Greg Cunt schrieb am Samstag, 18. September 2021 um 01:55:24 UTC+2:
> On Friday, September 17, 2021 at 9:43:23 PM UTC+2, WM wrote:
>
> Let n e IN.
> > n is not a concrete natural number
> It's certainly a "concrete" natural number, though we don't KNOW which one.

For every natural number we know it. Concrete means known.
Matheologians are simplistic minds who believe even the most stupid ideas like
- There are unknown known numbers
- All fractions can be enumerated.
- There are uncountably many real numbers although only countably many can be distinguished.
- There are more paths in the Binary Tree than nodes.
- The bankruptcy of McDuck.
- aleph_0 rational points separate uncountably many irrational points such that never two are existing without rational between them.

But the matter with "number" n is even clearer than all the other nonsense.

> > but every natural number can be put at its place.
> Nope it can't. Hint n+1 =/= n.

n+1 is not a natural number too.
>
> > Let n ∈ ℕ, then n has a unique prime decomposition.
>
> Exactly (if n > 1).
> > You cannot know what this prime decomposition is.
> Right.
>
But for every natural number we know it.
>
> Right.

Regards, WM

Greg Cunt schrieb am Samstag, 18. September 2021 um 01:49:32 UTC+2:
> On Friday, September 17, 2021 at 9:55:31 PM UTC+2, WM wrote:
>
> > Or is the set of endsegments infinite?
> Errrr, right, it is infinite.

Then its indices exhaust all natural numbers. None are remaining for the contents. Or would you know one natural number that is not used as index but belongs to the infinitely many remaining in all infinite endsegments?
>
> Hence card({E(n) : n e IN}) = card(IN) = aleph_0. Which means that {E(n) : n e IN} is infinite. qed

Thought so. What are the infinitely many natural numbers remaining as indices in all endsegments?

Regards, WM

On Saturday, September 18, 2021 at 5:56:43 PM UTC+2, WM wrote:

Let n e IN.

> > > n is not a concrete natural number [WM]
> > >
> > It's certainly a "concrete" natural number, though we don't KNOW which one.
> >
> For every natural number we know it.

Huh?! So you personally know ALL natural numbers? Wow - that's rather cool!

> Concrete means known.

Nope.

You may feel a coin in your pocket without knowing which one it is. This coin isn't "concret"?

> > Hint: n+1 =/= n.
> >
> n+1 is not a natural number too.

Oh, I stand corrected!

So n is in IN and n+1 is in IN, but neither n nor n+1 are natural numbers? Is that your claim? So what are they? Pink elephants?

On 9/18/2021 11:01 AM, WM wrote:
> Greg Cunt schrieb am Samstag, 18. September 2021 um 01:49:32 UTC+2:
>> On Friday, September 17, 2021 at 9:55:31 PM UTC+2, WM wrote:
>>
>>> Or is the set of endsegments infinite?
>> Errrr, right, it is infinite.
>
> Then its indices exhaust all natural numbers. None are remaining for the contents. Or would you know one natural number that is not used as index but belongs to the infinitely many remaining in all infinite endsegments?
>>
>> Hence card({E(n) : n e IN}) = card(IN) = aleph_0. Which means that {E(n) : n e IN} is infinite. qed
>
> Thought so. What are the infinitely many natural numbers remaining as indices in all endsegments?
>
> Regards, WM
>

think of a two dimensional matrix of rocks at the beach, each row is infinite, each column is infinite, each row uses N aleph_0, each column uses N aleph_0

and aleph_0 * aleph_0 = aleph_0

On Saturday, September 18, 2021 at 6:01:26 PM UTC+2, WM wrote:
> Greg Cunt schrieb am Samstag, 18. September 2021 um 01:49:32 UTC+2:
> > On Friday, September 17, 2021 at 9:55:31 PM UTC+2, WM wrote:
> >
> Then its indices exhaust all natural numbers.

Usually the index set of an infinite sequence is just IN. And right, a function is defined for all elements in its domain.

> None are remaining for the contents.

Huh?! No dual use allowed in Mückenwolkenheim?

Hint: In the context of mathematics we have: Ak e IN: card(E(k)) = aleph_0. So each and every endsegment is "filled" with infinitely many natural numbers, even though there are infinitely many endsegments. Why does this work? Because each and every natural number has infinitelly many successors.

> would you know one natural number that [...] belongs to [...] all infinite endsegments?

No, I wouldn't, since there is/are no such number/s.

> What are the [...] natural numbers remaining [...] in all endsegments?

There are no natural numbers "remaining" in all endsegments.

Proof: Let n be an arbitrary natural number. Then n is not an element in the endsegment E(n+1) (and all endsegements with an index k > n+1). Hence n is not an endsegment in _all_ endsegments (since it isn't an element, say, in E(n+1)). Since n was an arbitrary natural number, this holds for all natural numbers. Hence there is _no_ natural number which is "remaining" in all endsegments. qed

How about a proof by induction? (Maybe you've heard of that proof method.)

Theorem: ~Ek e IN: Ak' e IN: k e E(k').

This theorem is logically equivalent with Ak e IN: Ek' e IN: k !e E(k').

Proof (by induction): 1 !e E(2). Since 2 e IN, this implies Ek' e IN: 1 !e E(k'). So we have shown the base case. Now the step from n to n+1. Let's assume that Ek' e IN: n !e E(k') holds for some n e IN. Then n+1 !e E(n+2) with n+2 e IN. This implies Ek' e IN: n+1 !e E(k'). qed

On Sunday, September 19, 2021 at 2:33:58 AM UTC+2, Serg io wrote:

> think of a two dimensional matrix of rocks at the beach, each row is infinite, each column is infinite, each row uses N aleph_0, each column uses N aleph_0
>
> and aleph_0 * aleph_0 = aleph_0

Where is this beach? :-)

After serious thinking Serg io wrote :
> On 9/18/2021 11:01 AM, WM wrote:
>> Greg Cunt schrieb am Samstag, 18. September 2021 um 01:49:32 UTC+2:
>>> On Friday, September 17, 2021 at 9:55:31 PM UTC+2, WM wrote:
>>>
>>>> Or is the set of endsegments infinite?
>>> Errrr, right, it is infinite.
>>
>> Then its indices exhaust all natural numbers. None are remaining for the
>> contents. Or would you know one natural number that is not used as index but
>> belongs to the infinitely many remaining in all infinite endsegments?
>>>
>>> Hence card({E(n) : n e IN}) = card(IN) = aleph_0. Which means that {E(n) :
>>> n e IN} is infinite. qed
>>
>> Thought so. What are the infinitely many natural numbers remaining as
>> indices in all endsegments?
>>
>> Regards, WM
>>
>
> think of a two dimensional matrix of rocks at the beach, each row is
> infinite, each column is infinite, each row uses N aleph_0, each column uses
> N aleph_0
>
> and aleph_0 * aleph_0 = aleph_0

Walking on sand and moving rocks around is exhausting.

Greg Cunt formulated on Saturday :
> On Sunday, September 19, 2021 at 2:33:58 AM UTC+2, Serg io wrote:
>
>> think of a two dimensional matrix of rocks at the beach, each row is
>> infinite, each column is infinite, each row uses N aleph_0, each column uses
>> N aleph_0
>>
>> and aleph_0 * aleph_0 = aleph_0
>
> Where is this beach? :-)

Maybe you can use shores algorithm to locate beaches. :)

On 9/18/2021 9:06 PM, FromTheRafters wrote:
> After serious thinking Serg io wrote :
>> On 9/18/2021 11:01 AM, WM wrote:
>>> Greg Cunt schrieb am Samstag, 18. September 2021 um 01:49:32 UTC+2:
>>>> On Friday, September 17, 2021 at 9:55:31 PM UTC+2, WM wrote:
>>>>> Or is the set of endsegments infinite?
>>>> Errrr, right, it is infinite.
>>>
>>> Then its indices exhaust all natural numbers. None are remaining for the contents. Or would you know one natural number that is not used as index but
>>> belongs to the infinitely many remaining in all infinite endsegments?
>>>>
>>>> Hence card({E(n) : n e IN}) = card(IN) = aleph_0. Which means that {E(n) : n e IN} is infinite. qed
>>>
>>> Thought so. What are the infinitely many natural numbers remaining as indices in all endsegments?
>>>
>>> Regards, WM
>>>
>>
>> think of a two dimensional matrix of rocks at the beach, each row is infinite, each column is infinite, each row uses N aleph_0, each column uses N
>> aleph_0
>>
>> and aleph_0 * aleph_0 =  aleph_0
>
> Walking on sand and moving rocks around is exhausting.

Waimea Bay Beach, the ocean can be flat, but due the way the bottom is shaped, there are 6 foot perfect waves near the shore,
no big rocks, but almost, not quite, aleph_0 of tiny tiny sand size rocks.

Serg io has brought this to us :
> On 9/18/2021 9:06 PM, FromTheRafters wrote:
>> After serious thinking Serg io wrote :
>>> On 9/18/2021 11:01 AM, WM wrote:
>>>> Greg Cunt schrieb am Samstag, 18. September 2021 um 01:49:32 UTC+2:
>>>>> On Friday, September 17, 2021 at 9:55:31 PM UTC+2, WM wrote:
>>>>>> Or is the set of endsegments infinite?
>>>>> Errrr, right, it is infinite.
>>>>
>>>> Then its indices exhaust all natural numbers. None are remaining for the
>>>> contents. Or would you know one natural number that is not used as index
>>>> but belongs to the infinitely many remaining in all infinite endsegments?
>>>>>
>>>>> Hence card({E(n) : n e IN}) = card(IN) = aleph_0. Which means that {E(n) :
>>>>> n e IN} is infinite. qed
>>>>
>>>> Thought so. What are the infinitely many natural numbers remaining as
>>>> indices in all endsegments?
>>>>
>>>> Regards, WM
>>>>
>>>
>>> think of a two dimensional matrix of rocks at the beach, each row is
>>> infinite, each column is infinite, each row uses N aleph_0, each column
>>> uses N aleph_0
>>>
>>> and aleph_0 * aleph_0 =  aleph_0
>>
>> Walking on sand and moving rocks around is exhausting.
>
>
> Waimea Bay Beach, the ocean can be flat, but due the way the bottom is
> shaped, there are 6 foot perfect waves near the shore, no big rocks, but
> almost, not quite, aleph_0 of tiny tiny sand size rocks.

Rocks you can dive off, and teeny tiny nearly inexhaustable rocks.

Greg Cunt schrieb am Samstag, 18. September 2021 um 20:20:00 UTC+2:
> On Saturday, September 18, 2021 at 5:56:43 PM UTC+2, WM wrote:
>
> Let n e IN.
> > > > n is not a concrete natural number [WM]
> > > >
> > > It's certainly a "concrete" natural number, though we don't KNOW which one.
> > >
> > For every natural number we know it.
> Huh?! So you personally know ALL natural numbers? Wow - that's rather cool!

I know, in principle, every number that you can mention. n does not belong to that set.

> > > Hint: n+1 =/= n.
> > >
> > n+1 is not a natural number too.
> Oh, I stand corrected!
>
> So n is in IN and n+1 is in IN, but neither n nor n+1 are natural numbers? Is that your claim? So what are they? Pink elephants?

They are placeholders. "Let n in |N, then P(n)" is an abbreviation for: "Any natural number can be put in place of n and will satisfy P."

Regards, WM

Greg Cunt schrieb am Sonntag, 19. September 2021 um 03:39:21 UTC+2:
> On Saturday, September 18, 2021 at 6:01:26 PM UTC+2, WM wrote:
> > Greg Cunt schrieb am Samstag, 18. September 2021 um 01:49:32 UTC+2:
> > > On Friday, September 17, 2021 at 9:55:31 PM UTC+2, WM wrote:
> > >
> > Then its indices exhaust all natural numbers.
> Usually the index set of an infinite sequence is just IN. And right, a function is defined for all elements in its domain.
> > None are remaining for the contents.
> Huh?! No dual use allowed in Mückenwolkenheim?

Not for endsegments. The contents of E(n) continues the indices used up to n-1.
>
> Hint: In the context of mathematics we have: Ak e IN: card(E(k)) = aleph_0. So each and every endsegment is "filled" with infinitely many natural numbers, even though there are infinitely many endsegments. Why does this work? Because each and every natural number has infinitelly many successors.

Why does this not work? Because the set of all indices is |N and has no successors.
>
> > What are the [...] natural numbers remaining [...] in all endsegments?
>
> There are no natural numbers "remaining" in all endsegments.

You claim that infinitely many are in every endsegment. But aleph_0 endsegments with contents aleph_0 numbers are not possible. Therefore one of your claims is wrong.

Regards, WM

Serg io schrieb am Sonntag, 19. September 2021 um 02:33:58 UTC+2:
> On 9/18/2021 11:01 AM, WM wrote:

> > Thought so. What are the infinitely many natural numbers remaining as indices in all endsegments?
> >
> think of a two dimensional matrix of rocks at the beach, each row is infinite, each column is infinite, each row uses N aleph_0, each column uses N aleph_0
>
> and aleph_0 * aleph_0 = aleph_0

Endsegments have contents that succeeds their indices. aleph_0 endsegments containing aleph_0 natnumbers each is impossible: 1, 2, 3, ..., n | n, n+1, n+2, ... .

Regards, WM

On Sunday, September 19, 2021 at 4:40:47 PM UTC+2, WM wrote:
> Greg Cunt schrieb am Samstag, 18. September 2021 um 20:20:00 UTC+2:

> > Let n e IN.
> >
> n does not belong to that set.

To the set of all natural numbers, IN?

So we have n e IN, but in your psychotic kingdom, n !e IN at the same time?

Cool!

I mean, THAT would explain A LOT!

> > > > Hint: n+1 =/= n. [GC]
> > > >
> > > n+1 is not a natural number too. [WM]

See?!

> > So n is in IN and n+1 is in IN, but neither n nor n+1 are natural numbers? Is that your claim?

No answer?

> > So what are they? Pink elephants?
> >
> They are placeholders.

So placeholders are elements in IN?

Cool!

So if these placeholders aren't natural numbers, there are objects in IN which aren't natural numbers? THAT'S FASCINATING, MÜCKENHEIM!

"Let n in IN, then ...n..." is an abbreviation for <bla bla>

It's not a abbreviation of anything, dumbo. But It might be the beginning of a proof.

On Sunday, September 19, 2021 at 4:51:19 PM UTC+2, WM wrote:

> aleph_0 endsegments with contents aleph_0 numbers are not possible.

False claim.

Hint: In the context of set theory (say ZFC) thhese are theorems:

1. card({E(n) : n e IN} = aleph_0

and

2. An e IN: card(E(n)) = aleph_0.

On Sunday, September 19, 2021 at 4:55:10 PM UTC+2, WM wrote:

> aleph_0 endsegments containing aleph_0 natnumbers each is impossible

Wrong.

Hint: In the context of set theory (say ZFC) these are theorems:

1. card({E(n) : n e IN}) = aleph_0

and

2. An e IN: card(E(n) n IN) = aleph_0.

On 9/19/2021 9:51 AM, WM wrote:
> Greg Cunt schrieb am Sonntag, 19. September 2021 um 03:39:21 UTC+2:
>> On Saturday, September 18, 2021 at 6:01:26 PM UTC+2, WM wrote:
>>> Greg Cunt schrieb am Samstag, 18. September 2021 um 01:49:32 UTC+2:
>>>> On Friday, September 17, 2021 at 9:55:31 PM UTC+2, WM wrote:
>>>>
>>> Then its indices exhaust all natural numbers.
>> Usually the index set of an infinite sequence is just IN. And right, a function is defined for all elements in its domain.
>>> None are remaining for the contents.
>> Huh?! No dual use allowed in Mückenwolkenheim?
>
> Not for endsegments. The contents of E(n) continues the indices used up to n-1.
>>
>> Hint: In the context of mathematics we have: Ak e IN: card(E(k)) = aleph_0. So each and every endsegment is "filled" with infinitely many natural numbers, even though there are infinitely many endsegments. Why does this work? Because each and every natural number has infinitelly many successors.
>
> Why does this not work? Because the set of all indices is |N and has no successors.
>>
>>> What are the [...] natural numbers remaining [...] in all endsegments?
>>
>> There are no natural numbers "remaining" in all endsegments.
>
> You claim that infinitely many are in every endsegment. But aleph_0 endsegments with contents aleph_0 numbers are not possible.

False. you have failed to understand infinity. You are using only one pile of rocks.

between any pair of numbers on the real line there are aleph_0 numbers.

How many pairs of numbers are there on the real line ? aleph_0

aleph_0 * aleph_0 = aleph_0

>
> Regards, WM
>

WM used his keyboard to write :
> Greg Cunt schrieb am Sonntag, 19. September 2021 um 03:39:21 UTC+2:
>> On Saturday, September 18, 2021 at 6:01:26 PM UTC+2, WM wrote:
>>> Greg Cunt schrieb am Samstag, 18. September 2021 um 01:49:32 UTC+2:
>>>> On Friday, September 17, 2021 at 9:55:31 PM UTC+2, WM wrote:
>>>>
>>> Then its indices exhaust all natural numbers. Usually the index set of an
>>> infinite sequence is just IN. And right, a function is defined for all
>>> elements in its domain. None are remaining for the contents.
>> Huh?! No dual use allowed in Mückenwolkenheim?
>
> Not for endsegments. The contents of E(n) continues the indices used up to
> n-1.
>>
>> Hint: In the context of mathematics we have: Ak e IN: card(E(k)) = aleph_0.
>> So each and every endsegment is "filled" with infinitely many natural
>> numbers, even though there are infinitely many endsegments. Why does this
>> work? Because each and every natural number has infinitelly many successors.
>
> Why does this not work? Because the set of all indices is |N and has no
> successors.
>>
>>> What are the [...] natural numbers remaining [...] in all endsegments?
>>
>> There are no natural numbers "remaining" in all endsegments.
>
> You claim that infinitely many are in every endsegment. But aleph_0
> endsegments with contents aleph_0 numbers are not possible. Therefore one of
> your claims is wrong.

Stop lying.

It happens that Serg io formulated :
> On 9/19/2021 9:51 AM, WM wrote:
>> Greg Cunt schrieb am Sonntag, 19. September 2021 um 03:39:21 UTC+2:
>>> On Saturday, September 18, 2021 at 6:01:26 PM UTC+2, WM wrote:
>>>> Greg Cunt schrieb am Samstag, 18. September 2021 um 01:49:32 UTC+2:
>>>>> On Friday, September 17, 2021 at 9:55:31 PM UTC+2, WM wrote:
>>>>>
>>>> Then its indices exhaust all natural numbers. Usually the index set of an
>>>> infinite sequence is just IN. And right, a function is defined for all
>>>> elements in its domain. None are remaining for the contents.
>>> Huh?! No dual use allowed in Mückenwolkenheim?
>>
>> Not for endsegments. The contents of E(n) continues the indices used up to
>> n-1.
>>>
>>> Hint: In the context of mathematics we have: Ak e IN: card(E(k)) = aleph_0.
>>> So each and every endsegment is "filled" with infinitely many natural
>>> numbers, even though there are infinitely many endsegments. Why does this
>>> work? Because each and every natural number has infinitelly many
>>> successors.
>>
>> Why does this not work? Because the set of all indices is |N and has no
>> successors.
>>>
>>>> What are the [...] natural numbers remaining [...] in all endsegments?
>>>
>>> There are no natural numbers "remaining" in all endsegments.
>>
>> You claim that infinitely many are in every endsegment. But aleph_0
>> endsegments with contents aleph_0 numbers are not possible.
>
>
> False. you have failed to understand infinity. You are using only one pile
> of rocks.
>
> between any pair of numbers on the real line there are aleph_0 numbers.

At least that. Could be many many many more though.

Crank Wolfgang Mueckenheim, WM wrote:
> Greg Cunt schrieb am Samstag, 18. September 2021 um 20:20:00 UTC+2:
>> On Saturday, September 18, 2021 at 5:56:43 PM UTC+2, WM wrote:
>>
>> Let n e IN.
>>>>> n is not a concrete natural number [WM]
>>>>>
>>>> It's certainly a "concrete" natural number, though we don't KNOW which one.
>>>>
>>> For every natural number we know it.
>> Huh?! So you personally know ALL natural numbers? Wow - that's rather cool!
>
> I know, in principle, every number that you can mention. n does not belong to that set.
>
>>>> Hint: n+1 =/= n.
>>>>
>>> n+1 is not a natural number too.
>> Oh, I stand corrected!
>>
>> So n is in IN and n+1 is in IN, but neither n nor n+1 are natural numbers? Is that your claim? So what are they? Pink elephants?
>
> They are placeholders. "Let n in |N, then P(n)" is an abbreviation for: "Any natural number can be put in place of n and will satisfy P."

Which in turns (following your "logic") is an abréviation for "Any
natural number can be put in place of n and will satisfy 'Any natural
number can be put in place of n and will satisfy'", ... Crank Wolfgang
Mueckenheim from Hochschule Augsburg.

Greg Cunt schrieb am Sonntag, 19. September 2021 um 19:21:44 UTC+2:
> On Sunday, September 19, 2021 at 4:51:19 PM UTC+2, WM wrote:
>
> > aleph_0 endsegments with contents aleph_0 numbers are not possible.
> False claim.

Two consecutive infinite sets in 1, 2, 3, ... ?
>
> Hint: In the context of set theory (say ZFC) thhese are theorems:
>
> 1. card({E(n) : n e IN} = aleph_0
>
> and
>
> 2. An e IN: card(E(n)) = aleph_0.

So ZFC proves itself inconsistent. Two consecutive infinite sets in 1, 2, 3, ... are impossible.

Regards, WM

On Monday, September 20, 2021 at 6:20:21 PM UTC+2, WM wrote:
> Greg Cunt schrieb am Sonntag, 19. September 2021 um 19:21:44 UTC+2:
> > On Sunday, September 19, 2021 at 4:51:19 PM UTC+2, WM wrote:
> > >
> > > aleph_0 endsegments with contents aleph_0 numbers are not possible. (*)
> > >
> > False claim.
> >
> Two consecutive infinite sets in 1, 2, 3, ... ? (**)

1. (*) is false.

2. What's your question - PRECISELY?

3. What's its relevamc concerning your false claim (*)? None, it seems.

You see:

> > In the context of set theory (say ZFC) these are theorems:
> >
> > 1. card({E(n) : n e IN} = aleph_0
> >
> > and
> >
> > 2. An e IN: card(E(n)) = aleph_0.
> >
> So ZFC proves itself inconsistent.

Where? What are the contradicting theorems and where is the proof for your silly claim?

You know you have to prove a statements A and a statement ~A from the axioms (and definitions) of ZFC (where A is a formula formulated in the language of ZFC + the defined notions).

> Two consecutive infinite sets in 1, 2, 3, ... are impossible.

4. What's your claim - PRECISELY?

I guess you mean the following:

| There are no two sets A c IN and B c IN such that AaAb(a e A & b e B --> a < b) _and_ both sets, A and B are infinite. (***)

I'd say that's a theorem in ZFC. Proof?

Well, I'd proceed the following way:

The simpler claim

| For all sets A c IN and B c IN: if AaAb(a e A & b e B --> a < b) and A is infinite, then B = {}.

implies (***).

Hence it suffices to prove the simpler claim.

Proof: A c IN, B c IN, AaAb(a e A & b e B --> a < b) and A is infinite. Now let's assume that B is not empty. Then B is a nonempty set of natural numbers. Hence there is at least one natural number in B. Let m be a natural number in B. Then there must be a number in A which is larger than m, say n. (Otherwise there were only finitely many numbers in A, i. e. numbers <= m, if any at all. But A is infinite.) Hence n e A & m e B and n > m, contradicting AaAb(a e A & b e B --> a < b). Hence B is empty. qed

Hint: The theorems:

1. card({E(n) : n e IN} = aleph_0

and

2. An e IN: card(E(n)) = aleph_0

do not imply that there are sets A c IN and B c IN such that AaAb(a e A & b e B --> a < b) _and_ both sets, A and B are infinite.

Clearly, for any two ensegments E and E': E n E' =/= {}. Hence ~AaAb(a e E & b e E' --> a < b).

On 9/20/2021 11:20 AM, WM wrote:
> Greg Cunt schrieb am Sonntag, 19. September 2021 um 19:21:44 UTC+2:
>> On Sunday, September 19, 2021 at 4:51:19 PM UTC+2, WM wrote:
>>
>>> aleph_0 endsegments with contents aleph_0 numbers are not possible.
>> False claim.
>
> Two consecutive infinite sets in 1, 2, 3, ... ?

1, 2, 3,... = 1,3,5,7,... + 2,4,6,8,...

aleph_0 = aleph_0 + aleph_0

>>
>> Hint: In the context of set theory (say ZFC) thhese are theorems:
>>
>> 1. card({E(n) : n e IN} = aleph_0
>>
>> and
>>
>> 2. An e IN: card(E(n)) = aleph_0.
>
> So ZFC proves itself inconsistent. Two consecutive infinite sets in 1, 2, 3, ... are impossible.
>
> Regards, WM
>

On Monday, September 20, 2021 at 6:54:35 PM UTC+2, Serg io wrote:
> On 9/20/2021 11:20 AM, WM wrote:
> >
> > Two consecutive infinite sets in 1, 2, 3, ... ?
> >
> {1, 2, 3, ...} = {1, 3, 5, 7, ...} u {2, 4, 6, 8, ...}

Sure. But _not_

for Ax e {1, 3, 5, 7, ...} and Ay e {2, 4, 6, 8, ...}: x < y .

I guess, WM is asking for two infinite sets A and B (with A c IN and B c IN) such that for all a e A and for all b e B: a < b.

On Monday, 20 September 2021 at 14:13:50 UTC-3, Greg Cunt wrote:
> On Monday, September 20, 2021 at 6:54:35 PM UTC+2, Serg io wrote:
> > On 9/20/2021 11:20 AM, WM wrote:
> > >
> > > Two consecutive infinite sets in 1, 2, 3, ... ?
> > >
> > {1, 2, 3, ...} = {1, 3, 5, 7, ...} u {2, 4, 6, 8, ...}
>
> Sure. But _not_
>
> for Ax e {1, 3, 5, 7, ...} and Ay e {2, 4, 6, 8, ...}: x < y .
>
> I guess, WM is asking for two infinite sets A and B (with A c IN and B c IN) such that for all a e A and for all b e B: a < b.

This, of course, depends on the definition of '<'. If a < b is defined properly, e.g., for a, b in |N, a < b iff [ (a in A and b in B) OR ( (b-a)/2 in |N ], Serg io's sets fit the bill. (Of course, WM can't fathom that there could be more than one way to order a set, but that's another story.)