What Does the Mathematical Notation XN∞ Represent in Set Theory?

[ShellWillis]

TL;DR Summary

im wondering how to turn this into a functional equation on python

could someone please explain this to me? I'm kind of looking at it and skimming it in different ways I need some help.

 

[PeterDonis]

Your equation as it appears in the title doesn't really make sense to begin with. Titles of threads are not good places to put equations. I would suggest posting the equation using the PF LaTeX feature in the body of a post. Also a reference to where you got the equation would help.

 

[Baluncore]

@ShellWillis
Welcome to PF.
I think this thread will be worth watching.

 

[Mark44]

Summary:: I am wondering how to turn this into a functional equation on python

could someone please explain this to me? I'm kind of looking at it and skimming it in different ways I need some help.

No doubt the formula you're trying to implement in Python comes from the paper you linked to in another post:

https://www.cs.bham.ac.uk/~mhe/papers/omniscient-journal-revised.pdf

In this paper, the author gave an example of an infinite omniscient set (? -- this is might be a term known only to the author of the paper) defined in this way:
$$\mathbb N_\infty = \{x \in 2^{\mathbb N}~|~ \forall i \in \mathbb N(x_i \ge x_{i + 1}\} $$

Note that this is not quite the same as what you wrote in the title of this thread. Before you can write a program that implements this set, you need to understand what this equation is saying. Please "unpack" the equation above and tell us four or five of the numbers that belong to this set.

 

[pasmith]

The author introduces this set as the one-point compactification of the natural numbers.

Earlier in the paper the author uses the notation $p: X \to 2$ in defining a function, so I suspect 2 actually means $\{0,1\}$ and we are looking at the set of all decreasing sequences taking values in $\{0,1\}$. You can identify such a sequence by $$I(x) = \begin{cases} 0 & \mbox{x_i = 1 for all i \in \mathbb{N},}\\ (\min\{ i \in \mathbb{N} : x_i = 0\})^{-1} & \mbox{otherwise.} \end{cases}$$
since once such a sequence hits zero it must stay there. This creates a bijection between $\mathbb{N}_\infty$ and $$K = \{0\} \cup \{n^{-1} : n \in \mathbb{Z}, n > 0 \}$$

 

[pbuk]

What does a functional equation look like in Python?

 

[Mark44]

Earlier in the paper the author uses the notation $p: X \to 2$ in defining a function, so I suspect 2 actually means $\{0,1\}$ and we are looking at the set of all decreasing sequences taking values in $\{0,1\}$.

That's my take as well, that $p: X \to 2$ maps integers to a number in the set {0, 1}. Seems like weird notation to me.

 

[pbuk]

That's my take as well, that $p: X \to 2$ maps integers to a number in the set {0, 1}. Seems like weird notation to me.

Yes, and it looks like a bit of a cheat to me in a constructivist context, although I can see its attraction: if we have such a function (and we can rely on the usual properties of a function) then we don't need LOM.

I'm also very wary of applying computational methods to constructive mathematics in general: $(P \lor \neg P)$ is built into the hardware.

I'd like to see a lot more effort from the OP demonstrating how they expect to tackle this, and what the answer they expect to see looks like.

 

[Mark44]

I'd like to see a lot more effort from the OP demonstrating how they expect to tackle this, and what the answer they expect to see looks like.

Same here.
Let's all hold off any more replies until we hear back from the OP.

 

[Jarvis323]

That's my take as well, that $p: X \to 2$ maps integers to a number in the set {0, 1}. Seems like weird notation to me.

I hope I'm not conflicting with the idea to wait for the OP to clarify. But I just have some insight on the notation to offer.

It makes some sense to use this notation because the notation for the Cantor space is $2^{\mathbb{N}}$ or $2^{\omega}$, while in more explicit notation, this means $\{0,1\}^{\mathbb{N}}$. The notation for the cantor set using 2 to represent $\{0,1\}$ is weird, but I guess it makes sense to be consistent with it.

Note that the other part of the notation (aside from 2) comes from regular expressions. $\{0,1\}^{\mathbb{N}}$ means the set of infinite binary strings, $\{0,1\}^*$ means the set of all finite binary strings including the empty string. $\{0,1\}^n$ means all binary strings of length $n$, $1^n$ is short for $\{1\}^n$ which is the set with 1 element, $111...$ , etc.

The set the OP is asking about seems to be $1^*0^{\mathbb{N}} \cup 1^{\mathbb{N}}$.

 

[ShellWillis]

From what I gather...

"2" is unwanted divergence

N is every pattern of union amongst empty sets

i means every set

x, i are each individual empty set aligned

And thatN infinity has the greatest magnitude of power

 

[Mark44]

For reference, here's the definition I wrote in post #4:
$$\mathbb N_\infty = \{x \in 2^{\mathbb N}~|~ \forall i \in \mathbb N(x_i \ge x_{i + 1}\} $$

From what I gather...

"2" is unwanted divergence

No. See posts #7 and #10.

N is every pattern of union amongst empty sets

No. What you wrote makes no sense at all. $\mathbb N$ is commonly used to mean the set of natural numbers 1, 2, 3, ..., with the possible inclusion of 0.

i means every set

No. i is a particular but unspecified member of $\mathbb N$.

x, i are each individual empty set aligned

No.

And thatN infinity has the greatest magnitude of power

$N_\infty$ is defined in the paper as well as at the top of this post.

It seems clear to me that you do not understand what the paper is trying to convey, so there is no possibility that you will be able to implement what the paper is discussing in Python or any other programming language. For this reason, I am closing this thread.

If at some point you come to a better understanding of what the paper is saying, please send me a PM and I will reopen the thread.