Mathematical axioms.

[Antonio Speltzu]

Are mathematical axioms physical laws?
That is, the universe would be different if, for example, the union of two sets did not exist, in some cases. Or it would be one way or another depending on whether the continuum hypothesis was true or not.

[WM]

Antonio Speltzu schrieb am Freitag, 23. September 2022 um 11:29:16 UTC+2:
> Are mathematical axioms physical laws?
> That is, the universe would be different if, for example, the union of two sets did not exist, in some cases. Or it would be one way or another depending on whether the continuum hypothesis was true or not.

The last question deserves a resounding no.

There is no countability. ==> There is no uncountability. ==> The continuum hypothesis is meaningless.

Proof:

Cantor enumerates the positive fractions m/n

1/1, 1/2, 1/3, 1/4, ...
2/1, 2/2, 2/3, 2/4, ...
3/1, 3/2, 3/3, 3/4, ...
4/1, 4/2, 4/3, 4/4, ...
5/1, 5/2, 5/3, 5/4, ...
...

using indices obtained from k = (m + n - 1)(m + n - 2)/2 + m. The result is the sequence 1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, 2/4, 3/3, 4/2, 5/1, 1/6, 2/5, 3/4, 4/3, 5/2, 6/1, ...

The indices are used first to index the Integer fractions m/1. If indexed fractions are denoted by X's and not indexed fractions are denoted by O's, we get the following matrix:

XOOO...
XOOO...
XOOO...
XOOO...
...

Then the indices are taken from their initial positions and are distributed by Cantor's formula. The indexed fractions are denoted by X, the fractions without indices are denoted by O. Index 1 remains at 1/1. The next step takes the index 2 from 2/1 and attaches it to 1/2:

XXOO...
OOOO...
XOOO...
XOOO...
...

Then the index 3 it taken from 3/1 and is attached to 2/1:

XXOO...
XOOO...
OOOO...
XOOO...
...

In the end, when all exchanges of X and O have been carried through and all X's have settled at their destination prescribed by Cantor's formula,, we have

XXXX...
XXXX...
XXXX...
XXXX...
...

but exchanging never deletes an an element. No O has left the matrix. Where are they? If they have not left, then they are within the matrix but they cannot be seen. They are at dark positions. Dark elements cannot be identified and cannot be mapped. Therefore countability of infinite sets has been contradicted.

Regards, WM

[Sergi o]

On 9/23/2022 6:57 AM, WM wrote:
> Antonio Speltzu schrieb am Freitag, 23. September 2022 um 11:29:16 UTC+2:
>> Are mathematical axioms physical laws?

of course not.

>> That is, the universe would be different if, for example, the union of two sets did not exist, in some cases. Or it would be one way or another depending on whether the continuum hypothesis was true or not.

"Cantor believed the continuum hypothesis to be true and for many years tried in vain to prove it.[3] It became the first on David Hilbert's list of
important open questions that was presented at the International Congress of Mathematicians in the year 1900 in Paris. Axiomatic set theory was at that
point not yet formulated. Kurt Gödel proved in 1940 that the negation of the continuum hypothesis, i.e., the existence of a set with intermediate
cardinality, could not be proved in standard set theory.[2] The second half of the independence of the continuum hypothesis – i.e., unprovability of the
nonexistence of an intermediate-sized set – was proved in 1963 by Paul Cohen.[4]"

>
> The last question deserves a resounding no.
>
> There is no countability. ==> There is no uncountability. ==> The continuum hypothesis is meaningless.
>

> Spoof:

>
> Cantor enumerates the positive fractions m/n
>
> 1/1, 1/2, 1/3, 1/4, ...
> 2/1, 2/2, 2/3, 2/4, ...
> 3/1, 3/2, 3/3, 3/4, ...
> 4/1, 4/2, 4/3, 4/4, ...
> 5/1, 5/2, 5/3, 5/4, ...
> ...
>
> using indices obtained from k = (m + n - 1)(m + n - 2)/2 + m. The result is the sequence 1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, 2/4, 3/3, 4/2, 5/1, 1/6, 2/5, 3/4, 4/3, 5/2, 6/1, ...
>
> The indices are used first to index the Integer fractions m/1. If indexed fractions are denoted by X's and not indexed fractions are denoted by O's, we get the following matrix:
>
> XOOO...
> XOOO...
> XOOO...
> XOOO...
> ...

That erases the matrix of fractions, and replaces it with O pasties, and X stickies.

>
> Then the indices are taken from their initial positions and are distributed by Cantor's formula. The indexed fractions are denoted by X, the fractions without indices are denoted by O. Index 1 remains at 1/1. The next step takes the index 2 from 2/1 and attaches it to 1/2:
>
> XXOO...
> OOOO...
> XOOO...
> XOOO...
> ...

Now WM has taken off the X pastie at 1,2 and put an O stickie on it.
And then WM takes off the O pastie at 2,1 and puts a X pastie on it.
there are no swaps here.

>
> Then the index 3 it taken from 3/1 and is attached to 2/1:
>
> XXOO...
> XOOO...
> OOOO...
> XOOO...
> ...

Now WM has taken off the X pastie at 1,3 and put an O stickie on it.
And then WM takes off the O pastie at 1,2 and puts a X pastie on it.
there are no swaps here.

just continual replacement of stickers and pasties


>
> In the end,

there is no end, these are infinite sets.

> when all exchanges of X and O have been carried through and all X's have settled at their destination prescribed by Cantor's formula,

Liar, Cantor did not use pasties and stickers.

, we have
>
> XXXX...
> XXXX...
> XXXX...
> XXXX...
> ...
>

What did you do with all the O stickers ??

Where did you get all the X and O stickers ?



>
> Regards, WM
>

[Ross A. Finlayson]

Maybe instead you should start "there is an infinite collection,
half X's and half O's".

You need to define "half infinity".

Then, you can build a very very large finite space, where,
it's not so much that "at random the sequence will be half zeros
and half ones", that the uniform space would give, but,
a given sample determines where it is in the space, between
0.0 infinities and 1.0 infinities, or wholes.

I.e., here the sequences having two ends and an infinite middle,
it's just that simple then for whether 'in this theory the one-sided
sequences are built from two-sided sequences not the other way
around", it's not some big problem only some "idiot's bridge",
"giant bridge stupid shall not pass", basically to explain the
pons asinorum or ass' bridge, that puts the idiot not just,
not passing the bridge, but yelling "you shall not pass",
in his marble pit.

I.e. the entire point of random sampling is that "each of
the sequences is having the same probability of existing
the sample", that infinite sampling is expected to produce
a copy, meaning, that sampling the samples, also has the
property hold.

Then, there's for building that from 0 to 1, it results the
expected value is 0 or 1, that around 1/2, it's 1/2.

This is about the _simplest_ things to put together for
the probability distributions the uniform and normal,
then _simply_ getting rid both troubles "can't count"
and "can't complete".

[Sergi o]

we could have an infinite collection of coins, one side with an X, the other with an O.
When passed from hand to hand one turns it over.

[Ross A. Finlayson]

How is that any different from three people passing
one coin with one side, what supertask is this toggle.

The "expected ergodicity", ergodic is equated as a property
with fair, that for two, always toggling is same as never toggling,
while, for three, it's always expected to switch.

That's for a fundamental difference between 2 and 3 that though
of course in the asymptotic is erased: in this example from some
simplest expectations, it's a kind of thought lever knowing that
fair or unfair it's same or different.

How is half 1s/0s or Xs/Os, no longer same with thirds 0/1/2 or
for example X/Y/Z, given units of coordinates in XYZ for example?

You figure it's all same all their infinite sequences, but,
the ways they vary, what write and build them all, with the opportunistic
that only exists for the least number of values: two.

Or as a space of tally marks, "self-consistent count and measure", one.

Then, given it's all blind people how many sides does the coin have
and how do you flip it, to guess how many people are in the room?

Given that somebody eventually returns it, ....

[Ross A. Finlayson]

Is motion a supertask? Isn't it?

"The proof of equivalence is very abstract;
understanding the result is not:
by adding one at each time step,
every possible state of the odometer is visited,
until it rolls over, and starts again."
-- https://en.wikipedia.org/wiki/Ergodicity#Processes



What's ergodic and parameterized by t: physics.

[Ross A. Finlayson]

"... the definition of ergodicity carries over unchanged
if one replaces invariant measures by quasi-invariant measures. "
-- https://en.wikipedia.org/wiki/Ergodicity#Generalisations

The quasi-invariant is mostly singular not "not invariant".
Singular in exchange, ..., including wave dissipation.


About defines "quasi-invariant measure theory".


Though, I only heard about "shift spaces" and "sofic systems" today,
I've already been talking about them and "spaces of infinite sequences".

"... measure-preserving, ..., angle-preserving, ..., truth-preserving".



"In fact, both the one- and two-side shift spaces are compact metric spaces."


So anyways, such magic of infinity in mathematics lives directly in
constructing the numbers as I usually make it _simple_, why what
according facts in measure theory live directly in GEOMETRY.