Am 15.06.2024 um 18:19 schrieb Ross Finlayson:

On 06/15/2024 08:58 AM, Moebius wrote:

Am 14.06.2024 um 20:52 schrieb Jim Burns:

On 6/14/2024 12:39 PM, WM wrote:

Just seen here:

"number(s)" (WM) seems to refer to "natural number(s)" in this context.

WM (Proof by contradiction):
[Assume:] Every number has ℵo successors.

Actually, we do not have to assume that, since it can be proved (in the
context of mathematics/set theory).

An e IN: card({m e IN : m > n}) = ℵo.

If every number is subtracted the successors remain.

Huh?! Just a silly (psychotic) claim. If _every_ number "is subtracted"
(based on "the set of numbers+their successors"), then NO numbers (and
hence no successors) "remain" [in the new/resulting set]. (After all,
the successors of any number are numbers too.*)

What did WM prove here? That he's a complete idiot?

_____________________________________________

*) An e IN: {m e IN : m > n} c IN.

If oo - oo = 0, or,

oo - oo usually is undefined (see: https://en.wikipedia.org/wiki/Extended_real_number_line#Arithmetic_operations)

While on the other hand:

N - N = 0 for a large number N [=/= oo],

Right.

Is there anything you want do say, Ross?

Something which is RELATED to my post you quoted?

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Am 15.06.2024 um 18:45 schrieb Ross Finlayson:

On 06/15/2024 09:35 AM, Moebius wrote:

Am 15.06.2024 um 18:19 schrieb Ross Finlayson:

On 06/15/2024 08:58 AM, Moebius wrote:

Am 14.06.2024 um 20:52 schrieb Jim Burns:

On 6/14/2024 12:39 PM, WM wrote:

Just seen here:

"number(s)" (WM) seems to refer to "natural number(s)" in this context. >>>>

WM (Proof by contradiction):
[Assume:] Every number has ℵo successors.

Actually, we do not have to assume that, since it can be proved (in the >>>> context of mathematics/set theory).

An e IN: card({m e IN : m > n}) = ℵo.

If every number is subtracted the successors remain.

Huh?! Just a silly (psychotic) claim. If _every_ number "is subtracted" >>>> (based on "the set of numbers+their successors"), then NO numbers (and >>>> hence no successors) "remain" [in the new/resulting set]. (After all,
the successors of any number are numbers too.*)

What did WM prove here? That he's a complete idiot?

_____________________________________________

*) An e IN: {m e IN : m > n} c IN.

If oo - oo = 0, or,

oo - oo usually is undefined (see:
https://en.wikipedia.org/wiki/Extended_real_number_line#Arithmetic_operations)

While on the other hand:

N - N = 0 for a large number N [=/= oo],

Right.

Is there anything you want do say, Ross?

Something which is RELATED to my post you quoted?

Sometimes instead of "undefined" we say "indeterminate form".

Sometimes "indeterminate forms", are, "defined".

The expressions oo - oo, 0 x (+/-oo), +/-oo/+/-oo (called indeterminate
forms) are usually left undefined.

Is there anything you want do say, Ross?

Something which is RELATED to my post you quoted?

Obviously not.

Hence EOD.

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Am 15.06.2024 um 18:45 schrieb Ross Finlayson:

On 06/15/2024 09:35 AM, Moebius wrote:

Am 15.06.2024 um 18:19 schrieb Ross Finlayson:

On 06/15/2024 08:58 AM, Moebius wrote:

Am 14.06.2024 um 20:52 schrieb Jim Burns:

On 6/14/2024 12:39 PM, WM wrote:

Just seen here:

"number(s)" (WM) seems to refer to "natural number(s)" in this context. >>>>

WM (Proof by contradiction):
[Assume:] Every number has ℵo successors.

Actually, we do not have to assume that, since it can be proved (in the >>>> context of mathematics/set theory).

An e IN: card({m e IN : m > n}) = ℵo.

If every number is subtracted the successors remain.

Huh?! Just a silly (psychotic) claim. If _every_ number "is subtracted" >>>> (based on "the set of numbers+their successors"), then NO numbers (and >>>> hence no successors) "remain" [in the new/resulting set]. (After all,
the successors of any number are numbers too.*)

What did WM prove here? That he's a complete idiot?

_____________________________________________

*) An e IN: {m e IN : m > n} c IN.

If oo - oo = 0, or,

oo - oo usually is undefined (see:
https://en.wikipedia.org/wiki/Extended_real_number_line#Arithmetic_operations)


While on the other hand:

N - N = 0 for a large number N [=/= oo],

Right.

Is there anything you want do say, Ross?

Something which is RELATED to my post you quoted?

Sometimes instead of "undefined" we say "indeterminate form".

Sometimes "indeterminate forms", are, "defined".

"The expressions oo - oo, 0 x (+/-oo), ±oo/+/-oo (called indeterminate
forms) are usually left undefined."

Source: https://en.wikipedia.org/wiki/Extended_real_number_line#Arithmetic_operations

Is there anything you want do say, Ross?

Something which is RELATED to my post you quoted?

Obviously not.

Hence EOD.

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On 6/15/2024 12:45 PM, Ross Finlayson wrote:

On 06/15/2024 09:35 AM, Moebius wrote:

[...]

I.e., singularities in closed theories,
are multiplicities in more open theories.

It is an essential aspect of mathematics that,
when we are discussing a thing or things,
we are not discussing a different thing or things.

That's a principle with wider application, I think,
but mathematics is simply impossible without it.

Consider the set of finite von Neumann ordinals.
None of them are infinite.
Amazing! Fantastic! A tour de force!
Not.

But,
using the knowledge that none of them are infinite,
we explore the infinity of them by means of
our not.first.false telescope.

That is not an exploration of more open theories
about infinite von Neumann ordinals as well,
even though those are theories we could explore.
We _could_ explore them, but we _aren't_
Not right now.

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Am 15.06.2024 um 19:30 schrieb Jim Burns:

It is an essential aspect of mathematics that,
when we are discussing a thing or things,
we are not discussing a different thing or things.

Right. The context should be a certain/unique theory.

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Am 16.06.2024 um 01:45 schrieb Moebius:

Am 15.06.2024 um 19:30 schrieb Jim Burns:

It is an essential aspect of mathematics that,
when we are discussing a thing or things,
we are not discussing a different thing or things.

Right. The context should be a certain/unique theory.

For example, in real analysis there are no infinitesimals, on the other
hand, in non-standard analysis there are.

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Am 16.06.2024 um 21:20 schrieb Chris M. Thomasson:

x = .00000000000000000000000000000000000018973240008372

is radically different (wrt the resulting render) than:

x = .000000000000000000000000000000000000189732400083720400000001

Right. It's the difference that makes the difference.

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