Problem with Cantors Diagonalization Proof.

[Brian_G]

There is no real number between 1 and 0.9999… recurring. This is only true for equal numbers.

Thanks, I can grasp that now.

But it's still saying to me that the tiniest difference, at infinite scale, is no different from having no difference. However we paint that, it's still weird.

 

[cguy]

Thanks, I can grasp that now.

But it's still saying to me that the tiniest difference, at infinite scale, is no different from having no difference. However we paint that, it's still weird.

Perhaps, think of it this way: For a distance function d(x,y), if d(x,y) = 0, then x = y. This is axiomatic on all metric spaces. So, assume x and y are distinct. This means that a distinct number (x+y)/2 must lie equidistant between them. Since in the case of x=1 and y=0.999..., we know that no number lies between them, the numbers cannot be distinct.

 

[talanum1]

"x is defined as sum(((f(i)+1) mod 10).10^-i)"

Same problem: you don't have access to f(infinity) to compute x.

 

[cguy]

"x is defined as sum(((f(i)+1) mod 10).10^-i)"

Same problem: you don't have access to f(infinity) to compute x.

I don’t need to compute x, I just need to define it.

 

[Brian_G]

Thanks @cguy , will give that further thought... it's on the edge of my understanding though.
I'm left with the impression that space may be an illusion :- )

You're stating you can change pi [or a similar type number] digit by digit for all it's digits.

Yes.

Although we can't properly create / observe the process in physical reality due to it being infinite, we can still show it could be done.
You would need to show why you think the digit "at" infinity could otherwise influence that.

I can't see anything else I or others could contribute.

 

[talanum1]

I don’t need to compute x, I just need to define it.

If you can't compute it, it is ill defined! How else are you going to define "well defined".

For example f(x) = 1/x is undefined at x = 0!

 

[cguy]

If you can't compute it, it is ill defined! How else are you going to define "well defined".

For example f(x) = 1/x is undefined at x = 0!

It’s not the same thing. What I am doing is the equivalent of saying that 2 x PI is 2PI. I don’t need to perform an addition on every digit of PI to show this. It is true by definition.

The ill definedness of your example is due to division by zero being undefined.

 

[talanum1]

With 2Pi you can compute it approximately and you don't need access to the digit "at" infinity, but in this case you do.

 

[Brian_G]

, but in this case you do.

Again... show us why


Not

If you can't compute it, it is ill defined! How else are you going to define "well defined".

but rather why you classify it as actually needing that further definition?

If I bought a cow to have some milk, why would it matter if the cow has black spots or brown spots? Just to make my choice "well defined"?


Or you can ignore my question again..

 

[talanum1]

You are computing a number that is different from pi in every digit: then you need f(infinity).

What further definition?

The black or brown spots determines if the cows milk is sour or not.

 

[Brian_G]

You are computing a number that is different from pi in every digit: then you need f(infinity).

There is no computing involved, it's a simple random-options alteration experiment process, for all the digits, with a random but mathematically meaningful outcome, which only proves that a certain strange phenomena exists.
It also has no relation to pi at any stage, if that seems to still be a thing for you. Rather pick another meaningless irrational number example.

What further definition?

The black or brown spots determines if the cows milk is sour or not.

Not sure if you're being funny here or not.
Good luck either way.

 

[cguy]

With 2Pi you can compute it approximately and you don't need access to the digit "at" infinity, but in this case you do.

2PI is exactly 2xPI, not an approximation. The same way the entire infinitely long number is multiplied or added exactly and instantaneously, the prior definitions of PI differing at each digit and uncounted real exist exactly and instantaneously.

 

[talanum1]

It all requires a completed infinity and I don't accept such an infinity. Such a list existing is supernatural and should be carried through to other results to test for inconsistency.

 

[EmmanuelGoldstein]

It all requires a completed infinity and I don't accept such an infinity. Such a list existing is supernatural and should be carried through to other results to test for inconsistency.

Ask the aliens then…seeing that you said you have contact with those grey people

 

[cguy]

It all requires a completed infinity and I don't accept such an infinity. Such a list existing is supernatural and should be carried through to other results to test for inconsistency.

Disputing the last 62-150 years of mathematical research is your prerogative.

 

[Brian_G]

a completed infinity

Oxymoron.

Some infinite behaviour is observable, no completion required.
Nothing indicates any possible funny business about "reaching" infinity in this case.
Infinity doesn't ever complete, end of argument (pun unintended).

You're in need of flexibility
But thanks for the mental gymnastics,

 

[Brian_G]

aliens

[are]

grey people

Only the bad ones

 

[cguy]

Oxymoron.

Some infinite behaviour is observable, no completion required.
Nothing indicates any possible funny business about "reaching" infinity in this case.
Infinity doesn't ever complete, end of argument (pun unintended).

You're in need of flexibility
But thanks for the mental gymnastics,

I expect that he may have been referring to this;

Actual and potential infinity - Wikipedia

en.wikipedia.org

Although that may be too much of a benefit of the doubt, given that he was going on about the digit at infinity.

 

[talanum1]

Dirac starts with the energy dispersion equation: E^2 =(pc)^2+(mc^2)^2, but then he draws the positive square root of this. Thus it seems he tried to exclude negative energy. Then he derives the Dirac Equation which can be shown to have negative energy solutions, even though he tried to exclude negative energy.

 

[RedViking]

Why do you create a new thread for each post. Why not just put it all in one thread.