Confusion of Set and Element of Set

[William]

>> > But if there are infinitely many starting elements, then there are no successors. You cannot count beyond all infinitely many starting elements.
>> There are infinitely many starting elements but each starting element is finite.

>If all are used, then none remains.

Indeed, no starting element remains so there is no problem

>> There is no starting element that does not have a successor, no starting element that you cannot "count beyond".

>That is irrelevant. ℕ has no finite successor,

And again you confuse a set with the element of a set. *The set* |N_F has no finite successor but each *element of the set |N* has a finite successor.

--
William Hughes

[WM]

William schrieb am Mittwoch, 22. September 2021 um 19:45:23 UTC+2:

> > ℕ has no finite successor,
>
> And again you confuse a set with the element of a set. *The set* |N_F has no finite successor but each *element of the set |N* has a finite successor.

I do not confuse them but I understand that then the set must contain more than its elements which have successors each.

Regards, WM

[William]

On Wednesday, September 22, 2021 at 3:41:46 PM UTC-3, WM wrote:
> William schrieb am Mittwoch, 22. September 2021 um 19:45:23 UTC+2:
>
> > > ℕ has no finite successor,
> >
> > And again you confuse a set with the element of a set. *The set* |N_F has no finite successor but each *element of the set |N* has a finite successor.
> I do not confuse them

Piffle

> but I understand that then the set must contain more than its elements

No, you have shown that the elements of the set must have successors, You have shown nothing about the set.

--
William Hughes

[zelos...@gmail.com]

A set by definition is all its members. There cannot be "more" than it.

[Eram semper recta]

How can a set be confused with an element? LMAO. There is no definition of set and an element is also called a "set".

>
> Regards, WM

[Eram semper recta]

And moron shit bag Malum says in another thread:

A set cannot contain itself. LMAO.

N = {1, 2, 3, ...} is a subset of N.

When is N a subset of N? After 5 trillion elements? Maybe 10 trillion? Maybe more? Chuckle.

Only problem is that N never contains all its "members" because there is greatest end.

Mainstream math Flat-earthers are hilarious!

"Infinity" is the opiate of fundamentalist mainstream math academics. Without it, not a single one would get a degree, because bullshit is not possible without "infinity". LMAO.

[zelos...@gmail.com]

Read up on axiom of regularity that forbids it :)

>N = {1, 2, 3, ...} is a subset of N.

Correct, N is not a member of itself there so it is all fine!

>When is N a subset of N? After 5 trillion elements? Maybe 10 trillion? Maybe more? Chuckle.

There is no "when", it ALWAYS is and ALWAYS was.

>Only problem is that N never contains all its "members" because there is greatest end.

It contains all its members. All sets contain all its members.

>Mainstream math Flat-earthers are hilarious!

You are one funny guy indeed! You are the flat-earther of mathematics!

>"Infinity" is the opiate of fundamentalist mainstream math academics. Without it, not a single one would get a degree, because bullshit is not possible without "infinity". LMAO.

Bullshit is possible with or without it :) You bullshit all the time and reject it!

[WM]

William schrieb am Mittwoch, 22. September 2021 um 21:37:56 UTC+2:
> On Wednesday, September 22, 2021 at 3:41:46 PM UTC-3, WM wrote:
> > William schrieb am Mittwoch, 22. September 2021 um 19:45:23 UTC+2:
> >
> > > > ℕ has no finite successor,
> > >
> > > And again you confuse a set with the element of a set. *The set* |N_F has no finite successor but each *element of the set |N* has a finite successor.
> > I do not confuse them

> > but I understand that then the set must contain more than its elements
> No, you have shown that the elements of the set must have successors,

Every definable element of |N has infinitely many successors.

> You have shown nothing about the set.

"All elements n" or "the set |N" has no natural numbers as successors.

This proves that there are dark natural numbers.

Regards, WM

[WM]

zelos...@gmail.com schrieb am Donnerstag, 23. September 2021 um 07:52:59 UTC+2:
> onsdag 22 september 2021 kl. 20:41:46 UTC+2 skrev WM:
> > William schrieb am Mittwoch, 22. September 2021 um 19:45:23 UTC+2:
> >
> > > > ℕ has no finite successor,
> > >
> > > And again you confuse a set with the element of a set. *The set* |N_F has no finite successor but each *element of the set |N* has a finite successor.
> > I do not confuse them but I understand that then the set must contain more than its elements which have successors each.
> >

> A set by definition is all its members. There cannot be "more" than it.

But there can be more than every definable element. In fact every definalbe element is followed by infinitely many. All elements or the set has no successors. This proves that not all elements can be defined.

Regards, WM

[William]

On Thursday, September 23, 2021 at 9:36:41 AM UTC-3, WM wrote:

> William schrieb am Mittwoch, 22. September 2021 um 21:37:56 UTC+2:

> > You have shown nothing about the set.
> "All elements n"

Standard use of the ambiguous "All"

--
William Hughes

[William]

On Thursday, September 23, 2021 at 9:38:25 AM UTC-3, WM wrote:

> All elements

Standard use of ambiguous all.

--
William Hughes

[WM]

All means all. That is not ambiguous. The set is all its elements.

Regards, WM

[William]

On Thursday, September 23, 2021 at 10:04:11 AM UTC-3, WM wrote:
> William schrieb am Donnerstag, 23. September 2021 um 14:52:18 UTC+2:
> > On Thursday, September 23, 2021 at 9:36:41 AM UTC-3, WM wrote:
> > > William schrieb am Mittwoch, 22. September 2021 um 21:37:56 UTC+2:
> >
> > > > You have shown nothing about the set.
> > > "All elements n"
> > Standard use of the ambiguous "All"
> >
> All means all.

All elements of N have property P can mean

The set N has property P
Every element of the set N has property P

The two statements are different. "All" is ambiguous.

--
William Hughes

[Gus Gassmann]

On Thursday, 23 September 2021 at 09:38:25 UTC-3, WM wrote:
> But there can be more than every definable element. In fact every definalbe element is followed by infinitely many. All elements or the set has no successors.

Why the *FUCK* would you think that a set has a successor?

[zelos...@gmail.com]

Your "definable" is meaningless and "followed" does not apply to all sets.

[WM]

William schrieb am Donnerstag, 23. September 2021 um 15:11:53 UTC+2:
> On Thursday, September 23, 2021 at 10:04:11 AM UTC-3, WM wrote:

> > All means all.
> All elements of N have property P can mean
>
> The set N has property P

No. The set, as you treat it, is an individual different from any natural number. Here we consider the set |N simply as "all natural numbers". They have no successors (except omega).

Every definable finite initial segment of |N has infinitely many successors.

You should be able to comprehend these two statements. You should further be able to see that "all natural numbers" comprehend more than "all definable natural numbers", namely all successors of definable naqtural numbers too.

Regards, WM

[WM]

Because the set is simply all its numbers, all definable of which have successors.

Regards, WM

[Transfinity]

> Your "definable" is meaningless and "followed" does not apply to all sets.

Let "definable number" simply amount to "number that has successors". All natural numbers together have no successors.

Regards, WM

[Serg io]

All vs Each vs Every vs Any

"Each elements of N have property P"....

[Serg io]

That invalidates your definition of "definable" entirely, you have made it meaningless, which is was anyway.

[William]

On Thursday, September 23, 2021 at 10:42:01 AM UTC-3, WM wrote:

> all



as noted "all" is ambiguous.

--
William Hughes

[Gus Gassmann]

So, just for the record, your heinous: What, in your opinion, is "the successor" of |N? What kind of object is it, and if it is a set, what are its elements?

[Jim Burns]

On 9/23/2021 9:46 AM, Transfinity wrote:

> Let "definable number" simply amount to
> "number that has successors".

For each real number x, x+1 exists.
There are more real numbers than definitions.

> All natural numbers together have no successors.

Each natural number has its successor.
Different natural numbers have different successors.

All natural numbers together is not a natural number.

If you would like to have a successor for
all natural numbers together, instead of whining about how,
when the successor operation on natural numbers was defined,
it wasn't defined on things NOT a natural number,
extend it.

A good choice would be von Neumann's x+1 = (x U {x})
Then N+1 = (N U {N}) which would be
the successor of N, all natural numbers together.

Simple. And too clear to be what you mean.
Your (WM's) point is to be as unclear as possible,
I suspect that the purpose is only to hide your
inadequacies.

[Greg Cunt]

On Thursday, September 23, 2021 at 7:52:59 AM UTC+2, zelos...@gmail.com wrote:

> A set by definition is all its members.

No, that's wrong. If it were true (in a certain sense) then the set {a} would have to be /a/ and the empty set would/could not exist.

We certainly may claim that a set is "specified" by (just) its elements. For any two sets A, B: A = B <-> Ax(x e A <-> x e B).

That's why we may refer to a set by a symbol like, say, "{1, 2, 3}". This symbol refers to the set which contains the numbers 1, 2, 3, as elements, and nothing else.

But there's a reason why we do not refer to this set with a symbol like "1, 2, 3" (i.e. just a list of all its members).

{1, 2, 3} =/= 1, 2, 3 (whatever this may mean)

{1, 2} =/= 1, 2 (whatever this may mean)

and especially (at least in the context of ZF)

{1} =/= 1.

( Again, which of its members is {}? )

[Greg Cunt]

On Thursday, September 23, 2021 at 8:03:49 AM UTC+2, Eram semper recta wrote:
.
> How can a set be confused with an element?

By writing/claiming {a} = a (in the context of ZF), for example.

Ask your buddy WM for this.

[Greg Cunt]

On Thursday, September 23, 2021 at 8:10:01 AM UTC+2, Eram semper recta wrote:

> A set cannot contain itself [as an element].

Right, at least not in ZF(C).

> N = {1, 2, 3, ...} is a subset of N.

Right. For each and every set A: A c A.

> When is N a subset of N?

Huh?! When is 1 + 2 = 3?

> Only problem is

that you don't understand set theory.

Hint: IN c IN since Ax(x e IN -> x e IN).

> Mainstream <bla>

Yeah, whatever.

[Greg Cunt]

On Thursday, September 23, 2021 at 3:04:11 PM UTC+2, WM wrote:

> The set is all its elements.

Please formulate this claim using the symbolic/formal language of set theory Thanks in advance!

[Greg Cunt]

On Thursday, September 23, 2021 at 3:46:12 PM UTC+2, Transfinity wrote:

> All natural numbers together [whatever that may mean] have no successors.

But each and every natural numbers has a successor. In other words, all natural numbers have successors [there is no natural number which doesn't have a successor].

See: https://www.sgipt.org/wisms/analogik/alle/hermes41.GIF

[zelos...@gmail.com]

All natural numbers have successors ergo all members of N is definable.

The successor function is defined on N, but not for N so your latter sentence is meaningless

[zelos...@gmail.com]

Sloppy wording on my part, you are correct.

[Eram semper recta]

On Thursday, 23 September 2021 at 15:38:25 UTC+3, WM wrote:
> zelos...@gmail.com schrieb am Donnerstag, 23. September 2021 um 07:52:59 UTC+2:
> > onsdag 22 september 2021 kl. 20:41:46 UTC+2 skrev WM:
> > > William schrieb am Mittwoch, 22. September 2021 um 19:45:23 UTC+2:
> > >
> > > > > ℕ has no finite successor,
> > > >
> > > > And again you confuse a set with the element of a set. *The set* |N_F has no finite successor but each *element of the set |N* has a finite successor.
> > > I do not confuse them but I understand that then the set must contain more than its elements which have successors each.
> > >
> > A set by definition is all its members. There cannot be "more" than it.

I guess what the local idiot Malum does not realise, is that "ALL" has a special meaning. ALL does not mean a general pattern followed by an ellipsis or an authoritative decree.

The idiot talks about the "axiom" of regularity that "forbids it". Chuckle.

Poor fool is clueless as to what the bullshit axiom means to begin with....

The bullshit "axiom of foundation" states: "every non-empty set A contains an element that is disjoint from A."

Not only are elements mixed with membership operators (minimal element ∈), but there are also elements that are not actually elements but still have the same property as elements.

One not only has to be severely brain damaged to take ZFC and set theory seriously, but one has also to be a religious fanatic which describes Zelos exactly.

Like you've said Wolfgang, set theory is not only an IF-THEN occupation, it is a religion where you must not question the beliefs. After all, what would a religion be if it did not consist of a basic doctrine of beliefs?

> But there can be more than every definable element. In fact every definalbe element is followed by infinitely many. All elements or the set has no successors. This proves that not all elements can be defined.

I think using the expression "set successor" is not strictly correct. Perhaps, in set theory terminology it would be better to say, an extended set? "New from Old lingo". LMAO. Every Tom, Dick and Moron set theory teacher loves to use that phrase, but ONLY when it suits them. Chuckle.

So much for the belief of "regularity" which explains why Malum's brain is so constipated all the time. You simply cannot convince a cult member that his beliefs are wrong for otherwise he would be excommunicated.

>
> Regards, WM

[zelos...@gmail.com]

>I guess what the local idiot Malum does not realise, is that "ALL" has a special meaning.

I know what All means just fine :)

>ALL does not mean a general pattern followed by an ellipsis or an authoritative decree.

No one says it, the pattern and ellipsis just illustrates the elements and tells the reader to continue the pattern to get all the elements.

>The idiot talks about the "axiom" of regularity that "forbids it". Chuckle.

The idiot is you whom do not understand anything in mathematics :) Axiom of regularity do make it so no set can contain itself, aka have itself as a member of itself.

>Poor fool is clueless as to what the bullshit axiom means to begin with....

Know it far better than you :)

>The bullshit "axiom of foundation" states: "every non-empty set A contains an element that is disjoint from A."

Correct, written as https://wikimedia.org/api/rest_v1/media/math/render/svg/b186acc3628667ed14e8c169107244c8c5a1085c
in formal logic.

>Not only are elements mixed with membership operators (minimal element ∈), but there are also elements that are not actually elements but still have the same property as elements.

elements are defined by the membership relation, there is no mixing there. All that is in the set are members so what you're on about I have no clue.

>One not only has to be severely brain damaged to take ZFC and set theory seriously, but one has also to be a religious fanatic which describes Zelos exactly.

Religious? No one is talking about gods or deities here so no religoin.

>Like you've said Wolfgang, set theory is not only an IF-THEN occupation, it is a religion where you must not question the beliefs. After all, what would a religion be if it did not consist of a basic doctrine of beliefs?

Given it has no deities it cannot by definition be a religion :) And yes, it is IF-THEN because it is built on first order logic which uses implication to deduce things.

But there are many set theories that are used and all are equally valid. ZFC is not my favourite. I much prefer NBG

>So much for the belief of "regularity" which explains why Malum's brain is so constipated all the time. You simply cannot convince a cult member that his beliefs are wrong for otherwise he would be excommunicated.

You could convince me and all of mathematics, if you could find a legitimate contradiction where you prove that P and ~P are both true at the same time at which it would be revised. But the issue is you cannot do that. You either misrepresent things, hence making it invalid because what you address is a strawman, or you bitch and moan based on it not fitting your system based on Euclids which is invalid because no one cares about it anymore except for its historical significance.

[WM]

zelos...@gmail.com schrieb am Freitag, 24. September 2021 um 07:04:24 UTC+2:
> torsdag 23 september 2021 kl. 15:46:12 UTC+2 skrev Transfinity:

> > Let "definable number" simply amount to "number that has successors". All natural numbers together have no successors.
> >

> All natural numbers have successors ergo all members of N is definable.
>
> The successor function is defined on N, but not for N so your latter sentence is meaningless

All natural numbers together cannot be considered? Don't all exist?

Regards, WM

[WM]

zelos...@gmail.com schrieb am Freitag, 24. September 2021 um 11:16:00 UTC+2:

> You could convince me and all of mathematics, if you could find a legitimate

But every true contradiction is classified by you as illegitimate.

> contradiction where you prove that P and ~P are both true at the same time at which it would be revised.

Every path of the Binary Tree differes from every other path by at least one node. Therefore the number of nodes is at least as large as the number of paths. Contradiction with the claim that there are more houses than bricks, err more paths than nodes.

Regards, WM

[WM]

zelos...@gmail.com schrieb am Freitag, 24. September 2021 um 11:16:00 UTC+2:

> All that is in the set are members

All these members together have no successors. Contradiction with all members have successors.

Regards, WM

[WM]

Greg Cunt schrieb am Freitag, 24. September 2021 um 06:17:09 UTC+2:
> On Thursday, September 23, 2021 at 3:46:12 PM UTC+2, Transfinity wrote:
>
> > All natural numbers together [whatever that may mean]

It means what you read: All.

> have no successors.
>
> But each and every natural numbers has a successor. In other words, all natural numbers have successors [there is no natural number which doesn't have a successor].

All together have none {1, 2, 3, ...}. Contradiction.

Regards, WM

[WM]

William schrieb am Donnerstag, 23. September 2021 um 16:28:58 UTC+2:
> On Thursday, September 23, 2021 at 10:42:01 AM UTC-3, WM wrote:
>
> > all
>
> as noted "all" is ambiguous.
>

These are all natural numbers: 1, 2, 3, ... . They form an infinite sequence. None is following upon all. This is not ambiguous but shows a contradiction.

Regards, WM

[Gus Gassmann]

On Friday, 24 September 2021 at 10:35:15 UTC-3, WM wrote:
> Greg Cunt schrieb am Freitag, 24. September 2021 um 06:17:09 UTC+2:

[...]

> > But each and every natural numbers has a successor. In other words, all natural numbers have successors [there is no natural number which doesn't have a successor].
> All together have none {1, 2, 3, ...}. Contradiction.

You've gotta be pretty far gone to think that there is a successor operator that works on arbitrary sets. In particular, the set {1, 2, 3, ...} does not have a successor. I invite you to cook up an operator that defines S(|N) coherently. While you're at it, how about defining S({red, green, blue})?

[William]

On Friday, September 24, 2021 at 10:36:52 AM UTC-3, WM wrote:
> William schrieb am Donnerstag, 23. September 2021 um 16:28:58 UTC+2:
> > On Thursday, September 23, 2021 at 10:42:01 AM UTC-3, WM wrote:
> >
> > > all
> >
> > as noted "all" is ambiguous.
> >
> all

as noted "all" is ambiguous. An easy check is not to use natural language. E.g.

Let P(x) be, x has a successor.

P(|N_F) is false.
Forall x element of |N_F, P(x) is true.

No ambiguity. If you have to use natural language it is because you need the ambiguity

--
William Hughes

[Greg Cunt]

On Friday, September 24, 2021 at 4:17:03 PM UTC+2, Gus Gassmann wrote:
> On Friday, 24 September 2021 at 10:35:15 UTC-3, WM wrote:
> > Greg Cunt schrieb am Freitag, 24. September 2021 um 06:17:09 UTC+2:
> [...]
> > > But each and every natural numbers has a successor. In other words, all natural numbers have successors [there is no
> > > natural number which doesn't have a successor].
> > >
> > All together have none {1, 2, 3, ...}. Contradiction.

*sigh*

> You've gotta be pretty far gone to think that there is a successor operator that works on arbitrary sets.

There is such an operator in set theory: s(A) := A u {A}, where A is a set.

> In particular, the set {1, 2, 3, ...} does not have a successor.

Well, actually, {1, 2, 3, ..., {1, 2, 3, ...}} would be its successor.

Well, better stick to IN = {0, 1, 2, 3, ...} as defined due to von Neumann.

In such a context we (usually) have: omega = IN.

> I invite you to cook up an operator that defines S(IN) coherently.

s(IN) = {0, 1, 2, 3, ... {0, 1, 2, 3, ...}} = s(omega) = omega + 1.

But even if this weren't possible, WHY ON EARTH should

(a) each and every natural numbers has a successor. In other words, all natural numbers have successors [there is no natural number which doesn't have a successor].

(b) _the set_ of all natural numbers does not have a successor

imply a contradiction (as claimed by Mückenheim)?!

[Greg Cunt]

On Friday, September 24, 2021 at 3:35:15 PM UTC+2, WM wrote:
> Greg Cunt schrieb am Freitag, 24. September 2021 um 06:17:09 UTC+2:
> >

> > Each and every natural numbers has a successor. In other words, all natural numbers have successors [there is no natural number which doesn't have a successor].

> >
> All together have none {1, 2, 3, ...}. Contradiction.

WHY ON EARTH should

(a) each and every natural numbers has a successor. In other words, all natural numbers have successors [there is no natural number which doesn't have a successor].

(b) _the set_ of all natural numbers does not have a successor

imply a contradiction?!

[Serg io]

as usual you are Wrong again. Using your logic;

"Every path of the Binary Tree differes from every other path by at least one path. Therefore the number of paths is at least as large as the number of
nodes."

Which directly conflicts with your conclusion.


Facts:
Each Node has 3 paths connected to it.
Each Path has only 2 nodes connected to it.


All (each, every) bifurcating arborescence agree.

>
> Regards, WM
>

https://en.wikipedia.org/wiki/Binary_tree

[WM]

Jim Burns schrieb am Donnerstag, 23. September 2021 um 21:42:06 UTC+2:

> All natural numbers together is not a natural number.

That is a bit too clumsy.
All sets of definable natural numbers leave natural numbers, for instance all FISONs F(n) have successors, meaning ℕ \ F(n) =/= { }
ℕ does not leave natural numbers, meaning: ℕ \ ℕ = { }.
Seems ℕ contains more than definable numbers.

>
> If you would like to have a successor for
> all natural numbers together,

I am satisfied to have it for every set of definabe numbers.

> when the successor operation on natural numbers was defined,
> it wasn't defined on things NOT a natural number,
> extend it.

It is defined for definable natnumbers only.

Regards, WM

[WM]

There is none. Only sets S of definable numbers have successors, meaning ℕ \ S =/= { }. They have aleph_0 successors like every definable element of ℕ. Most successors are undefinable. Therefore they can only be subtracted collectively: ℕ \ ℕ = { }.

Regards, WM

[WM]

Greg Cunt schrieb am Freitag, 24. September 2021 um 05:44:04 UTC+2:
> On Thursday, September 23, 2021 at 7:52:59 AM UTC+2, zelos...@gmail.com wrote:
>
> > A set by definition is all its members.
> No, that's wrong. If it were true (in a certain sense) then the set {a} would have to be /a/ and the empty set would/could not exist.

But as they like to pervert mathematics, matheologians have decided to add some "spirit" or "soul" by writing the curly brackets.

Regards, WM

[William]

On Friday, September 24, 2021 at 1:30:30 PM UTC-3, WM wrote:

> Only sets S of definable numbers have successors,

Piffle. A element of the set |N_F has a successor even if you cannot write it down. |N_F is a Peano set.

--
William Hughes

[William]

On Friday, September 24, 2021 at 1:29:05 PM UTC-3, WM wrote:

> It is defined for definable natnumbers only.

Piffle. |N_F is a Peano set. If x is an element of |N_F then x has a successor whether or not you can write x down.

--
William Hughes

[Greg Cunt]

On Friday, September 24, 2021 at 6:31:31 PM UTC+2, WM wrote:

> mathematicians have decided to [use] curly brackets [when specifying sets].

See: https://en.wikipedia.org/wiki/Set-builder_notation

[Serg io]

{cute, but no cigar, baby mathman}

[Jim Burns]

1a. j > k
1b. j =< k

1a and 1b contradict each other.

2a. for all j, [1b]
2b. exists j, not [1b]

2a and 2b contradict each other.

Since 1a and 1b contradict each other,
"[1a]" can be substituted for "not [1b]", and
2a' and 2b' still contradict each other.

2a'. for all j, j =< k
2b'. exists j, j > k

3a. for all k, [2b']
3b. exists k, not [2b']

3a and 3b contradict each other.

Since 2a' and 2b' contradict each other,
"[2a']" can be substituted for "not [2b']", and
3a' and 3b' still contradict each other.

3a'. for all k, exists j, j > k
3b'. exists k, for all j, j =< k

3a' (all members have successors) contradicts
3b' (all these members together have a successor).

On the other hand, you (WM) will insult people who
believe their own eyes and disagree with you.

ToMAYto, toMAHto.
I guess we'll never know what's correct.

[Gus Gassmann]

On Friday, 24 September 2021 at 13:30:30 UTC-3, WM wrote:
> Gus Gassmann schrieb am Donnerstag, 23. September 2021 um 17:11:44 UTC+2:
> > On Thursday, 23 September 2021 at 10:43:07 UTC-3, WM wrote:
>
> > > Because the set is simply all its numbers, all definable of which have successors.
> > So, just for the record, your heinous: What, in your opinion, is "the successor" of |N? What kind of object is it, and if it is a set, what are its elements?
> There is none. Only sets S of definable numbers have successors

I presume 1, 2, and 3 are definable natural numbers in your goofy system. What then, in your opinion, is the successor of {1, 2, 3}?

[Transfinity]

Gus Gassmann schrieb am Freitag, 24. September 2021 um 20:59:37 UTC+2:
> On Friday, 24 September 2021 at 13:30:30 UTC-3, WM wrote:
> > Gus Gassmann schrieb am Donnerstag, 23. September 2021 um 17:11:44 UTC+2:
> > > On Thursday, 23 September 2021 at 10:43:07 UTC-3, WM wrote:
> >
> > > > Because the set is simply all its numbers, all definable of which have successors.
> > > So, just for the record, your heinous: What, in your opinion, is "the successor" of |N? What kind of object is it, and if it is a set, what are its elements?
> > There is none. Only sets S of definable numbers have successors

> I presume 1, 2, and 3 are definable natural numbers in your system. What then, in your opinion, is the successor of {1, 2, 3}?
4. Since 1, 2, 3, is same as {1, 2, 3}.

Regards, WM

[Transfinity]

Jim Burns schrieb am Freitag, 24. September 2021 um 19:08:22 UTC+2:
> On 9/24/2021 9:32 AM, WM wrote:
> > zelos...@gmail.com schrieb
> > am Freitag, 24. September 2021 um 11:16:00 UTC+2:
>
> >> All that is in the set are members
> >
> > All these members together have no successors.
> > Contradiction with all members have successors.
> 1a. j > k
> 1b. j =< k
>
> 1a and 1b contradict each other.

Find a FISON without successors.
Find a successor of |N.
Trust your eyes.

Regards, WM

[Transfinity]

Greg Cunt schrieb am Freitag, 24. September 2021 um 18:59:42 UTC+2:
> On Friday, September 24, 2021 at 6:31:31 PM UTC+2, WM wrote:
>
> > mathematicians have decided to [use] curly brackets [when specifying sets].
>

Matheologians have breathed the breath of life into the curly brackets and made them a living soul. Even the nothing became something.

Regards, WM

[Transfinity]

Then every element is definable and has successors. But all elements are mostly undefinable.

FISON F(n):
∀n ∈ ℕ: ℕ \ F(n) =/= { }
ℕ \ ℕ = { }

Regards, WM

[William]

On Friday, September 24, 2021 at 5:59:13 PM UTC-3, Transfinity wrote:
> William schrieb am Freitag, 24. September 2021 um 18:45:58 UTC+2:
> > On Friday, September 24, 2021 at 1:30:30 PM UTC-3, WM wrote:
> >
> > > Only sets S of definable numbers have successors,

> > Piffle. An element of the set |N_F has a successor even if you cannot write it down. |N_F is a Peano set.

> Then every element is definable

Piffle Not every element of a Peano set can be written down.

--
William Hughes

[WM]

In principle it can. But Peano "sets" are potentially infinite collections only.

Regards, WM

[William]

On Friday, September 24, 2021 at 6:13:27 PM UTC-3, WM wrote:
> William schrieb am Freitag, 24. September 2021 um 23:07:44 UTC+2:
> > On Friday, September 24, 2021 at 5:59:13 PM UTC-3, Transfinity wrote:
> > > William schrieb am Freitag, 24. September 2021 um 18:45:58 UTC+2:
> > > > On Friday, September 24, 2021 at 1:30:30 PM UTC-3, WM wrote:
> > > >
> > > > > Only sets S of definable numbers have successors,
> > > > Piffle. An element of the set |N_F has a successor even if you cannot write it down. |N_F is a Peano set.
> > > Then every element is definable
> > Piffle Not every element of a Peano set can be written down.
> >
> In principle it can.

Piffle.

>But Peano "sets" are potentially infinite collections only.

Piffle. A Peano set is a *set*, not something that changes. "Potentially infinite" is nonsense when applied to something that changes.
A "Potentially infinite set" is not a set.
--
William Hughes

[Serg io]

WM has ditched sets !

[Serg io]

you do not need sets when counting rocks, or sheeps

[Jim Burns]

On 9/24/2021 4:49 PM, Transfinity wrote:
> Jim Burns schrieb
> am Freitag, 24. September 2021 um 19:08:22 UTC+2:
>> On 9/24/2021 9:32 AM, WM wrote:
>>> zelos...@gmail.com schrieb
>>> am Freitag, 24. September 2021 um 11:16:00 UTC+2:

>>>> All that is in the set are members
>>>
>>> All these members together have no successors.
>>> Contradiction with all members have successors.
>>
>> 1a. j > k
>> 1b. j =< k
>>
>> 1a and 1b contradict each other.
>
> Find a FISON without successors.

Each FISON has a successor.
Different FISONs have different successors.

> Find a successor of |N.

Each element of N has a successor in N.
N is not an element of N.

> Trust your eyes.

Okay.
A natural number which cannot be counted to
even in principle is not a natural number.
And vice versa.

Define
{0,...,k} to be _the FISON from 0 to k_ iff
{0,...,k} is a collection
with a transitive and connected order '<' such that
{0,...,k} begins at 0, ends at k, and
for each _split_ BEFORE,AFTER of {0,...,k},
a _crossing-pair_ j,j+1 exists.

Define
that k can be counted to in principle iff
the FISON {0,...,k} from 0 to k exists.

Lemma.
If the FISON {0,...,k} from 0 to k exists,
then the FISON {0,...,k+1} from 0 to k+1 exists,

( Assume FISON {0,...,k} exists.
(
( Consider ({0,...,k}U{k+1})
( with the order '<' for which,
( for each j in {0,...,k}, j < k+1 and ~(k+1 < j)
( and the same as in {0,...,k}
(
( ({0,...,k}U{k+1}) is a collection
( with a transitive and connected order '<' such that
( ({0,...,k}U{k+1}) begins at 0, ends at k+1, and
( for each _split_ BEFORE,AFTER of ({0,...,k}U{k+1}),
( a _crossing-pair_ j,j+1 exists.
(
( ({0,...,k}U{k+1}) is the FISON {0,...,k+1}.
( It exists.

Note that
for each FISON {0,...,k}, FISON {0,...,k+1} exists,
and *equivalently*
no FISON {0,...,k} exists such that
all FISONs {0,...,j} sub {0,...,k}

[WM]

William schrieb am Samstag, 25. September 2021 um 00:09:33 UTC+2:
> On Friday, September 24, 2021 at 6:13:27 PM UTC-3, WM wrote:
> > William schrieb am Freitag, 24. September 2021 um 23:07:44 UTC+2:
> > > On Friday, September 24, 2021 at 5:59:13 PM UTC-3, Transfinity wrote:
> > > > William schrieb am Freitag, 24. September 2021 um 18:45:58 UTC+2:
> > > > > On Friday, September 24, 2021 at 1:30:30 PM UTC-3, WM wrote:
> > > > >
> > > > > > Only sets S of definable numbers have successors,
> > > > > Piffle. An element of the set |N_F has a successor even if you cannot write it down. |N_F is a Peano set.
> > > > Then every element is definable
> > > Piffle Not every element of a Peano set can be written down.
> > >
> > In principle it can.
> Piffle.
> >But Peano "sets" are potentially infinite collections only.
> Piffle. A Peano set is a *set*, not something that changes.

The "Peano set" comprises only elements that can be defined in principle.

> A "Potentially infinite set" is not a set.

Therefore "Peano sets" are not sets. They are collections only.

Regards, WM

[WM]

Jim Burns schrieb am Samstag, 25. September 2021 um 01:58:51 UTC+2:
> On 9/24/2021 4:49 PM, Transfinity wrote:

> > Find a FISON without successors.
> Each FISON has a successor.
> Different FISONs have different successors.
> > Find a successor of |N.
> Each element of N has a successor in N.
> N is not an element of N.

But all elements of |N are elements in |N like the elements of every FISON are elements in |N. Therefore they all together, if definable, must have successors. And when you remove them, the successors must remain.

>
> > Trust your eyes.
>
> Okay.
> A natural number which cannot be counted to
> even in principle is not a natural number.
> And vice versa.

Then your "set" |N is a potentially infinite collection only.

Regards, WM

[mitchr...@gmail.com]

No quantity of elements is empty set.
Quantities of elemental no quantities
are a set of zeros...
Zero as set math...

Mitchell Raemsch

[Serg io]

WM, you seem to be very confused. State the axioms in the language of formal logic, and using formal logic rather than the informal way you argue in
English may be needed here. Several books contain details of this formalized presentation, and it may be to your advantage to look for one, and read it
carefully. If you are not familiar with formal logic, start with a book on the subject, before specializing to one covering (formal) Peano Arithmetic. I
suggest Klenne's "Introduction to metamathematics", as more modern treatments may skip some of the formal details needed here

[William]

On Saturday, September 25, 2021 at 3:37:17 PM UTC-3, WM wrote:
> William schrieb am Samstag, 25. September 2021 um 00:09:33 UTC+2:
> > On Friday, September 24, 2021 at 6:13:27 PM UTC-3, WM wrote:
> > > William schrieb am Freitag, 24. September 2021 um 23:07:44 UTC+2:
> > > > On Friday, September 24, 2021 at 5:59:13 PM UTC-3, Transfinity wrote:
> > > > > William schrieb am Freitag, 24. September 2021 um 18:45:58 UTC+2:
> > > > > > On Friday, September 24, 2021 at 1:30:30 PM UTC-3, WM wrote:
> > > > > >
> > > > > > > Only sets S of definable numbers have successors,
> > > > > > Piffle. An element of the set |N_F has a successor even if you cannot write it down. |N_F is a Peano set.
> > > > > Then every element is definable
> > > > Piffle Not every element of a Peano set can be written down.
> > > >
> > > In principle it can.
> > Piffle.
> > >But Peano "sets" are potentially infinite collections only.
> > Piffle. A Peano set is a *set*, not something that changes.
> The "Peano set" comprises only elements that can be defined in principle.

Piffle. A Peano set contains elements that cannot be written down.

--
William Hughes

[William]

On Saturday, September 25, 2021 at 3:43:13 PM UTC-3, WM wrote:
> all

As noted "all" is ambiguous

--
William Hughes

[FromTheRafters]

WM expressed precisely :

Stop lying.

[WM]

Does every element of your set have infinitely many successors?
If you have a set, then all elements can be subtracted such that no successors remain.

Regards, WM

[WM]

William schrieb am Samstag, 25. September 2021 um 21:34:49 UTC+2:
> On Saturday, September 25, 2021 at 3:43:13 PM UTC-3, WM wrote:
> > all
>
> As noted "all" is ambiguous

All elements of a set are within this set and can be considered together without considering the set. This is not ambiguous. (Ambiguity enters only when "all" is used instead of every.) It turns out that "all elements together" have other properties than a finite subset.

Regards, WM

[William]

On Saturday, September 25, 2021 at 4:56:52 PM UTC-3, WM wrote:
> William schrieb am Samstag, 25. September 2021 um 21:28:41 UTC+2:
> > On Saturday, September 25, 2021 at 3:37:17 PM UTC-3, WM wrote:
>
> > > > >But Peano "sets" are potentially infinite collections only.
> > > > Piffle. A Peano set is a *set*, not something that changes.
> > > The "Peano set" comprises only elements that can be defined in principle.
> > Piffle. A Peano set contains elements that cannot be written down.

> Does every element of [a Peano] set have infinitely many successors?

Let P(x) be "has infinitely many successors.
P(|N_F) is false
Forall x element of |N_F, P(x) is true.

> all

As noted "all" is ambiguous.

--
William Hughes

[William]

On Saturday, September 25, 2021 at 5:00:09 PM UTC-3, WM wrote:
> William schrieb am Samstag, 25. September 2021 um 21:34:49 UTC+2:
> > On Saturday, September 25, 2021 at 3:43:13 PM UTC-3, WM wrote:
> > > all
> >
> > As noted "all" is ambiguous
> All

Still ambiguous.

--
William Hughes

[Jim Burns]

On 9/25/2021 2:43 PM, WM wrote:
> Jim Burns schrieb
> am Samstag, 25. September 2021 um 01:58:51 UTC+2:
>> On 9/24/2021 4:49 PM, Transfinity wrote:

>>> Find a FISON without successors.
>>
>> Each FISON has a successor.
>> Different FISONs have different successors.
>>
>>> Find a successor of |N.
>>
>> Each element of N has a successor in N.
>> N is not an element of N.
>
> But all elements of |N are
> elements in |N like
> the elements of every FISON are
> elements in |N.
> Therefore they all together,

Each element of N is in a FISON.
All elements together are not an element of N.

> Therefore they all together, if definable,
> must have successors.

Each element of N, definable or not, has a successor.
All elements together are not an element of N.

> And when you remove them, the successors must remain.

_They_ include all the successors.
When you remove them, the successors among them are removed.

>>> Trust your eyes.
>>
>> Okay.
>> A natural number which cannot be counted to
>> even in principle is not a natural number.
>> And vice versa.
>
> Then your "set" |N is a potentially infinite collection only.

| If k can be counted to in principle, k is in N.
| If k cannot be counted to in principle, k is not-in N.

Does potential-infiniteness disagree?
Then potential-infiniteness is wrong.

Does potential-infiniteness agree?
Great. Now, can we move on?

[Serg io]

On 9/25/2021 3:00 PM, WM wrote:
> William schrieb am Samstag, 25. September 2021 um 21:34:49 UTC+2:
>> On Saturday, September 25, 2021 at 3:43:13 PM UTC-3, WM wrote:
>>> all
>>
>> As noted "all" is ambiguous
>
> All elements of a set are within this set and can be considered together without considering the set.

so you take all the elements out of a set, the set is {}| , and all your elements are in a pile somewhere ?

So you do not have a set anymore.

This is not ambiguous. (Ambiguity enters only when "all" is used instead of every.) It turns out that "all elements together" have other properties than
a finite subset.

wrong, you do not have a set, nor a subset, nor a finite subset, anymore. Just a pile of steaming elements over there.


>
> Regards, WM
>




Just a pile of steaming Ants over there.

[FromTheRafters]

Serg io has brought this to us :

> On 9/25/2021 3:00 PM, WM wrote:
>> William schrieb am Samstag, 25. September 2021 um 21:34:49 UTC+2:
>>> On Saturday, September 25, 2021 at 3:43:13 PM UTC-3, WM wrote:
>>>> all
>>>
>>> As noted "all" is ambiguous
>>
>> All elements of a set are within this set and can be considered together
>> without considering the set.
>
> so you take all the elements out of a set, the set is {}| , and all your
> elements are in a pile somewhere ?

Very untidy. Better to consider those elements to be in a set called a
subset. That way set theory can apply. I'm sure he can pick up some
curly brackets at the hardware store or see a dentist for the braces.

[WM]

It can be made unique. All elements of |N = {1, 2, 3, ...} are 1, 2, 3, ... . They are natural numbers but not all can have successors.

Regards, WM

[Gus Gassmann]

Again you show your utter and total ignorance of all things infinity. In particular, you "prove" by assertion and intimidation. Shame on you!

[Wasell]

On Sat, 25 Sep 2021 12:34:44 -0700 (PDT), in article <eb254b24-
c840-47bf-902b...@googlegroups.com>, William wrote:
>
> On Saturday, September 25, 2021 at 3:43:13 PM UTC-3, WM wrote:
> > all
>
> As noted "all" is ambiguous

Oh, FFS! No, it really, really *isn't*! In this context
(mathematics and logic) "all" is exactly synonymous with "each".
It never, ever means "the collection of every". It is completely
unambiguous. I am pretty sure that WM knows this. He is nothing
but a despicable troll.

[William]

On Sunday, September 26, 2021 at 11:59:11 AM UTC-3, WM wrote:
> William schrieb am Samstag, 25. September 2021 um 22:09:04 UTC+2:
> > On Saturday, September 25, 2021 at 5:00:09 PM UTC-3, WM wrote:
> > > William schrieb am Samstag, 25. September 2021 um 21:34:49 UTC+2:
> > > > On Saturday, September 25, 2021 at 3:43:13 PM UTC-3, WM wrote:
> > > > > all
> > > >
> > > > As noted "all" is ambiguous
> > > All
> > Still ambiguous.
> It can be made unique.

"All elements" can mean "the set" or "the elements of the set". You are not consistent as to the meaning. Hence ambiguous. If you wish to avoid ambiguity do not use natural language. E.g.

P(X) is true if X has a successor

P(|N_F) is false
Forall x element of |N_F, P(x) is true.

--
William Hughes

[Greg Cunt]

On Sunday, September 26, 2021 at 4:59:11 PM UTC+2, WM wrote:

> [The] elements [in] IN = {1, 2, 3, ...} are 1, 2, 3, ... .

>
> They are natural numbers but not all can have successors.

Oh, so the Peano axioms do not hold for IN in Mückenmath?

So in Mückenmath there is an element in IN , say, WM such that EITHER WM+1 is identical with WM or a predecessor of WM OR WM+1 !e IN. FASCINATING!

Well at least in mathematics we have:

An e IN: n+1 e IN
and
for no n e IN the number n+1 is identical with n or a predecessor of n.

So all elements in IN DO have /successors/ (in the context of mathematics).

[Jim Burns]

On 9/24/2021 12:28 PM, WM wrote:
> Jim Burns schrieb

> am Donnerstag, 23. September 2021 um 21:42:06 UTC+2:

>> All natural numbers together is not a natural number.
>
> That is a bit too clumsy.
> All sets of definable natural numbers leave natural numbers,

Not all sets.
Let k be in N iff
k can be counted to in principle.

> for instance all FISONs F(n) have successors,
> meaning ℕ \ F(n) =/= { }

That's not a very good way to say
"all FISONs F(n) have successors"
| ℕ \ F(n) =/= { }

Try this:
| for each FISON F, exists FISON F' such that
| F'\F is a singleton, F\F' is {}.

> ℕ does not leave natural numbers, meaning: ℕ \ ℕ = { }.
> Seems ℕ contains more than definable numbers.

It seems that way to you because
| for each k, exists j: j > k
seems to you to "contradict"
| not exists k, for each j: j =< k

>> If you would like to have a successor for
>> all natural numbers together,
>
> I am satisfied to have it for every set of definabe numbers.

Sets of (definable) natural numbers are not
(definable) natural numbers.

>> when the successor operation on natural numbers was defined,
>> it wasn't defined on things NOT a natural number,
>> extend it.
>
> It is defined for definable natnumbers only.

(Definable) natural numbers are not
sets of (definable) natural numbers.

----
However, extend "successor" or don't extend it,
whatever gives you joy. It won't matter to whatever
(definable) natural numbers are.

For each j in {0,...,k}, {0,...,j} exists.
Anything in any {0,...,k} is in N_def,
so any dark "extras" are not in any {0,...,k}.
When we discuss natural numbers (which each can be
counted to in principle) your dark "extras" are not
part of the discussion.

[Serg io]

nope, {you} are confused again, one is a set, the other is a sequence.

[zelos...@gmail.com]

fredag 24 september 2021 kl. 15:25:52 UTC+2 skrev WM:
> zelos...@gmail.com schrieb am Freitag, 24. September 2021 um 07:04:24 UTC+2:
> > torsdag 23 september 2021 kl. 15:46:12 UTC+2 skrev Transfinity:
>
> > > Let "definable number" simply amount to "number that has successors". All natural numbers together have no successors.
> > >
> > All natural numbers have successors ergo all members of N is definable.
> >
> > The successor function is defined on N, but not for N so your latter sentence is meaningless
> All natural numbers together cannot be considered? Don't all exist?
>
> Regards, WM

They can but talking about "successor" to N is meaningless.

>But every true contradiction is classified by you as illegitimate.

Because they are not legitimate. You make up shit that isn't in the frame and make up your own rules and claim contradiction but thats not a contradiction because it contradicts nothing in ZFC.

>Every path of the Binary Tree differes from every other path by at least one node. Therefore the number of nodes is at least as large as the number of paths. Contradiction with the claim that there are more houses than bricks, err more paths than nodes.

Is only true for finite positions. Not when you look at all of them together. That is where your "contradiction" fails and you fail. you try to think on it in terms of the finite and apply it to the infinite, which is a non-sequitor.

>All these members together have no successors. Contradiction with all members have successors.

All natural numbers have successors. You state therea re those without successors but you have proven nothing of the sort.

[WM]

William schrieb am Sonntag, 26. September 2021 um 18:13:30 UTC+2:
> On Sunday, September 26, 2021 at 11:59:11 AM UTC-3, WM wrote:

> > > Still ambiguous.
> > It can be made unique.
> "All elements" can mean "the set" or "the elements of the set".

When I say "all elements together", then I mean all elements with no exception, but not the set of all elements, because matheologians seem to ascribe some soul or spirit to the braces. For instance the empty set is more than nothing, in their eyes. Therefore I refrain from sets but simply consider all elements which belong to the set. All natural numbers together are 1, 2, 3, ... They have no natnumbers as successors although every definable natnumber has infinitely many natnumbers as successors.

> You are not consistent as to the meaning. Hence ambiguous. If you wish to avoid ambiguity do not use natural language. E.g.
>
> P(X) is true if X has a successor
> P(|N_F) is false
> Forall x element of |N_F, P(x) is true.

P(|N_F) is false. Small wonder, the set is something else than a natural number or some natural numbers. What should be its successor? But if I say that all natural numbers together have no successor, then it is clear that there are natural numbers with less than aleph_0 successors.

Regards, WM

[Ross A. Finlayson]

On Thursday, September 23, 2021 at 6:11:53 AM UTC-7, William wrote:
> On Thursday, September 23, 2021 at 10:04:11 AM UTC-3, WM wrote:
> > William schrieb am Donnerstag, 23. September 2021 um 14:52:18 UTC+2:
> > > On Thursday, September 23, 2021 at 9:36:41 AM UTC-3, WM wrote:
> > > > William schrieb am Mittwoch, 22. September 2021 um 21:37:56 UTC+2:
> > >
> > > > > You have shown nothing about the set.
> > > > "All elements n"
> > > Standard use of the ambiguous "All"
> > >
> > All means all.
> All elements of N have property P can mean
>
> The set N has property P
> Every element of the set N has property P
>
> The two statements are different. "All" is ambiguous.
>
> --
> William Hughes


If you consider a theory that has for example only the finite
ordinals, then, apply "Russell's paradox" as it were, then,
what results is a set that carries along this "sputnik of quantification",
that results that the set of naturals contains itself (is compact, ...).

Then, when you wonder that that's been "defined away" in ZF,
still, there is more than less framing in a theory "there exists
some standard ground model of the finite integers without an
infinite", that otherwise sees a result that there isn't.

So, it's not il-logical to consider frameworks where infinites
are part of finites, ordinals with their own successor as limit
ordinals in the infinite, ..., extra-ordinary "sets" what result.

[Serg io]

On 9/27/2021 8:57 AM, WM wrote:
> William schrieb am Sonntag, 26. September 2021 um 18:13:30 UTC+2:
>> On Sunday, September 26, 2021 at 11:59:11 AM UTC-3, WM wrote:
>
>>>> Still ambiguous.
>>> It can be made unique.
>> "All elements" can mean "the set" or "the elements of the set".
>
> When I say "all elements together", then I mean all elements with no exception, but not the set of all elements, because matheologians seem to ascribe some soul or spirit to the braces.

WM's utterance that proves he knows nothing about sets, nor Math notation.

[Greg Cunt]

On Monday, September 27, 2021 at 3:57:45 PM UTC+2, WM wrote:
> William schrieb am Sonntag, 26. September 2021 um 18:13:30 UTC+2:

> When I say "all elements together" then I mean all elements with no exception,

What are they elements of?

Hint: Your idea is clearly to adopt some sort of mereology. See: https://en.wikipedia.org/wiki/Mereology

We might keep your term "All natural numbers together" to refer to the whole which "consists" of all natural numbers.

> All natural numbers together [...] have no natnumbers as successors although every definable natnumber has infinitely many natnumbers as successors.

Makes sense. Right.

> All natural numbers together, are 1, 2, 3, ...

I'd propose to write "[1, 2, 3, ...]" in ths case, to avoid any missunderstanding.

Did you learn any formal mereology? We CERTAINLY would need that for a reasonable discussion!

Things aren't "that simple" in mereology.

Hint: "A basic choice in defining a mereological system, is whether to consider things to be parts of themselves. In naive set theory a similar question arises: whether a set is to be considered a "subset" of itself. In both cases, "yes" gives rise to paradoxes analogous to Russell's paradox: Let there be an object O such that every object that is not a proper part of itself is a proper part of O. Is O a proper part of itself? No, because no object is a proper part of itself; and yes, because it meets the specified requirement for inclusion as a proper part of O. In set theory, a set is often termed an improper subset of itself. Given such paradoxes, mereology requires an axiomatic formulation." (Wikipedia)

But when you are talking about mereology, you COMPLETELY changed the topic.

Doesn't make ANY sense in the present discussion (related to set theory).

Btw.: "Husserl never claimed that mathematics could or should be grounded in part-whole rather than set theory. Lesniewski consciously derived his mereology as an alternative to set theory as a foundation of mathematics, but did not work out the details. Goodman and Quine (1947) tried to develop the natural and real numbers using the calculus of individuals, but were mostly unsuccessful; Quine did not reprint that article in his Selected Logic Papers. In a series of chapters in the books he published in the last decade of his life, Richard Milton Martin set out to do what Goodman and Quine had abandoned 30 years prior. A recurring problem with attempts to ground mathematics in mereology is how to build up the theory of relations while abstaining from set-theoretic definitions of the ordered pair. Martin argued that Eberle's (1970) theory of relational individuals solved this problem." (Wikipedia)

True in the context mereology as well:

> > You are not consistent as to the meaning. Hence ambiguous. If you wish to avoid ambiguity do not use natural language. E.g.

Hint: "A mereological "system" is a first-order theory (with identity) whose universe of discourse consists of wholes and their respective parts, collectively called objects. Mereology is a collection of nested and non-nested axiomatic systems, not unlike the case with modal logic."

> > P(X) is true if X has a successor
> > P(|N_F) is false
> > Forall x element of |N_F, P(x) is true.
> >

> P(|N_F) is false. Small wonder, the set is something else than a natural number [...]

Exactly. You see, it's HELPFUL to use the standard language of set theory. (So why DON'T YOU use it?)

> But if I say that all natural numbers together have no successor, then it is clear that there are natural numbers with less than aleph_0 successors.

WHY ON EARTH DO YOU THINK THAT THIS IS CLEAR?

Hint: This would even be FALSE in any mereological system used as a basis for math, allowing for the definition of "the natural numbers".

Hint: It's a FUNDAMENTAL property of the objects that WE (like to) call "natural numbers", that each and every natural number has infinitely many successors.

Otherwiese there would have to be/exist a "natural number" that does not have a successor - NO ONE (except you, it seems) would accept to call such an object a "natural number".

Holy shit!

[William]

On Monday, September 27, 2021 at 10:57:45 AM UTC-3, WM wrote:


>[If] I say that all natural numbers together have no successor,

you do not mean
Forall x element of |N_F, x has no successor
nor do you mean
|N_F has no successor

you are saying something that has no meaning.

--
William Hughes

[Jim Burns]

On 9/27/2021 10:24 AM, Ross A. Finlayson wrote:

> If you consider a theory that has for example only the finite
> ordinals, then, apply "Russell's paradox" as it were,

Russell's paradox --
If R is the set of each set not containing itself,
then R can't be in R and R must be in R.

In a theory with only the finite ordinals,
there is no Russell's set R.

I'm assuming that, by finite ordinals, you mean von Neumann's
0, 1, 2, 3, ...
represented by
{}, {0}, {0,1}, {0,1,2}, ...

None of them contain themselves.
All of them would be in R, if R existed.

Each finite ordinal does not contain all of them.
(There is at least (kU{k}) not-in k.)
No finite ordinal is R.

> then, what results is a set that carries along this
> "sputnik of quantification", that results that
> the set of naturals contains itself

In a theory with only finite ordinals,
there is no set of naturals.

The closest reasonable thing I see to what you're saying
would be the "set" of all ordinals. Since, in the von Neumann
formulation, an ordinal, finite or infinite, is the set of
all ordinals before it, the "set" of all ordinals would be
an ordinal, making it a member of itself.

In any set theory with which I am the tiniest bit familiar,
a set cannot be a member of itself -- and so there is no "set"
of all ordinals, though there might be a proper class of
all ordinals.

But you said "only finite", so never mind.

> the set of naturals contains itself (is compact, ...).

https://en.wikipedia.org/wiki/Compact_space
|
| In mathematics, specifically general topology, compactness is
| a property that generalizes the notion of a subset of Euclidean
| space being closed (containing all its limit points) and bounded
| (having all its points lie within some fixed distance of each
| other).[1][2] Examples of compact spaces include a closed real
| interval, a union of a finite number of closed intervals, a
| rectangle, or a finite set of points. This notion is defined
| for more general topological spaces in various ways, which are
| usually equivalent in Euclidean space but may be inequivalent
| in other spaces.

There does not appear to be any role for compactness in this discussion.

[Serg io]

it may depends on what "together" means
are they welded together ?
Stuck together ?
Inside a candy roll together ?
inside a mental institution together ?
{together in one set} ?



yep, in all cases, it is a meaningless utterance

[Gus Gassmann]

Well, what we /do/ know is that WM is no longer there altogether. And in that case it is a *very* meaningful utterance.

[WM]

Jim Burns schrieb am Sonntag, 26. September 2021 um 20:54:13 UTC+2:
> On 9/24/2021 12:28 PM, WM wrote:
> > Jim Burns schrieb

> > for instance all FISONs F(n) have successors,
> > meaning ℕ \ F(n) =/= { }
>
> That's not a very good way to say
> "all FISONs F(n) have successors"
> | ℕ \ F(n) =/= { }

But it is a way that can be understood.

> > ℕ does not leave natural numbers, meaning: ℕ \ ℕ = { }.
> > Seems ℕ contains more than definable numbers.
>
> It seems that way to you because
> | for each k, exists j: j > k
> seems to you to "contradict"
> | not exists k, for each j: j =< k

No, why should that contradict anything? What I claim ist this: There exists omega, for each n: n < omega. This is not a natural successor.

>
> >> If you would like to have a successor for
> >> all natural numbers together,
> >
> > I am satisfied to have it for every set of definabe numbers.
>
> Sets of (definable) natural numbers are not
> (definable) natural numbers.

Good observartion. Relevance?

>
> >> when the successor operation on natural numbers was defined,
> >> it wasn't defined on things NOT a natural number,
> >> extend it.
> >
> > It is defined for definable natnumbers only.
>
> (Definable) natural numbers are not
> sets of (definable) natural numbers.

Relevance?
Both, numbers and sets, have successors as explained above.

> When we discuss natural numbers (which each can be
> counted to in principle)

then there are successors. But if we discuss all natural numbers, then there are no successors (because there are not more than all).

Regards, WM

[WM]

Greg Cunt schrieb am Montag, 27. September 2021 um 18:15:00 UTC+2:
> On Monday, September 27, 2021 at 3:57:45 PM UTC+2, WM wrote:

> > All natural numbers together [...] have no natnumbers as successors although every definable natnumber has infinitely many natnumbers as successors.
>
> Makes sense. Right.
>
> > All natural numbers together, are 1, 2, 3, ...
>
> I'd propose to write "[1, 2, 3, ...]" in ths case, to avoid any missunderstanding.
>

> But when you are talking about mereology, you COMPLETELY changed the topic.

I am talking correct mathematics. That is the only topic.

> > But if I say that all natural numbers together have no successor, then it is clear that there are natural numbers with less than aleph_0 successors.
> WHY ON EARTH DO YOU THINK THAT THIS IS CLEAR?

If each one has successors, then removing each one will leave successors.

>
> Hint: It's a FUNDAMENTAL property of the objects that WE (like to) call "natural numbers", that each and every natural number has infinitely many successors.

This is only possible if the successors remain in every case, i.e., if not all natural numbers can be considered / manipulated / removed together.

Regards, WM

[WM]

William schrieb am Montag, 27. September 2021 um 18:49:08 UTC+2:
> On Monday, September 27, 2021 at 10:57:45 AM UTC-3, WM wrote:
>
>
> >[If] I say that all natural numbers together have no successor,
>
> you do not mean
> Forall x element of |N_F, x has no successor

No, I say there is no successor succeeding all elements.

Regards, WM

[William]

do you mean

there does not exist x such that forall y element of |N_F x>y

?

--
William Hughes

[Greg Cunt]

On Monday, September 27, 2021 at 9:19:15 PM UTC+2, WM wrote:
> Greg Cunt schrieb am Montag, 27. September 2021 um 18:15:00 UTC+2:

> I am talking <bla>

Yeah, whatever Mückenheim.

> > > it is clear that there are natural numbers with less than aleph_0 successors.
> > >
> > WHY ON EARTH DO YOU THINK THAT THIS IS CLEAR?
> >
> If each one has successors, then removing each one will leave successors.

WTF?!?

> > Hint: It's a FUNDAMENTAL property of the objects that WE (like to) call "natural numbers", that each and every natural
> > number has infinitely many successors.
> >

> This is only possible if the successors remain in every case, i.e. <bla>

Look, dumbo, successors are no moving or vanishing objects.

Holy shit!

[Jim Burns]

On 9/27/2021 9:57 AM, WM wrote:
> William schrieb
> am Sonntag, 26. September 2021 um 18:13:30 UTC+2:

>> "All elements" can mean "the set" or "the elements of the set".
>
> When I say "all elements together", then I mean
> all elements with no exception,
> but not the set of all elements,
> because matheologians seem to ascribe some soul
> or spirit to the braces.

I think that the general opinion among mathematicians
is that their work is already challenging enough.
A set containing things not in that set seems like
an unnecessary complication, sucking up energy that
could be put to better use elsewhere.

> For instance the empty set is
> more than nothing, in their eyes.

The empty set is an individual among all the individuals
which have and don't have members.
It is the one which has no members.

When we describe these member-having-not-having individuals,
we could say "but not the empty set", if we wanted to.
However, it is very often better to refer to _all_ the
member-having-not-having individuals, without setting apart
the member-only-not-having individual for special attention.
So we don't want to.

> Therefore I refrain from sets
> but simply consider all elements which belong to the set.
> All natural numbers together are 1, 2, 3, ...
> They have no natnumbers as successors
> although every definable natnumber has
> infinitely many natnumbers as successors.

Each natural number has a number after it.
No natural number has all other numbers before it.

That doesn't have anything to do with sets.
It's enough to consider j and k such that
(j > k) xor (j =< k)
(for all j, [claim1]) xor (exists j, not [claim1])
(for all k, [claim2]) xor (exists k, not [claim2])

> But if I say that all natural numbers together have no successor,
> then it is clear that there are natural numbers with
> less than aleph_0 successors.

(j > k) xor (j =< k)
(for all j, [claim1]) xor (exists j, not [claim1])
(for all k, [claim2]) xor (exists k, not [claim2])

If all numbers together have no successor (none are >= all),
then no numbers have less than aleph_0 successors.

( If a number had less then aleph_0 successors,
( then a later number >= all exists.
( Since there is no such >= all number,
( there is no number with less than aleph_0 successors.

[Serg io]

On 9/27/2021 2:19 PM, WM wrote:
> Greg Cunt schrieb am Montag, 27. September 2021 um 18:15:00 UTC+2:
>> On Monday, September 27, 2021 at 3:57:45 PM UTC+2, WM wrote:
>
>>> All natural numbers together [...] have no natnumbers as successors although every definable natnumber has infinitely many natnumbers as successors.
>>
>> Makes sense. Right.
>>
>>> All natural numbers together, are 1, 2, 3, ...
>>
>> I'd propose to write "[1, 2, 3, ...]" in ths case, to avoid any missunderstanding.
>>
>> But when you are talking about mereology, you COMPLETELY changed the topic.
>
> I am talking correct mathematics. That is the only topic.

no. you rejected the meaning of {1,2,3...},

you do not utter correct mathematics at all.

We are here to correct your math.

>
>>> But if I say that all natural numbers together have no successor, then it is clear that there are natural numbers with less than aleph_0 successors.
>> WHY ON EARTH DO YOU THINK THAT THIS IS CLEAR?
>
> If each one has successors, then removing each one will leave successors.

homework: express above in equations.

>>
>> Hint: It's a FUNDAMENTAL property of the objects that WE (like to) call "natural numbers", that each and every natural number has infinitely many successors.
>
> This is only possible if the successors remain in every case, i.e., if not all natural numbers can be considered / manipulated / removed together.

"considered, manipulated, removed together" are not terms in Math, and have no meaning in Math.


>
> Regards, WM
>

[Kye Fox]

Serg io wrote:

>> I am talking correct mathematics. That is the only topic.
>
> no. you rejected the meaning of {1,2,3...},
> you do not utter correct mathematics at all.
> We are here to correct your math.

welcome to capitalist america, the country of dreams. Man down, shoot as
a dog.

MAN GETS SHOT MULTIPLE TIMES BY POLICE!
https://www.bitchute.com/video/Ir5tCP6PtLp6/

[FromTheRafters]

After serious thinking Serg io wrote :

> On 9/27/2021 2:19 PM, WM wrote:
>> Greg Cunt schrieb am Montag, 27. September 2021 um 18:15:00 UTC+2:
>>> On Monday, September 27, 2021 at 3:57:45 PM UTC+2, WM wrote:
>>>> All natural numbers together [...] have no natnumbers as successors
>>>> although every definable natnumber has infinitely many natnumbers as
>>>> successors.
>>>
>>> Makes sense. Right.
>>>
>>>> All natural numbers together, are 1, 2, 3, ...
>>>
>>> I'd propose to write "[1, 2, 3, ...]" in ths case, to avoid any
>>> missunderstanding.
>>>
>>> But when you are talking about mereology, you COMPLETELY changed the topic.
>>
>> I am talking correct mathematics. That is the only topic.
>
> no. you rejected the meaning of {1,2,3...},

You missed out on a comma there methinks.