tables will
expose both that truth tables can be put into
isomorphism with the Cartesian product of the
subdirectly irreducible de Morgan algebra on
four symbols with itself and truth tables can be
organized into a finite affine plane over 16
elements.

Just because you "can go some way" in philosophy
does not make it mathematical.

mitch

[mitch]

On 10/14/2017 09:33 PM, George Greene wrote:

It is called combinatorial topology. There is an excellent
and comprehensive text by Alexandroff available at Dover
books.

mitch

[mitch]

On 10/14/2017 09:33 PM, George Greene wrote:

Occam's razor includes the provision concerning
all things being equal.

Things are not equal here. Mathematical logic
should be grounded in mathematical constructions.

mitch

[mitch]

Learn some mathematics.

The author recommendation is in the other reply.

mitch

[mitch]

On 10/14/2017 09:36 PM, George Greene wrote:

Look here:

http://www.iue.tuwien.ac.at/phd/heinzl/node37.html

mitch

[mitch]

On 10/14/2017 03:27 AM, WM wrote:
> Am Samstag, 14. Oktober 2017 02:29:20 UTC+2 schrieb mitch:
>
>
>>
>> Your "potential infinity" falls out of the quantitative interpretation
>> of universal quantifiers.
>

> Yes. But that's how mathematics works. Therefore the formulation "for all n in |N" is not mathematics. I prefer "for every n in |N". It leaves the completeness question where it belongs.
>

I noticed the play on words in your earlier statement. But, I had
not connected it with Russell's distinction between "all" and "any"
until now. Because of Zuhair's threads on impredicativity I have
been looking at Russell's paper on the theory of types.

Russell's interpretation of a universal quantifier in

Ax( phi(x) )

is not

"for every value that x can take, phi(x)"

but, rather

"phi(x), always"

One of his telling passages is

< begin quote >

Even if there were such an object as "all men",
it is plain that it is not this object to which
we attribute mortality when we say "all men are
mortal". If we were attributing mortality to this
object, we should have to say "/all/ /men/ is mortal".
Thus the supposition that there is an object as
"all men" will not help us to interpret "all men
are mortal"

< end quote >

Russell's distinction between "all" and "any" is
explained in relation to the use of an "ambiguous
individual" to arrive at a universal generalization.
He refers to the variables used to present this
ambiguity as "real variables".

His argument is based on an observation I made for
myself many years ago. Proofs in mathematics are
very similar to "transactions" in databases. Mathematical
proofs begin with closed formulas (true in all models)
and end with closed formulas (true in all models). So,
when free variables appear in a proof, they never occur
as an antecedent assumption or a succedent conclusion.

The example he uses corresponds with the proof,

001 | Ax( Fx -> Gx ) ................ assumption
002 | | Ax( Fx ) ...................... assumption
003 | | Fy -> Gy ...................... Universal elimination, step 001
004 | | Fy ............................ Universal elimination, step 002
005 | | Gy ............................ Conditional elimination, steps
003 and 004
006 | | Ay( Gy ) ...................... Universal introduction, step 005


Universal introduction is a "structural rule" in the
standard presentation of first-order logic. It depends
on there being no free occurrences of the "apparent
variable" used in the quantifier introduced at step 006
in any of the assumption steps preceding step 006. In the
proof above, strokes in the form of '|' are used to keep
track of assumptions in force at any step. So, both
assumptions from 001 and 002 are in force at 006.

The general rule may be stated as

< begin >

If the last step obtained so far has as its ordered
assumption wffs phi_1, phi_2, ..., phi_n (where n may
be 0) and if a step preceding the last step or identical
with the last step has the same ordered assumption steps
as the last step and has as its wff phi and if x is not
free in phi_1, phi_2, ..., phi_n, then as a new step
you may extend all vertical lines of the last step and
write "Ax( phi )".

< end >

with the general form

.
:
| phi_1
| .
| :
| . . . | phi_n
| | .
| | :
k | | phi
| | .
| | :
i | | psi
i + 1 | | Ax( phi ), where x is not free in phi_1, phi_2, ..., phi_n


The problem here from the standpoint of first-order
logic is an observation made by Mr. Percival in this
forum. Namely, Russell had no semantic theory. His
philosophy of mathematics had been based upon a theory
of knowledge (epsitemology). I personally find no
problem with this. But, the difficulty arises because
what is supposed to be preserved at every step of a
proof is "truth".

What this means for those dissatisfied with such an
account is that one needs a theory for how formulas
with free variables in the middle of a proof could
be "true".

This is why first-order logic interprets universal
quantifiers as

"for every value that x can take, phi(x)"

in the context of satisfaction maps as described by
Tarski.

This, however, is why it is significant that you
have a different theory of truth. In numbered
sentence (8) from the link,

https://plato.stanford.edu/entries/truth-identity/

you will find the statement,

(8) All facts are identical with true thinkables

The SEP link openly states that it will not consider
this statement in its presentation. However, they
refer to a number of theorists who hold this position
among which is Hornsby. A paper of hers is available
at JSTOR,

https://www.jstor.org/stable/4545250?seq=1#page_scan_tab_contents

although one would have to have an account to read
it. Accounts with limited access are free.

I have never, myself, considered an epistemic conception
of truth in conjunction with an identity theory of
truth, but that seems to be your position.

Martin-Lof's paper,

https://www.andrew.cmu.edu/user/ulrikb/80-518-818/MartinLof83.pdf

discusses how intuitionism has led to an interpretation
of logical constants that restores the kind of epistemic
notion of truth underlying a Russellian theory of
knowledge.

mitch

[WM]

Am Samstag, 14. Oktober 2017 21:50:02 UTC+2 schrieb George Greene:
> On Wednesday, October 11, 2017 at 8:37:32 AM UTC-4, WM wrote:
> > - Axiom of well-ordering: Every set can be well-ordered. (This axiom is not constructive. In most cases provably no set theoretic definition of a well-order can be found.)
> THAT'S *WHY*IT*HAS* to be *AN* *A*X*I*O*M*,

No. An axiom establishes what can be done. An axiom that establishes what cannot be done is nonsense. By the way Zermelo expected and "proved" and said that it can be done. Every set can be well-oredered, Fraenkel said that this had not yet been accomplished.

> If a definition were available then you would USE THAT, INSTEAD,

A definition of n + 1 is available and yields a natural number for every n. Nevertheless there is an axiom stating precisely this.

Of course all set theorists expected that well-ordering could be done for ever set. They were just stupid. The modern ones including you are deceivers.

> The construction would constitute THE PROOF of the existence AS A THEOREM,
> in which case there COULD BE no AXIOM! Axioms HAVE to be INDEPENCENT OF the
> OTHER AXIOMS! They have to NOT be provable from the other axioms!!

But they must no be contradicted by other axioms and theorems derived from them. Here we have the proof that countability is same as well-orderability.

>

> > - Axiom of three points on a line: Every triple of points belongs to a straight line. (But in most cases provably no geometrical construction can be given.)
>
> You are completely mis-stating this. In the plane or in space, three points DON'T have to lie on a straight line. WHAT axiom did you EVEN MEAN?

An axiom as silly and stupid as the well-ordering axiom.

>
> > - Axiom of ten even primes: There are 10 even prime numbers. (But provably no arithmetical method to find them is available.)
>
> That is just false. It is disprovable in PA.

Like the well-ordering of uncountable sets is disprovable in set theory: There is no set-theoretically definable well-ordering of the real numbers.

Regards, WM

[WM]

Am Samstag, 14. Oktober 2017 21:51:45 UTC+2 schrieb George Greene:
> On Wednesday, October 11, 2017 at 12:49:10 PM UTC-4, WM wrote:

> > Here is an important statement:
> > For every n in |N: Every n belongs to a finite initial segment
>
> Of course
>
> > and does neither require nor allow to consider the set |N as actually closed.
>
> That statement is NOT important. It is also not coherent. It is also not mathematically.

It is all of that.

> YOU CANNOT DEFINED whether any purported "set" is or is not "actually closed".

I do not define it. I prove that for every natural number the set is not actually infinite. More is neither required nor possible.

Regards, WM

[WM]

Am Sonntag, 15. Oktober 2017 04:29:42 UTC+2 schrieb Shobe, Martin:
> The giveaways are he's asking for a
> second one and he's calling (3,5,7) one. This falls in the category of
> things that can easily be proven false.
>

Just like the distinguishability and hence finite expressability of uncountably many ideas.

Regards, WM

[George Greene]

Oh, you're right.
I guess (belatedly) that beyond 3,5,7, you CAN prove that
3 consecutive odd numbers must have ONE with a factor OF THREE, Duh!

> This falls in the category of
> things that can easily be proven false.

You're exactly right. I was indeed giving him credit for being able
to prove that if 3,5,7 were the kind of thing he was talking about,
he would BE ABLE to prove a simple theorem about divisibility by 3, if
one was relevant. I gave him credit for telling the truth about it not
being that easy. But he was lying and it was.

[George Greene]

On Sunday, October 15, 2017 at 3:59:40 PM UTC-4, WM wrote:
> Am Samstag, 14. Oktober 2017 21:50:02 UTC+2 schrieb George Greene:
> > On Wednesday, October 11, 2017 at 8:37:32 AM UTC-4, WM wrote:
> > > - Axiom of well-ordering: Every set can be well-ordered. (This axiom is not constructive. In most cases provably no set theoretic definition of a well-order can be found.)
> > THAT'S *WHY*IT*HAS* to be *AN* *A*X*I*O*M*,
>
> No. An axiom establishes what can be done.
> An axiom that establishes what cannot be done is nonsense.

You really do just need to go fuck yourself.
The word "nonsense" IS NONSENSE in this context!
IT DOES NOT MATTER how "nonsensical" YOU think something may look!
THE ONLY thing that COUNTS AROUND HERE
is DERIVING A LOGICAL CONTRADICTION from something!
That is the ONLY limitation on ANYbody else's freedom that YOU get to impose!

If you can't derive a contradiction from it then the fact that it is non-
constructive simply DOES NOT MATTER.

[George Greene]

On Sunday, October 15, 2017 at 3:59:40 PM UTC-4, WM wrote:

> Like the well-ordering of uncountable sets is disprovable in set theory:

It IS NOT, dumbass.
You don't know what a proof is.

> There is no set-theoretically definable well-ordering of the real numbers.

The fact that something is not definable DOES NOT preclude its existence, YOU IDIOT! Given that THE DEFINITIONS ARE COUNTABLE and that that enumeration IS DIAGONALIZABLE, it is A TRIVIAL theorem that UNdefinable things MUST exist!!

[Ben Bacarisse]

Ben Bacarisse <ben.u...@bsb.me.uk> writes:

> WM <wolfgang.m...@hs-augsburg.de> writes:
>
>> Am Freitag, 13. Oktober 2017 22:41:55 UTC+2 schrieb Ben Bacarisse:
>>> WM <wolfgang.m...@hs-augsburg.de> writes:
>>>
>>> > Am Donnerstag, 12. Oktober 2017 21:08:59 UTC+2 schrieb Ben Bacarisse:
>>> >
>>> >> > Try to imagine a path that cannot be constructed by nodes. Note that
>>> >> > by definition a path is a construct by certain nodes.
>>> >>
>>> >> The left-hand edge is constructed by nodes. It is not in the image of
>>> >> your bijection.
>>> >
>>> > Most paths are not in the bijection because they cannot be defined.
>>>
>>> I just defined a path not in your supposed bijection: the left-hand
>>> edge. It is defined by nodes -- those that are powers of 2.
>>>
>> In potential infinity this path is only the set of its finite initial
>> segments. Each one has its own node.
>
> The left-hand edge is one such path.

Nothing about this I see, just a new thread as a distraction from your
obvious error. A wise move. You'd have to either admit that the
function you gave does /not/ include all paths, or you'd have to say the
left-hand edge is not a path in the tree, thereby admitting that it was
always about definitions (a WM-binary-tree) and not substance.

--
Ben.

[WM]

Am Montag, 16. Oktober 2017 00:19:36 UTC+2 schrieb George Greene:


> I guess (belatedly) that beyond 3,5,7, you CAN prove that
> 3 consecutive odd numbers must have ONE with a factor OF THREE, Duh!
>
> > This falls in the category of
> > things that can easily be proven false.

[WM]

Am Montag, 16. Oktober 2017 00:23:56 UTC+2 schrieb George Greene:


> THE ONLY thing that COUNTS AROUND HERE
> is DERIVING A LOGICAL CONTRADICTION from something!
> That is the ONLY limitation on ANYbody else's freedom that YOU get to impose!
>
> If you can't derive a contradiction from it then the fact that it is non-
> constructive simply DOES NOT MATTER.

But I can. The distinguishability and hence finite expressability of uncountably many ideas is a logical contradiction. They can neither be chosen nor be well-ordered.

Regards, WM

[WM]

Am Montag, 16. Oktober 2017 00:25:18 UTC+2 schrieb George Greene:

> On Sunday, October 15, 2017 at 3:59:40 PM UTC-4, WM wrote:
>
> > Like the well-ordering of uncountable sets is disprovable in set theory:

> > There is no set-theoretically definable well-ordering of the real numbers.
>
> The fact that something is not definable DOES NOT preclude its existence,

It does, if the thing to be defined is a definition like a real number.

Given that THE DEFINITIONS ARE COUNTABLE and that that enumeration IS DIAGONALIZABLE, it is A TRIVIAL theorem that UNdefinable things MUST exist!!

That seems so to those believing in finished infinity. But infinity is never finished, and diagonalization proves just and only this.

Regards, WM

[WM]

It is the sequence of its finite initial segments.

Concerning a simple proof of countability see A direct comparison test for the Binary Tree.

Regards, WM

[graham...@gmail.com]

I think George suggested "EACH"


The problem with above proof is there are NUMEROUS steps

(I will try to find my notes)

on Quantifier Elimination.


ELIMINATE NOT(), PRENEX FORM, CONJUNCTIVE NORMAL FORM ...



*MASSIVELY COMPLICATED*


HENCE we are all stuck here talking 20TH C Axiom Schema written in W.F.F. 2.0



So YES ..... QUANTIFIERS ARE PREDICATES!

but NO ..... DONT EVEN BOTHER!



ALL YOU NEED IS 1 QUANTIFIER!

JUST 1!



Right HERE!

ss(S,T) <- ALL(m) meS -> meT


* * * * * * * * * * * * * * *




then you can DO THE LOGIC!


e(X,effs) <- f(X) ........................ SPECIFICATION OF SET 'effs'
e(X,gees) <- g(X)

ss(effs,gees)

[George Greene]

On Monday, October 16, 2017 at 2:45:01 AM UTC-4, WM wrote:

> Just like the distinguishability
> and hence finite expressability of uncountably many ideas.

Damn, you're stupid.
Nobody has alleged that uncountably many ideas are finitely expressable.
Yes, they are finitarily distinguishable -- any 2 of them are distinguishable after finitely much effort -- but that's ONLY IF you in fact ALREADY KNOW that they in fact ARE distinct. THEY MIGHT NOT be. THEY MIGHT BE THE SAME.

[George Greene]

On Monday, October 16, 2017 at 2:57:50 AM UTC-4, WM wrote:
> It is the sequence of its finite initial segments.

NO, IT ISN'T.
NOTHING is the sequence of its finite initial segments. It is the sequence of ITS INDIVIDAL PARTS, YOU IDIOT. 1,2,3,4,5
is the sequence of 1,2,3,4,and 5. It IS NOT the sequence of
1,1,2,1,2,3,1,2,3,4,1,2,3,4,5. I am missing some "sequence-constructor delimiters" here but the WHOLE point is that it IS YOU, AND NOT me who is failing
to notice that sequences of individuals and individuals ARE INHERENTLY DIFFERENT kinds of things. Except, of course, in the case of the von Neumann encoding of the ordinals. But THOSE are SETS AND NOT sequences, so that's not relevant here.

[Ben Bacarisse]

If the left-hand edge is a path in the tree, it is not in the image of
the function you gave. If it not a path in the tree, your binary tree
is not the same as anyone else's and you are simply playing crank 101
with words.

Here's the mistake (which you have tried to hide by cutting it from the
quoted material):

me: It would be a great coup for you if you could write out the bijection
from N to the paths in the binary tree.

you: Map every finite path on its last node.

So, is the left-hand edge a path in the tree or not?

> Concerning a simple proof of countability see A direct comparison test
> for the Binary Tree.

But you did not make the error of claiming to state a bijection in that
thread. You made that error here, and people should see the point
cleared up here. (One way to do that is to stop posting here and
concentrate on the new thread you started when you realised you'd made
this mistake.)

--
Ben.

[Shobe, Martin]

The issue is existence.

Martin Shobe

[WM]

What exists can be found by chance. A well-ordering of uncountable sets cannot be found by chance. ==> It does not exist.

Regards, WM

[WM]

Am Dienstag, 17. Oktober 2017 01:50:45 UTC+2 schrieb George Greene:
> On Monday, October 16, 2017 at 2:45:01 AM UTC-4, WM wrote:
>
> > Just like the distinguishability
> > and hence finite expressability of uncountably many ideas.

> Nobody has alleged that uncountably many ideas are finitely expressable.

Everybody who believes in the existence of a well-ordering of uncountable sets does so.

> Yes, they are finitarily distinguishable -- any 2 of them are distinguishable after finitely much effort

No.

> -- but that's ONLY IF you in fact ALREADY KNOW that they in fact ARE distinct.

And for that task you must in fact already know what you are talking about. That means, you must lready have defined the elements.

Therefore the claim "any two of them are distinguishable after finitely much effort" is the summit of stupidness or deceiving. It is hard to distinguish whether the more intelligent set theorists are deliberate defrauders or simply as stupid as the average.

Regards, WM

[WM]

Am Dienstag, 17. Oktober 2017 03:01:17 UTC+2 schrieb Ben Bacarisse:


> me: It would be a great coup for you if you could write out the bijection
> from N to the paths in the binary tree.
>
> you: Map every finite path on its last node.
>
> So, is the left-hand edge a path in the tree or not?

You mean the path 0.000... that always goes left?

Yes it is a path of its own in actual infinity (in potential infinity it is only the sequence of its finite initial segments.)

>
> > Concerning a simple proof of countability see A direct comparison test
> > for the Binary Tree.
>
> But you did not make the error of claiming to state a bijection in that
> thread. You made that error here, and people should see the point
> cleared up here. (One way to do that is to stop posting here and
> concentrate on the new thread you started when you realised you'd made
> this mistake.)

It is not a mistake. The bijection is as I said in potential infinity. In actual infinity there are two different ways to obtain the path 0.000...

The Binary Tree contains paths like 0.010101... converging to 1/3 and paths converging to irrational numbers but not these limits themselves. However, there is an exception with limits that are fractions with denominator divisible by 2 like 0.1000... or 0.000...

Such numbers possess several representations. Consider 0 for instance. Besides being represented by the path 0.000... it is the limit of the sequence of paths 0.1t, 0.01t, 0.001t, ... --> 0,000... where t is an arbitrary tail, for instance t = 000... or t = 111... or t = 010101... etc., or a mixture of these.

When we assume that every path can be distinguished from all other paths, then the path 0.000... differs from all paths of a sequence like

0.111..., 0.0111..., 0.00111..., ... (S)

That means, when each path of this sequence is completely coloured, then the path 0.000... is not yet completely coloured. In other words, it is not possible to colour (or to cover) the Binary Tree by different sets of infinite paths. Each and every path is required.

On the one hand, this is clear, because every path of the sequence (S) has a tail of nodes consisting of bits 1 only, while 0.000... does not. Let's call this position A.

On the other hand, we cannot find a node 0 of the path 0.000... which is not covered by the sequence (S). That means, we cannot distinguish the path 0.000... from all other paths of the Binary Tree. We can colour or cover the whole Binary Tree by a set U of paths not containing the path 0.000... or by a set V containing it. Let's call this position B.

If (A) is correct, then there must be nodes in 0.000... that cannot be found and defined. That implies that actual infinity, the complete infinite Binary Tree, and its infinite paths do not exist. Because nodes that cannot be defined cannot yet exist. They only can "come into being".

If (B) is correct, then 0.000... cannot be distinguished from all paths of the sequence. That implies that in a Cantor-list like the following

0.1
0.01
0.001
0.0001
...

when replacing the diagonal digit 1 by 0, the resulting antidiagonal 0.000... cannot be distinguished from all entries. Therefore Cantor's diagonal argument fails in this special case and hence always, because it is based on a proof by contradiction which never must fail.

Result: There is no uncountable set of paths in the Binary Tree: In case (A) there is no actual infinity and therefore it cannot be surpassed. In case (B) there is actual infinity but it does not supply uncountable sets.

Regards, WM

[shio...@googlemail.com]

No, that is not even true in reality.
THere are a lot of things in physics that we will never be able to measure.

And there are things in math that exist but cannot be found.

[George Greene]

On Tuesday, October 17, 2017 at 5:44:56 AM UTC-4, WM wrote:
> What exists can be found by chance.

What a crock of shit.
Every finite natural number exists.
If you think you can find ANY of them -- even ONE -- by chance -- WHAT IS the probability of your finding it?
DAMN,you're stupid.

[George Greene]

On Tuesday, October 17, 2017 at 10:57:26 AM UTC-4, shio...@googlemail.com wrote:
> And there are things in math that exist but cannot be found.

I don't buy that either. Certainly, there are things that are not definable.
But "found" IS NOT even a mathematical CONCEPT! Anybody who is talking about whether something "can" be "found" is speaking metaphorically.
If the thing being sought is finitarily describable then one can find it just by going through all the finite descriptions in some known order. The probability that you have found it ALREADY may always be negligibly small but there is (in that case) a KNOWN Finding Algorithm that "finds" it with probability 1 EVENTUALLY. Things for which we don't know an order are HARDER to "find" but even if you can't say HOW, sometimes, well, it is STILL the case that YOU DID *GET* there!

[Julio Di Egidio]

On Tuesday, October 17, 2017 at 9:56:54 PM UTC+2, George Greene wrote:
> On Tuesday, October 17, 2017 at 10:57:26 AM UTC-4, shio...@googlemail.com wrote:
> > And there are things in math that exist but cannot be found.
>
> I don't buy that either. Certainly, there are things that are not definable.

Not definable in some system. In general, I would argue, things that are
not definable are not things at all.

> But "found" IS NOT even a mathematical CONCEPT!

I think they mean undecidable: and that too is relative. Indeed, does
absolute undecidability exist at all? Similarly to the above, I'd think
not.

Random link:

THEOREM 2:
There exists no L(PA)-definable bit sequence which is purely random at level Pi_1^0.
<https://www.cs.nyu.edu/pipermail/fom/2015-September/019144.html>

Julio

[George Greene]

On Tuesday, October 17, 2017 at 5:45:04 AM UTC-4, WM wrote:
> Am Dienstag, 17. Oktober 2017 01:50:45 UTC+2 schrieb George Greene:
> > Nobody has alleged that uncountably many ideas are finitely expressable.
>
> Everybody who believes in the
> existence of a well-ordering of uncountable sets does so.

You are LYING; we DO NOT.

[George Greene]

On Tuesday, October 17, 2017 at 5:45:04 AM UTC-4, WM wrote:

> Therefore the claim "any two of them are distinguishable
> after finitely much effort" is the summit of stupidness or deceiving

It is NEITHER of the above; it is A FACT.
IF THEY ARE in fact TWO then THEY DO DIFFER after A FINITE number of
places. If they don't then they ARE NOT two, THEY ARE ONE, so there IS NOTHING TO distinguish!

[Peter Percival]

WM wrote:
> - Axiom of well-ordering: Every set can be well-ordered. (This axiom is not constructive. In most cases provably no set theoretic definition of a well-order can be found.)
>

> - Axiom of three points on a line: Every triple of points belongs to a straight line. (But in most cases provably no geometrical construction can be given.)
>

> - Axiom of ten even primes: There are 10 even prime numbers. (But provably no arithmetical method to find them is available.)
>

> - Axiom of prime number triples: There is a second triple of prime numbers besides (3, 5, 7). (But provably this second triple is not arithmetically definable.)
>

> - Axiom of meagre sum (AMS): There is a set of n different positive natural numbers with sum n*n/2. (This axiom is not constructive. Provably no such set can be found.)
>
> - Axiom of ultimate mathematics simplification: All mathematical problems are solved by whatever I declare as the solution. (This axiom is guaranteed not less useful than the axiom of choice.)
>
> Regards, WM
>
The axiom of choice is provable consistent with the other axioms of set
theory (say ZF, for definiteness). The other "axioms" are not; in fact
they are demonstrably false.

--
Do, as a concession to my poor wits, Lord Darlington, just explain
to me what you really mean.
I think I had better not, Duchess. Nowadays to be intelligible is
to be found out. -- Oscar Wilde, Lady Windermere's Fan