Problem with Cantor's Diagonalization Proof.

[The Sage of Main Street]

Good question. You can name such a set, for example the set of all irrational numbers.

That's what leads to the dilemma, sets that aren't really sets at all

 

[The Sage of Main Street]

But that tells us nothing.

If infinity is multiply defined, the question is whether there are "degrees" of infinity or merely multiple "types".

It depends on how you define infinity. One concept is "larger than the largest number", so on the number line you have + and - infinity. Another concept has to do with distance (norm) and it means "sufficiently far away". In this case + infinity is the same as - infinity, all infinities are the same.

Cantor's contribution is showing these concepts are related numerically by a simple equation. Cantor intuitively explored recursive processes ("dynamic programming"), in his examples you can find many versions of DFS and BFS along with explanations for why they differ.



Does the set have a topology? Is it compact? There are reasons why people want to add points at infinity.

You get some peculiar behavior when you add points at infinity. For instance an Alexandroff 1-point compactification changes a real interval into a circle. You can now describe it with complex numbers, and it has commutative and non-commutative projection symmetries. How many times can you go around the circle? Count, starting at 1, the result is aleph 0, a "countable" infinity. This result, is what most people think of when they think of infinity. "A very large number, a number so big it's greater than anything you can write down".

"Numerable" means you can write it down. So by definition, we have to add a point at infinity to the set of reals. But when we do this, we change our real interval into a circle. UNLESS, we add two points at infinity, instead of just one. If we distinguish + and - infinity then our real interval can remain an interval (it is merely "extended").

{ } [ ] ​

It's a one-bracket set, i.e., a non-set, a one-parenthesis parentheses.

 

[ReinyDays]

That weirdo theory is spooky, goofy, irrational, and childish, just like Theism

That was Schödinger's point ... how can the cat be half-alive .... that's complete nonsense ...

“Well! I’ve often seen a cat without a grin,” thought Alice; “but a grin without a cat! It’s the most curious thing I ever saw in my life!”


History that deserves to be remembered

 

[scruffy]

Newton's definition of infinity isn't any good any more? ...

There was never a single definition, until Cantor.

Even Aristotle claimed there was no such thing as an "actual" infinity.

 

[ReinyDays]

There was never a single definition, until Cantor.

Even Aristotle claimed there was no such thing as an "actual" infinity.

Oh ... and what is Cantor's definition? ... and how is this more useful than Newton's infinitesimals? ...

"She had not gone much farther before she came in sight of the house of the March Hare: she thought it must be the right house, because the chimneys were shaped like ears and the roof was thatched with fur."

 

[scruffy]

Oh ... and what is Cantor's definition?

Aleph number - Wikipedia

en.wikipedia.org

... and how is this more useful than Newton's infinitesimals? ...

Infinitesimals are limits.

"She had not gone much farther before she came in sight of the house of the March Hare: she thought it must be the right house, because the chimneys were shaped like ears and the roof was thatched with fur."

Begin at the beginning," the King said, very gravely, "and go on till you come to the end: then stop".

 

[ReinyDays]

Aleph number - Wikipedia

en.wikipedia.org

"The aleph numbers differ from the infinity (

) commonly found in algebra and calculus, in that the alephs measure the sizes of sets, while infinity is commonly defined either as an extreme limit of the real number line (applied to a function or sequence that "diverges to infinity" or "increases without bound"), or as an extreme point of the extended real number line."

Doesn't sound like a very good definition of infinity ...

Infinitesimals are limits.

Which function has a differential limit? ...

"It was so large a house, that she did not like to go nearer till she had nibbled some more of the lefthand bit of mushroom, and raised herself to about two feet high: even then she walked up towards it rather timidly, saying to herself “Suppose it should be raving mad after all! I almost wish I’d gone to see the Hatter instead!”

 

[The Sage of Main Street]

That was Schödinger's point ... how can the cat be half-alive .... that's complete nonsense ...

“Well! I’ve often seen a cat without a grin,” thought Alice; “but a grin without a cat! It’s the most curious thing I ever saw in my life!”


History that deserves to be remembered

Motto of Post-clac Quacks: "If It's Weird, It's Wise"

Recognizing a fourth spatial dimension explains these phenomena rationally. For example, the quantum leap is impossible unless the electron goes there and comes back at a different place in its third dimensional setting.

 

[scruffy]

.
Doesn't sound like a very good definition of infinity ..

You have a better one?

Which function has a differential limit? ...

All derivatives are limits.

"It was so large a house, that she did not like to go nearer till she had nibbled some more of the lefthand bit of mushroom, and raised herself to about two feet high: even then she walked up towards it rather timidly, saying to herself “Suppose it should be raving mad after all! I almost wish I’d gone to see the Hatter instead!”

You missed the point of my quote.

COUNT.

Start at the beginning. Stop at the end. Don't take shortcuts or make wild guesses.

 

[toobfreak]

With all due respect ... the set of real numbers forms a vector space, whereas the set of natural numbers do not ... so there's no "one and only one" correspondence ... the problem is your "zero" value; it's a whole number, but not a natural number ...

Are you sure? I just looked up one definition of natural number which argued whether 0 was or wasn't natural.

Too many numbers for my book. Numbers is numbers, instead, we have this huge mess of whole numbers, natural numbers, counting numbers, real numbers, rational numbers, and on and on.

 

[ReinyDays]

You have a better one?

A better definition of infinity? ... better than the one used in algebra and calculus? ... Cantor seems to have just invented an new definition of infinity ... and if that's useful, that's fine, but it's a philosophical question ...

All derivatives are limits.

All derivatives are functions ... are you suggesting all functions are limits? ... I think it's better to say all functions have limits ... although related, they are two different things ... much like energy and power ...

You missed the point of my quote.

You must have missed the point of my quotes ... they are actually intended for Sage l'Main Street ... quite insane ...

 

[scruffy]

A better definition of infinity? ... better than the one used in algebra and calculus? ...

What is the definition of infinity in algebra and calculus?

Cantor seems to have just invented an new definition of infinity ... and if that's useful, that's fine, but it's a philosophical question ...

It's set theory. It has many ramifications in topology, which in turn affects quantum behavior.

All derivatives are functions ...

False. The derivative of the Dirac delta function is not a function. It's not even a limit.

are you suggesting all functions are limits? ... I think it's better to say all functions have limits ... although related, they are two different things ... much like energy and power ...

Derivatives are defined in terms of limits. Taylor series and etc.

You must have missed the point of my quotes ... they are actually intended for Sage l'Main Street ... quite insane ...

Count.

Counting is the basis of all mathematics.

Counting can be further subdivided into labeling and subsetting.

Try it yourself. On a set. How do you count the members of a set?

Well, you just do it. 1, 2, 3... those digits are just labels, they could be anything, ABC or red green blue or cat dog giraffe. But when you COUNT you are performing a sequential operation, because you're not allowed to count a member that's already been counted. So once a member of the set has a label ("any" label) it is de-facto moved from a subset called "unlabeled" to a subset called "labeled".

Furthermore - the act of counting invokes the concept of "sequential" labels and therefore conforms to number theory and the rules of algebra. We have defined a "path" (an ordering, even a group structure) on the set of labels. Which means 2 < 3 < 4.

Now, given this information, can you define infinity?

 

[ReinyDays]

What is the definition of infinity in algebra and calculus?

Webster's gives the quality of boundlessness ... otherwise dividing by zero is undefined ...

Counting is the basis of all mathematics.

Addition is the basis of all mathematics ... scalar multiplication is just an extension of addition ... and accounting is the highest form of mathematical thought ... money is the only thing worth counting ...

=====

What makes you think Dirac is a function? ... is the value of f(0) unique? ... no? ... then it's not a function over the real numbers ... are you using Dirac's "q-numbers"? ... then of course the diagonal of your matrix can equal anything you want it to ... giraffes, dogs, chocolate cake ...

I think you need to be more honest here ... Dirac operates over an non-communitive ring(?) space ... not a vector space ... which was my complaint from the beginning ... counting numbers do not form a vector space ... and the value of infinity doesn't exist in a vector space ... any definition is strictly for the philosopher ...

 

[scruffy]

Any definition is strictly for the philosopher ...

Your inability to define terms does not equate with impossibility.

Boundless is an incorrect definition. Clearly, if a set contains a point at infinity it is not "boundless". This is one of Cantor's realizations. Bounding is something completely different from the label assigned to a point.

You need to read Turing's thesis on tractability. It will clarify things for you.

 

[talanum1]

Boundless is the right definition. If we talk about the point at infinity: put the "at" in inverted commas because it is not really "at".

 

[toobfreak]

You need to read Turing's thesis on tractability. It will clarify things for you.

Turing? Turing? Would you be referring to Alan Turing???

 

[scruffy]

Boundless is the right definition.

No, it isn't.

Aleph 0 is bounded from above by Aleph 1.

If we talk about the point at infinity: put the "at" in inverted commas because it is not really "at".

 

[talanum1]

No, it isn't.

Aleph 0 is bounded from above by Aleph 1.

Aleph0 is not bounded from above by aleph1 because aleph1 is also a cardinality of an infinite set therefore aleph1 is boundless. To say: "aleph0 is bounded from above by aleph1" is artificial.

 

[scruffy]

Aleph0 is not bounded from above by aleph1 because aleph1 is also a cardinality of an infinite set therefore aleph1 is boundless. To say: "aleph0 is bounded from above by aleph1" is artificial.

You're confused.

"An infinite set" does not equate with "a set containing a point at infinity".