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Sep 23, 2022, 5:29:16 AMSep 23

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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.

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.

Sep 23, 2022, 7:57:41 AMSep 23

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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.
> 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.

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

Sep 23, 2022, 10:46:32 AMSep 23

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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.
> 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.

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.

>

>

> 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.
> 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...

> ...

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...

> ...

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,

, we have

>

> XXXX...

> XXXX...

> XXXX...

> XXXX...

> ...

>

Where did you get all the X and O stickers ?

>

> Regards, WM

>

Sep 23, 2022, 11:56:35 AMSep 23

to

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".

Sep 23, 2022, 12:19:16 PMSep 23

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When passed from hand to hand one turns it over.

Sep 24, 2022, 12:49:14 PMSep 24

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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, ....

Sep 24, 2022, 1:17:05 PMSep 24

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"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.

Sep 24, 2022, 1:30:14 PMSep 24

to

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.

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