Consecutive values in set theory

[Simmer]

So i have cantors equation doing the same thing

w^e=e

i added "and not all(not e)"

 

obviously is the relativization you were talking about, not a powerset

are there any other terms for relativization that might point me in the right direction to program it? 

[wtf]

44 minutes ago, Simmer said:

obviously is the relativization you were talking about, not a powerset

This refers to ordinal exponentiation, it's not relativization.

https://en.wikipedia.org/wiki/Ordinal_arithmetic#Exponentiation

Edited October 29, 2020 by wtf

[Simmer]

Oh it was different than I thought it was 

 

its funny because I see this and I think self become entity

and cantors opinion was that of God

 

so I got confused 

anyway im trying something that looks the same but using empty sets instead of numbers and using relativization instead of exponentialism

 

what a confidence

 

i added "and not all(not e)"

[wtf]

1 hour ago, Simmer said:

and using relativization instead of exponentialism

Best way to start would be to take the time to understand relativization then. 

[Simmer]

Yep

i think it's something like e of w replaces w

Edited October 30, 2020 by Simmer

[Simmer]

You know what it also looks like?

biochemistry

the outcome of a catalyst being the catalyst itself

[wtf]

2 hours ago, Simmer said:

Yep

i think it's something like e of w replaces w

It's ordinal exponentiation in this case. You need to start with learning about ordinal numbers and then ordinal arithmetic, working up to exponentiation. 

[Simmer]

no I'm looking for my original interpretation of the formula

not the usage stated

It was just so similar it inspired me

[wtf]

1 hour ago, Simmer said:

no I'm looking for my original interpretation of the formula

 

The original interpretation is ordinal exponentiation. Here's an overview.

If we "keep counting" after 0, 1, 2, 3, ... we can tack on an element after all those, called [math]\omega[/math]. So we have the ordered set 0, 1, 2, 3, 4, ..., [math]\omega[/math].

Now we keep on going: [math]\omega + 1, \omega + 2, ...[/math] and when we're done with all those we've reached [math]\omega + \omega[/math] or [math]\omega 2[/math]. Ordinal multiplication is notated backwards, [math]\omega 2[/math] means 2 copies of [math]\omega[/math], one after another, and NOT [math]\omega[/math] copies of 2, which would actually be equal to [math]\omega[/math]. [Exercise for you: figure out why [math]2 \omega = \omega[/math]. You have to learn about ordinals. This is not an easy exercise until you have worked through the elementary material on what ordinals are]. This example shows that ordinal addition is not commutative.

As a historical note, I tracked this backward notation down once. It turns out that Cantor himself went back and forth on whether to call [math]\omega + \omega[/math] [math]\omega 2[/math] or [math]2 \omega[/math]. In the end for whatever reason he went with the notation that I consider backwards. But like everything else in math, you just get used to it.

So now we have [math]\omega 2, \omega 3, \omega 4, ...[/math], and at the end of that we tack on [math]\omega \omega[/math] or [math]\omega^2[/math].

Then we have [math]\omega^3, \omega^4 [/math], and we keep on going till we get [math]\omega^\omega[/math]. If we keep on going we get 

[math]\omega^{\omega^\omega}, \omega^{\omega^{\omega^\omega}}[/math], and so forth. If we keep this process going, at the end we tack on a countably infinite power tower of [math]\omega[/math]'s. This infinite power tower is given the name [math]\epsilon_0[/math], and it is the smallest ordinal [math]\epsilon[/math] such that [math]\omega^\epsilon = \epsilon[/math]. Exercise: Can you see why this equation is true of [math]\epsilon_0[/math]?

This "tacking on at the end" idea can be formalized by letting the limit ordinal, as it's called, simply be the set of everything that's come before. So if the natural numbers are 0, 1, 2, 3, ..., then we define [math]\omega = \{0, 1, 2, 3, \dots \}[/math]. These are the von Neumann ordinals. https://en.wikipedia.org/wiki/Ordinal_number#Von_Neumann_definition_of_ordinals. By the way I recommend that you read this entire Wiki page to begin to learn about the ordinal numbers.

It's important, and mind boggling, to realize that [math]\epsilon_0[/math] is a countable ordinal; that is, it's a countable set. The countable ordinals are very strange and hard to get one's mind around. It's an amazing theorem that every countable ordinal can be embedded in some subset of the rationals in their usual order.

This is the subject of the theorem you are interested in. This is what it's about, not something else. You can be interested in something else if you like, but if you are interested in this particular idea, this is what it's about. 

https://en.wikipedia.org/wiki/Epsilon_numbers_(mathematics)

Interestingly in physics, the notation [math]\epsilon_0[/math] refers to something called the vacuum permitivity, but that has nothing to do with what we're talking about here.

I don't expect you to understand all this right now. It's taken me a long time to get even a little bit of understanding of [math]\epsilon_0[/math].

But if you want to learn this material, you have to make an attempt to learn it, starting with understanding what ordinal numbers are. Otherwise you are having a vacuous conversation with yourself.

 

 

 

Edited October 30, 2020 by wtf