Infinite Basis and Supernatural Numbers

My guess is that she meant that the result of the addition of three vectors is an infinite sequence all of whose components are 6.
  • #1
FallenApple
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I've read that it is unsatisfactory to consider infinitely many basis vectors to span an infinite dimensional space. For example, for the infinite dimensional Hilbert space, {e1,e2,e3...} we could use this to make an arbitrary infinite tuple (a,b,c,...). If this is looked down upon, then why are Supernatural Numbers, which are the product of infinitely many primes, considered ontologically valid ?

Surely if we do a transformation, we could express those supernatural numbers as vectors, since its just converting multiplication to vector addition.
 
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  • #2
FallenApple said:
I've read that it is unsatisfactory to consider infinitely many basis vectors to span an infinite dimensional space.
Citation needed. For example, the fact that the infinite set of basis vectors ##e^{ikx}, k\in \mathbb{Z}## spans the space of periodic functions is the cornerstone of Fourier theory. Also, the axiom of choice is equivalent to the statement that every vector space--even if it has uncountably infinite dimensions--has a basis.
 
  • #3
TeethWhitener said:
Citation needed. For example, the fact that the infinite set of basis vectors ##e^{ikx}, k\in \mathbb{Z}## spans the space of periodic functions is the cornerstone of Fourier theory. Also, the axiom of choice is equivalent to the statement that every vector space--even if it has uncountably infinite dimensions--has a basis.
InfiniteBasis.png

http://www.math.lsa.umich.edu/~kesmith/infinite.pdf
 
  • #4
For the question in your quoted text, I don't see the problem. It seems to me that (1, 1, 1, ..., ) + (2, 2, 2, ... , ) + (3, 3, 3, ... , ) = (6, 6, 6, ... , ).

IOW, component-wise vector addition, even though the vectors belong to an infinite-dimension vector space.

I wouldn't consider the sum of 3 vectors to be an infinite sum, being as there are only three of them being added. I will take a look at the link you gave to see if there is some context that is missing from what you quoted.
 
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  • #5
IMO, the author of the PDF from which you quoted is quite sloppy.
In Example 2 she says
The vector addition and scalar multiplication are defined in the natural way: the sum of ##(\alpha_1, \alpha_2, \alpha_3, \dots, \dots)## and
##(\beta_1, \beta_2, \beta_3, \dots, \dots)## is ##(\alpha_1 + \beta_1, \alpha_2 + \beta_2, \alpha_3 + \beta_3, \dots, \dots)##

Later in the same example, she says
Since the vector v = (1, 1, 1, 1, ..., ) can not be written as a linear combination of
the vectors ei = (0, ..., 0, 1; 0; 0, ...), it must be that together the ei's and v form
a linearly independent set.
In the sentence above she meant that v cannot be written as a finite linear combination, but neglected to write it.

Finally, from the text you quoted, she calls (1, 1, 1, ..., ) + (2, 2, 2, ..., ) + (3, 3, 3, ..., ) an infinite sum even though there are only three vectors. She asks what would be the sum of these vectors, a question she answers more generally in the first part of Example 2.
 
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  • #6
Mark44 said:
IMO, the author of the PDF from which you quoted is quite sloppy.
In Example 2 she saysLater in the same example, she says
In the sentence above she meant that v cannot be written as a finite linear combination, but neglected to write it.

Finally, from the text you quoted, she calls (1, 1, 1, ..., ) + (2, 2, 2, ..., ) + (3, 3, 3, ..., ) an infinite sum even though there are only three vectors. She asks what would be the sum of these vectors, a question she answers more generally in the first part of Example 2.

Ok that makes sense. I was thinking somewhere alone those lines of totaling up the coefficients per coordinate, which is example 2. That makes sense.

In an infinite vector space, it is a bit self explanatory that infinite combinations are needed unless the resultant vector is non zero in finitely many places; of which we still get around this by placing 0 in front of the remaining basis vectors. So it's strange for her to attach significance to the fact that some vectors wouldn't be able to be written as a finite linear combo, given the context of infinite dimensions.
 
  • #7
FallenApple said:
In an infinite vector space, it is a bit self explanatory that infinite combinations are needed unless the resultant vector is non zero in finitely many places; of which we still get around this by placing 0 in front of the remaining basis vectors. So it's strange for her to attach significance to the fact that some vectors wouldn't be able to be written as a finite linear combo, given the context of infinite dimensions.
Yes. Although it's obvious that you can't generate a particular vector with a finite linear combination of basis vectors, it seemed very sloppy to me to omit "finite".
And her example didn't make any sense, namely the one with (1, 1, ...) + (2, 2, ...) + (3, 3, 3, ...) and calling that an infinite sum.
 
  • #8
Mark44 said:
Yes. Although it's obvious that you can't generate a particular vector with a finite linear combination of basis vectors, it seemed very sloppy to me to omit "finite".
And her example didn't make any sense, namely the one with (1, 1, ...) + (2, 2, ...) + (3, 3, 3, ...) and calling that an infinite sum.
Thats true. It's a tri-nary sum producing a single answer infinitely many separate times , which is not an infinite sum of a sequence.
 
  • #9
Mark44 said:
For the question in your quoted text, I don't see the problem. It seems to me that (1, 1, 1, ..., ) + (2, 2, 2, ... , ) + (3, 3, 3, ... , ) = (6, 6, 6, ... , ).
I think there is a typo in the text quoted by OP, so that it should be "(1, 1, 1, ...) + (2, 2, 2, ...) + (3, 3, 3, ...) + ... "
 
  • #10
Erland said:
I think there is a typo in the text quoted by OP, so that it should be "(1, 1, 1, ...) + (2, 2, 2, ...) + (3, 3, 3, ...) + ... "
Maybe, but I would be embarrassed to put out something with such a glaring error.
 

FAQ: Infinite Basis and Supernatural Numbers

What are "Infinite Basis and Supernatural Numbers"?

Infinite Basis and Supernatural Numbers are mathematical concepts that deal with the idea of numbers that are infinitely large or infinitely small. These numbers go beyond the traditional real numbers and allow for calculations and concepts that were previously impossible to comprehend.

How are these numbers different from real numbers?

Real numbers are limited to a finite range, while infinite basis and supernatural numbers have no limits. This means that they can be infinitely larger or smaller than any real number. Additionally, supernatural numbers can have unusual properties, such as being both positive and negative at the same time.

What is the significance of these numbers?

Infinite basis and supernatural numbers have many applications in mathematics, physics, and other sciences. They allow for a deeper understanding of concepts such as infinity, limits, and the nature of space and time. They also have practical uses in fields such as computer science and cryptography.

Are these numbers just theoretical or do they have real-world applications?

While initially thought of as purely theoretical, infinite basis and supernatural numbers have been used in various real-world applications. For example, they have been used in computer algorithms to generate truly random numbers and in cryptography for secure communication. They also have potential uses in fields such as artificial intelligence and quantum computing.

Are there any limitations or challenges when working with these numbers?

Working with infinite basis and supernatural numbers can be challenging due to their abstract nature and the fact that they go beyond our traditional understanding of numbers. Additionally, there are still many unanswered questions and debates surrounding these numbers, so caution must be taken when using them in research or applications. Finally, the use of these numbers may not always provide practical or useful results, so careful consideration is needed in their application.

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