Re: Replacement of Cardinality

On 8/26/2024 10:08 AM, Moebius wrote:

Am 24.08.2024 um 00:04 schrieb Jim Burns:

x = 1/0  :⇔  0⋅x = 1

>
Using a slight variant of your "definition"

The issue is that you think my definition
makes claims it doesn't make.
You can show as many ways as you like that
_the claims you think it makes_ need proof.
My definition doesn't make them.
Another variation:
( x ≠ 1/0  :⇔  0⋅x ≠ 1
Is that more acceptable?
Why would it be?
It says the same thing.
Neither
( x ≠ 1/0  :⇔  0⋅x ≠ 1
nor
( x = 1/0  :⇔  0⋅x = 1
claims any of
x≠1/0 or 0⋅x≠1 or x≠1/0 or 0⋅x≠1

Using a slight variant of your "definition"
it's easy to show why your approach
(concerning definitions)
is not tenable in math:
>
Def.: x = 1/0  :⇔  x ∈ IR & 0⋅x = 1
>
Now from identity theory
(say in the context of FOPL=) we get
Ax(x = x)
>
and hence (by specialisiation, AE):
1/0 = 1/0 .

No.

[We may do this, because "1/0" is now -
after your definition -
a term in our system.]

No.
Being a term in the language is not sufficient.
The term needs to refer to an object in the domain.
1/0 doesn't refer to anything.
| [...] a valid rule of inference
| from a truth about each member of a CLASS of individuals
| to the truth about a particular individual of that CLASS.
|
https://en.wikipedia.org/wiki/Universal_instantiation
(Emphasis added.)
1/0 isn't a member of the relevant class of individuals,
something which you point out very thoroughly.
One consequence of it's non.membership is that
Universal Instantiation is out of reach.
Treating non.members as members and deriving nonsense,
which proves non.members aren't members,
has a long and glorious history in mathematics, but,
in order to do that,
it is necessary to reason using terms which don't refer.
You want to call that being.a.crank.
Have at it.
You will pry my 'ex falso quodlibet' from
my cold, dead hands.
----
In many treatments of the real numbers, integers, etc,
proofs of existence, uniqueness, etc, are
_closely positioned_ to definitions.
That is where they should be, if one intends to use
the existence and uniqueness of reals and integers,
and their sums, products, differences and quotients.
I think that your objection to my use of 1/0 is that
it _looks like_ something covered by those early proofs.
And it isn't.
But I don't say it is.
( x = 1/0  :⇔  0⋅x = 1
says
x=1/0 is as true AND AS FALSE as 0⋅x=1
and it doesn't say anything else.