An infinite amount of permutations,

In summary, the conversation discusses the concept of an "infinite" amount of permutations or swaps and how it applies to rearranging sequences. The example of rearranging the alternating harmonic series is given, and the question arises of whether or not this involves a non-finite amount of permutations. It is argued that the expression "infinite amount of permutations" may be the root of the confusion and that a clearer definition is needed. The conversation also touches on the difference between permutations and swaps, and how the rearrangement of a sequence can be well-defined even with an "infinite" amount of swaps.
  • #1
daniel_i_l
Gold Member
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an "infinite" amount of permutations,

Look at how the sum of the alternating harmonic series can be changed by rearranging the terms:
http://mathworld.wolfram.com/RiemannSeriesTheorem.html
But doesn't this involve a non-finite amount of permutations? If this counts as a rearrangement then why can't we change 0,1,0,1... into 0,0,0... by the following argument:
If we know that for a sequence a_n, for every N>0 for every n<N a_n = 0 then obviously
a_n = 0,0,0... right?
But suppose we take the sequence b_n = 1,0,1,0... Then with N permutations (it doesn't really matter how many) we can rearrange it so that the first N elements are 0. So with an "infinite" amount of permutations we can make the first N elements 0 for all N!
I'm pretty sure that the problem here is in the expression "infinite amount of permutations" but I can't see any way to describe the rearrangement on wolframs site. Can someone help me clear this up?
Thanks.
 
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  • #2
daniel_i_l said:
Look at how the sum of the alternating harmonic series can be changed by rearranging the terms:
http://mathworld.wolfram.com/RiemannSeriesTheorem.html
But doesn't this involve a non-finite amount of permutations? If this counts as a rearrangement then why can't we change 0,1,0,1... into 0,0,0... by the following argument:
If we know that for a sequence a_n, for every N>0 for every n<N a_n = 0 then obviously
a_n = 0,0,0... right?
But suppose we take the sequence b_n = 1,0,1,0... Then with N permutations (it doesn't really matter how many) we can rearrange it so that the first N elements are 0. So with an "infinite" amount of permutations we can make the first N elements 0 for all N!
That does not follow. For one thing you haven't defined what you mean by an "infinite" amount of permutations. For another, you cannot simply assert that something that is true "for finite n" becomes true for an "infinity"- whatever you choose to mean by that.

I'm pretty sure that the problem here is in the expression "infinite amount of permutations" but I can't see any way to describe the rearrangement on wolframs site. Can someone help me clear this up?
Thanks.

I, on the other hand, don't see how that at all describes it! That site is talking about all possible rearrangements- some of which might involve on a single swap of two numbers. You might well talk about "an infinite number of permutations" but that means, as is meant by the site, different results for an infinite number of different "regular permutations"- not a single permutation that involves an infinite number of swaps- which is what you appear to be thinking.
 
  • #3
First of all, "permutations" was the wrong word - I should have said "swaps". Where by a "swap" I mean switching between two elements.
Let's say that you rearrange the sequence a_n to sequence b_n so that
[tex]b_{2n} = a_{2n+1}[/tex] and [tex]b_{2n+1} = a_{2n}[/tex]for all n. Isn't this the same as performing an "infinite" amount of swaps on a_n (1-2, 3-4, 5-6 ...)? If so, why can't you use the an "infinite" amount of swaps on 0,1,0,1... to turn it into 0,0,0...?
I don't know how to define this "infinite" amount of swaps and this is the problem - but then what does it mean in the case of the a_n -> b_n rearrangement which clearly is well defined?
Thanks.
 

What is an infinite amount of permutations?

An infinite amount of permutations refers to the number of possible ways a set of elements can be arranged or ordered without repetition. This means that there is no limit to the number of ways the elements can be arranged, making it impossible to count them all.

How is an infinite amount of permutations calculated?

An infinite amount of permutations can be calculated using the formula n! (n factorial), where n represents the number of elements in the set. For example, if there are 5 elements in the set, there are 5! = 5 x 4 x 3 x 2 x 1 = 120 possible permutations.

What is the difference between an infinite amount of permutations and a finite amount of permutations?

The main difference between an infinite amount of permutations and a finite amount of permutations is that there is no limit to the number of possible arrangements in an infinite amount, while a finite amount has a limited number of possible arrangements. Additionally, an infinite amount of permutations does not allow for repetition, while a finite amount may include repeated elements in a permutation.

What is the significance of an infinite amount of permutations?

An infinite amount of permutations has significant implications in mathematics, computer science, and other fields. It allows for the exploration of all possible combinations and patterns, and is used in various algorithms and problem-solving techniques. It also helps in understanding the concept of infinity and its applications in real-world scenarios.

Can an infinite amount of permutations be visualized?

It is impossible to visualize an infinite amount of permutations because it is a concept that goes beyond our ability to comprehend or imagine. However, we can represent a finite amount of permutations using diagrams, tables, or other visual aids to better understand the concept and its applications.

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