What is the structure of the Ultra Power Space?

[phoenixthoth]

If U [i.e., set theory] were to be equippable with a vector space type morphology...Prolly more of a module than a v.s.. Yes, a field over a ring, perhaps, if that's possible...

dim(U)...:

0. emptiness
1. isolation
2. expansion
3. containment
4. transition
5. hyperspace
6. hyper-hyperspace
...
n. (n>4) n-space.
...
Ultra-"space" I == aleph-null
Ultra-"space" II == alpeh 1
...
Ultra-"space" n == aleph n

...

Ultra Power Space == Omega Set == Omega Cardinal == Omega Ordinal == The Entire Multi-Universe

... where == means morphomorphic.

That kind of seems like a combination of category theory and set theory to me.

It goes back to the cone. The tip is 0 - emptiness and a spiral is drawn to infinity whilst a line goes down from the origin. At each intersection is another "number" of some sort, leading to Omega.

 

[MathematicalPhysicist]

this reminds me of a poster called "doron shadmi", he wrote a lot but no one could actually understand him.

 

[phoenixthoth]

Indeed. Organic is a friend of mine. But I have the training to back this up (one day). This is the thesis of my Phd thesis, perhaps...

edit: maybe the ultra ultra power of N would be the basis for the "vector space" of set theory in which the vectors are axioms.

 

[Hurkyl]

Hrm. It's hard to figure out what you're looking for... so I'll toss out some random things.


Are you looking for a variant of logic where your language isn't a set of strings, but instead a vector space? (or some other linear structure)


We can talk about "Set-modules". They are some sort of generalization of the notion of a module over a ring, the simplest of which (and only one I really know about) are the categories Setⁿ, which is the category of n-tuples of sets.


We can talk about a topos object in the category of vector spaces. Since a topos is like a universe of sets, this would be like a set-theoretic universe that is simultaneously a vector space. I don't know if any nontrivial ones exist, though.


One of John Baez's conjectures is that many things are naturally analyzed using the category Hilb of Hilbert spaces as the "fundamental" object, rather than the category Set of sets.


John Baez talks a lot about higher category theory; sets are 0-"dimensional" categories (they consist of isolated "points"), ordinary categories are 1-dimensional (they have "points" and "lines" between points), then there are 2-categories (which have "sheets" between "lines"), and so forth.

 

[phoenixthoth]

always a pleasure, hurkyl.

John Baez. Where is he? does he author any texts / pdfs?

*googles...

 

[HallsofIvy]

John Baez is a well known physicist. He has posted a number of tutorials to various kinds of physics problems on the internet.

 

[sparkster]

edit: maybe the ultra ultra power of N would be the basis for the "vector space" of set theory in which the vectors are axioms.

What does "the ultra ultra power of N" mean? What is it's cardinality? I'm guessing that it is some infinity. That means that you have to have at least infinitely many vectors and therefore infinitely many axioms.

 

[Hurkyl]

I'm guessing that it is some infinity.

No. While it will surely be an infinite cardinal, we don't call cardinals "infinity".

 

[Hurkyl]

While I'm sure you found them, one of Baez's main series of papers can be found by searching for "Higher Dimensional Algebra".

One interesting related paper (though not by Baez himself) is titled "Categorified Algebra and Quantum Mechanics"

 

[sparkster]

No. While it will surely be an infinite cardinal, we don't call cardinals "infinity".

That's what I meant. Sorry for not being more precise.