|N.the set of natural numbers, is a Peano set (WM has a concept of a set of natural numbers that is not a Peano set. Call this non-standard thing |N_(WM))

There are many properties that stem from the fact that |N is a Peano set, the one that concerns us here is that every element of |N is the largest element of some FISON (the FISON F_n of course). So if you remove all the FISONs from N you remove every element of |N that is the largest element of some FISON. i.e. every element of the Peano set N.

Consider a process in which we remove FISON F_n at step n. At every finite step (and there are only finite steps, an infinite number of finite steps) there are an infinite number of elements of |N that still have to be removed. WM does not like this and calls it a contradiction. His "solution" is to add elements that are not elements of !N to the set |N forming the set |N_(WM). The problem is that |N_(WM) is not a Peano set thus is not the set of natural numbers.

--
William Hughes

William schrieb am Mittwoch, 19. Mai 2021 um 20:45:45 UTC+2:
> |N.the set of natural numbers, is a Peano set (WM has a concept of a set of natural numbers that is not a Peano set. Call this non-standard thing |N_(WM))

The set of definable natural numbers is a Peano set.
>
> There are many properties that stem from the fact that |N is a Peano set, the one that concerns us here is that every element of |N is the largest element of some FISON (the FISON F_n of course).

That is true for the definable natural numbers. But they are never more than can be counted by a FISON.

> So if you remove all the FISONs from N you remove every element of |N that is the largest element of some FISON. i.e. every element of the Peano set N.

Yes.
>
> Consider a process in which we remove FISON F_n at step n. At every finite step (and there are only finite steps, an infinite number of finite steps)

This is possible in potential infinity. In actual infinity we have a larger number of steps than any finite number. This implies that also the counter counts to a larger number. But that is impossible if the counter can handle only FISONs:

o
oo
ooo
oooo
ooooo
....

The number up to step n is counted by the FISON F(n).
The number of steps larger than every natnumber is counted by a FISON larger than every finite set. Contradiction.

> there are an infinite number of elements of |N that still have to be removed.

Start with the largest you can find. The largest that is available has more successors than predecessors.

Regards, WM

On Wednesday, May 19, 2021 at 4:03:58 PM UTC-3, WM wrote:
> William schrieb am Mittwoch, 19. Mai 2021 um 20:45:45 UTC+2:
> > |N.the set of natural numbers, is a Peano set (WM has a concept of a set of natural numbers that is not a Peano set. Call this non-standard thing |N_(WM))
> The set of definable natural numbers is a Peano set

Part of the definition of |N is that it is a Peano set. So the set of all natural numbers is a Peano set. If you are interested in a set which is not a Peano set, find a new name.

--
William Hughes

On 5/19/2021 3:03 PM, WM wrote:
> William schrieb
> am Mittwoch, 19. Mai 2021 um 20:45:45 UTC+2:

>> |N.the set of natural numbers, is a Peano set
>> (WM has a concept of a set of natural numbers that is
>> not a Peano set. Call this non-standard thing |N_(WM))
>
> The set of definable natural numbers is a Peano set.

The set of natural numbers is a Peano Set which DOES NOT
contain a Peano set as a proper subset.
(It's the _smallest_ Peano set in that sense.)

The set of definable natural numbers is a Peano set
and it is a subset of the natural numbers.

But no PROPER Peano subsets of the natural numbers exist.

Therefore, the definable natural number are a NOT-PROPER
subset of the natural numbers -- which is to say,
the definable natural numbers ARE the natural numbers.

William schrieb am Mittwoch, 19. Mai 2021 um 21:42:26 UTC+2:
> On Wednesday, May 19, 2021 at 4:03:58 PM UTC-3, WM wrote:
> > William schrieb am Mittwoch, 19. Mai 2021 um 20:45:45 UTC+2:
> > > |N.the set of natural numbers, is a Peano set (WM has a concept of a set of natural numbers that is not a Peano set. Call this non-standard thing |N_(WM))
> > The set of definable natural numbers is a Peano set
> Part of the definition of |N is that it is a Peano set. So the set of all natural numbers is a Peano set.

Every element of a Peano set is definable. Most natural numbers are undefinable.

U_{n∈ℕ} [1/(n+1), 1/n] = (0, 1].

For every definable natural number n, however we get [1/n, 1] =/= (0, 1].

Regards, WM

Jim Burns schrieb am Mittwoch, 19. Mai 2021 um 22:15:09 UTC+2:
> On 5/19/2021 3:03 PM, WM wrote:

> Therefore, the definable natural number are a NOT-PROPER
> subset of the natural numbers -- which is to say,
> the definable natural numbers ARE the natural numbers.

Repeating falsehoods does not make them true.

Every element of a Peano set is definable. Most natural numbers are undefinable.

U_{n∈ℕ} [1/(n+1), 1/n] = (0, 1].

For every definable natural number n, however we get [1/n, 1] =/= (0, 1].

Regards, WM

On Wednesday, May 19, 2021 at 5:34:16 PM UTC-3, WM wrote:
> William schrieb am Mittwoch, 19. Mai 2021 um 21:42:26 UTC+2:
> > On Wednesday, May 19, 2021 at 4:03:58 PM UTC-3, WM wrote:
> > > William schrieb am Mittwoch, 19. Mai 2021 um 20:45:45 UTC+2:
> > > > |N.the set of natural numbers, is a Peano set (WM has a concept of a set of natural numbers that is not a Peano set. Call this non-standard thing |N_(WM))
> > > The set of definable natural numbers is a Peano set
> > Part of the definition of |N is that it is a Peano set. So the set of all natural numbers is a Peano set.
> Every element of a Peano set is definable.

Indeed. And every natural number is an element of |N which is a Peano set

> Most natural numbers are undefinable.

Nope. All natural numbers are elements of a Peano set. If you are interested in a set which is not a Peano set, find a new name.

--
William Hughes

On Wednesday, 19 May 2021 at 16:03:58 UTC-3, WM wrote:

> This is possible in potential infinity. In actual infinity we have a larger number of steps than any finite number. This implies that also the counter counts to a larger number. But that is impossible if the counter can handle only FISONs:
>
> o
> oo
> ooo
> oooo
> ooooo
> ...
>
> The number up to step n is counted by the FISON F(n).
> The number of steps larger than every natnumber is counted by a FISON larger than every finite set. Contradiction.
"Repeating falsehoods does not make them true." (WM, sci.math, 19 May 2021)

On 5/19/2021 4:35 PM, WM wrote:

> Every element of a Peano set is definable.

I was mistaken about what
| P is a Peano set.
means. I thought it means
| 0 in P. If x in P, then x+1 in P.

| P is a Peano set.
apparently means
| 0 in P & if x in P, then x+1 in P,
| AND no proper subset C of P exists such that
| 0 in C & if x in C, then x+1 in C.

> Most natural numbers are undefinable.

A natural number is countable-to.

The intended use of natural numbers is to match uncounted
objects to successive natural numbers, after which such
a matched object is _counted_ We halt when there are no more
uncounted objects to match. The last natural number
matched to an object is the _count_ of the objects.
When the process has halted, we say that we have _counted_
the objects.

In order for a natural number k to be put to its intended use,
There must exists a linearly-ordered crowd 0,...,k of natural
numbers such that
each non-empty sub-crowd of 0,...,k contains a first and
a last, and,
for all adjacent i,j in 0,...,k, j = i+1.
k must be countable-to.

Also, in order for a natural number k to be put to its
intended use, the counting must never halt because there
are no more _natural numbers_ It must only halt because
there are no more _uncounted objects_ Therefore, in order
for k to be put to its intended use, k+1 must exist and
not occur in 0,...,k.
k must be countable-beyond.

> Most natural numbers are undefinable.

For "definable" == "countable-to",
all natural numbers are definable.
That's what makes them natural numbers.

> U_{n∈ℕ} [1/(n+1), 1/n] = (0, 1].
>
> For every definable natural number n,
> however we get [1/n, 1] =/= (0, 1].

| ... it is a tale
| Told by an idiot, full of sound and fury,
| Signifying nothing.
| -- Macbeth, "Macbeth"

onsdag 19 maj 2021 kl. 21:03:58 UTC+2 skrev WM:
> William schrieb am Mittwoch, 19. Mai 2021 um 20:45:45 UTC+2:
> > |N.the set of natural numbers, is a Peano set (WM has a concept of a set of natural numbers that is not a Peano set. Call this non-standard thing |N_(WM))
> The set of definable natural numbers is a Peano set.
> >
> > There are many properties that stem from the fact that |N is a Peano set, the one that concerns us here is that every element of |N is the largest element of some FISON (the FISON F_n of course).
> That is true for the definable natural numbers. But they are never more than can be counted by a FISON.
> > So if you remove all the FISONs from N you remove every element of |N that is the largest element of some FISON. i.e. every element of the Peano set N.
> Yes.
> >
> > Consider a process in which we remove FISON F_n at step n. At every finite step (and there are only finite steps, an infinite number of finite steps)
> This is possible in potential infinity. In actual infinity we have a larger number of steps than any finite number. This implies that also the counter counts to a larger number. But that is impossible if the counter can handle only FISONs:
>
> o
> oo
> ooo
> oooo
> ooooo
> ...
>
> The number up to step n is counted by the FISON F(n).
> The number of steps larger than every natnumber is counted by a FISON larger than every finite set. Contradiction.
> > there are an infinite number of elements of |N that still have to be removed.
> Start with the largest you can find. The largest that is available has more successors than predecessors.
>
> Regards, WM

The peano set si infinite so which is it?

Gus Gassmann laid this down on his screen :
> On Wednesday, 19 May 2021 at 16:03:58 UTC-3, WM wrote:
>
>> This is possible in potential infinity. In actual infinity we have a larger
>> number of steps than any finite number. This implies that also the counter
>> counts to a larger number. But that is impossible if the counter can handle
>> only FISONs:
>>
>> o
>> oo
>> ooo
>> oooo
>> ooooo
>> ...
>>
>> The number up to step n is counted by the FISON F(n).
>> The number of steps larger than every natnumber is counted by a FISON larger
>> than every finite set. Contradiction.
>
> "Repeating falsehoods does not make them true." (WM, sci.math, 19 May 2021)

Funny, I was just about to remind him of that very thing when he posted
that gem.

Crank Wolfgang Mueckenheim, aka WM wrote:
> Jim Burns schrieb am Mittwoch, 19. Mai 2021 um 22:15:09 UTC+2:
>> On 5/19/2021 3:03 PM, WM wrote:
>
>> Therefore, the definable natural number are a NOT-PROPER
>> subset of the natural numbers -- which is to say,
>> the definable natural numbers ARE the natural numbers.
>
> Repeating falsehoods does not make them true.

Yep! We'll make sure this to be the epitaph on your grave, Crank
Wolfgang Mueckenheim, from Hochschule Augsburg.

William schrieb am Mittwoch, 19. Mai 2021 um 22:44:41 UTC+2:
> On Wednesday, May 19, 2021 at 5:34:16 PM UTC-3, WM wrote:
> > William schrieb am Mittwoch, 19. Mai 2021 um 21:42:26 UTC+2:
> > > On Wednesday, May 19, 2021 at 4:03:58 PM UTC-3, WM wrote:
> > > > William schrieb am Mittwoch, 19. Mai 2021 um 20:45:45 UTC+2:
> > > > > |N.the set of natural numbers, is a Peano set (WM has a concept of a set of natural numbers that is not a Peano set. Call this non-standard thing |N_(WM))
> > > > The set of definable natural numbers is a Peano set
> > > Part of the definition of |N is that it is a Peano set. So the set of all natural numbers is a Peano set.
> > Every element of a Peano set is definable.
> Indeed. And every natural number is an element of |N which is a Peano set

Every definable natural number can be processed and fails to reach the state where all natural numbers have been processed: omega, omega + 1, ...

But according to set theory this state can be reached by collective processing.

> > Most natural numbers are undefinable.
> Nope. All natural numbers are elements of a Peano set.

All natural numbers together accomplish this:
U_{n∈ℕ} [1/(n+1), 1/n] = (0, 1].

For every definable natural number n however we get

[1/n, 1] =/= (0, 1].

Why do you not understand this?

Regards, WM

Crank Wolfgang Mueckenheim, aka WM wrote:
> William schrieb am Mittwoch, 19. Mai 2021 um 22:44:41 UTC+2:
>> On Wednesday, May 19, 2021 at 5:34:16 PM UTC-3, WM wrote:
>>> William schrieb am Mittwoch, 19. Mai 2021 um 21:42:26 UTC+2:
>>>> On Wednesday, May 19, 2021 at 4:03:58 PM UTC-3, WM wrote:
>>>>> William schrieb am Mittwoch, 19. Mai 2021 um 20:45:45 UTC+2:
>>>>>> |N.the set of natural numbers, is a Peano set (WM has a concept of a set of natural numbers that is not a Peano set. Call this non-standard thing |N_(WM))
>>>>> The set of definable natural numbers is a Peano set
>>>> Part of the definition of |N is that it is a Peano set. So the set of all natural numbers is a Peano set.
>>> Every element of a Peano set is definable.
>> Indeed. And every natural number is an element of |N which is a Peano set
>
> Every definable natural number can be processed and fails to reach the state where all natural numbers have been processed: omega, omega + 1, ...
>
> But according to set theory this state can be reached by collective processing.
>
>>> Most natural numbers are undefinable.
>> Nope. All natural numbers are elements of a Peano set.
>
> All natural numbers together accomplish this:
> U_{n∈ℕ} [1/(n+1), 1/n] = (0, 1].
>
> For every definable natural number n however we get
>
> [1/n, 1] =/= (0, 1].
>
> Why do you not understand this?

Why don't you understand that this is not a contradiction Crank
Wolfgang Mueckenheim from Hochschule Augsburg?

On Wednesday, 19 May 2021 at 14:45:45 UTC-4, William wrote:
> |N.the set of natural numbers, is a Peano set (WM has a concept of a set of natural numbers that is not a Peano set. Call this non-standard thing |N_(WM))
>
> There are many properties that stem from the fact that |N is a Peano set, the one that concerns us here is that every element of |N is the largest element of some FISON (the FISON F_n of course). So if you remove all the FISONs from N you remove every element of |N that is the largest element of some FISON. i.e. every element of the Peano set N.
>
> Consider a process in which we remove FISON F_n at step n. At every finite step (and there are only finite steps, an infinite number of finite steps) there are an infinite number of elements of |N that still have to be removed. WM does not like this and calls it a contradiction. His "solution" is to add elements that are not elements of !N to the set |N forming the set |N_(WM). The problem is that |N_(WM) is not a Peano set thus is not the set of natural numbers.
>
> --
> William Hughes

The idiot Peano is not even relevant. I find it incredibly telling that stupids like you subscribe to Peano's bullshit.

Peano was trying to NAME the natural numbers, but his 1=S(0), 2= S(S(0)), etc is simply a juvenile scheme that is accomplished by the use of any radix system formulated on well-formed concepts such as the digits 0-9 and the decimal system.

Laughable that morons like you still bother to call this joke of a "mathematician" anything but a fool who didn't accomplish shit.

1. 0 is a number?

LMAO. What the fuck is a number, you moron?!

2. If p is a number, the successor of p is a number ?

LMAO. What the fuck is a successor, you moron? Wait, what the fuck is a number first?

Students: Be very careful of this juvenile bullshit. It's not mathematics, but the joke of the twentieth century.

https://youtu.be/lAD-pfSNdSs

To learn how we really got number, you need ME:

https://drive.google.com/file/d/1hasWyQCZyRN3RkdvIB6bnGIVV2Rabz8w

Eram semper recta wrote:

> The idiot Peano is not even relevant. I find it incredibly telling that
> stupids like you subscribe to Peano's bullshit.
>
> Peano was trying to NAME the natural numbers, but his 1=S(0), 2=
> S(S(0)), etc is simply a juvenile scheme that is accomplished by the
> use of any radix system formulated on well-formed concepts such as the
> digits 0-9 and the decimal system.

HOW TO SELL A (FAKE) PANDEMIC - MARC VAN RANST - Belgian Flu Commissioner
at the ESWI/Chatham House

Python schrieb am Donnerstag, 20. Mai 2021 um 13:06:46 UTC+2:

> > All natural numbers together accomplish this:
> > U_{n∈ℕ} [1/(n+1), 1/n] = (0, 1].
> >
> > For every definable natural number n however we get
> >
> > [1/n, 1] =/= (0, 1].
> >
> > Why do you not understand this?
> Why don't you understand that this is not a contradiction

What contradiction? Why should that be a contradiction? Contradicting what?

It is simply a fact that [1/n, 1] =/= (0, 1] and similarly you cannot count individual natnumbers to reach omega. But we all believe or are convinced or even know that after having processed all of them collectively we can restart with omega and so on, don't we?

Regards, WM